SlideShare a Scribd company logo

Geometry Theorems 1 REMC Tutoring

β€’
β€’
R
REMCTutoring

Basic Geometry Theorems with Proofs. If you have any questions or corrections please email me at: remc.tutoring@gmail.com

Education
1 of 8
Download to read offline
Theorems Midpoint Theorem 𝐼𝑓 𝐡 𝑖𝑠 π‘‘β„Žπ‘’ π‘€π‘–π‘‘π‘π‘œπ‘–π‘›π‘‘ π‘œπ‘“ 𝐴𝐢 Μ…Μ…Μ…Μ…, π‘‘β„Žπ‘’π‘› 𝐴𝐡 Μ…Μ…Μ…Μ… β‰… 𝐡𝐢 Μ…Μ…Μ…Μ… Segment Congruence Theorems Reflexive Symmetric Transitive πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘π‘’ π‘“π‘œπ‘™π‘™π‘œπ‘€π‘  π‘‘β„Žπ‘’ π΄π‘™π‘”π‘’π‘π‘Ÿπ‘Žπ‘–π‘ π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘–π‘’π‘  π‘œπ‘“ 𝑅𝑒𝑓𝑙𝑒π‘₯𝑖𝑣𝑒, π‘†π‘¦π‘šπ‘šπ‘’π‘‘π‘Ÿπ‘–π‘, π‘Žπ‘›π‘‘ π‘‡π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘π‘–π‘£π‘’ 𝐴𝐡 Μ…Μ…Μ…Μ…Μ… β‰… 𝐴𝐡 Μ…Μ…Μ…Μ… 𝐼𝑓 𝐴𝐡 Μ…Μ…Μ…Μ… β‰… 𝐢𝐷 Μ…Μ…Μ…Μ…, π‘‘β„Žπ‘’π‘› 𝐢𝐷 Μ…Μ…Μ…Μ… β‰… 𝐴𝐡 Μ…Μ…Μ…Μ… 𝐼𝑓 𝐴𝐡 Μ…Μ…Μ…Μ… β‰… 𝐢𝐷 Μ…Μ…Μ…Μ… π‘Žπ‘›π‘‘ 𝐢𝐷 Μ…Μ…Μ…Μ… β‰… 𝐸𝐹 Μ…Μ…Μ…Μ…, π‘‘β„Žπ‘’π‘› 𝐴𝐡 Μ…Μ…Μ…Μ… β‰… 𝐸𝐹 Μ…Μ…Μ…Μ… 𝐴𝐡 Μ…Μ…Μ…Μ… β‰… 𝐢𝐷 Μ…Μ…Μ…Μ… π‘Žπ‘›π‘‘ 𝐢𝐷 Μ…Μ…Μ…Μ… β‰… 𝐸𝐹 Μ…Μ…Μ…Μ…, π‘‘β„Žπ‘’π‘Ÿπ‘’π‘“π‘œπ‘Ÿπ‘’ 𝐴𝐡 Μ…Μ…Μ…Μ… β‰… 𝐸𝐹 Μ…Μ…Μ…Μ… (π‘‘π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘π‘–π‘£π‘’ π‘π‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦ 𝑖𝑠 𝑒𝑠𝑒𝑑 π‘€β„Žπ‘’π‘› π‘œπ‘π‘—π‘’π‘π‘‘π‘  π‘Žπ‘Ÿπ‘’ π‘π‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ π‘‘π‘œ π‘Žπ‘›π‘œπ‘‘β„Žπ‘’π‘Ÿ) Supplement Theorem 𝐼𝑓 π‘Ž π‘™π‘–π‘›π‘’π‘Žπ‘Ÿ π‘π‘Žπ‘–π‘Ÿ 𝑖𝑠 π‘“π‘œπ‘Ÿπ‘šπ‘’π‘‘ 𝑏𝑦 π‘‘π‘€π‘œ π‘Žπ‘›π‘”π‘™π‘’π‘ , π‘‘β„Žπ‘’π‘› 𝑀𝑒 π‘˜π‘›π‘œπ‘€ π‘‘β„Žπ‘Žπ‘‘ π‘‘β„Žπ‘’ π‘‘π‘€π‘œ π‘Žπ‘›π‘”π‘™π‘’π‘  π‘Žπ‘Ÿπ‘’ π‘ π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦. ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ (π‘ π‘’π‘š π‘œπ‘“ π‘šβˆ 1 π‘Žπ‘›π‘‘ π‘šβˆ 2 π‘’π‘žπ‘’π‘Žπ‘™π‘  180Β°) Complement Theorem 𝐼𝑓 π‘‘β„Žπ‘’ π‘’π‘›π‘π‘œπ‘šπ‘šπ‘œπ‘› 𝑠𝑖𝑑𝑒𝑠 π‘“π‘Ÿπ‘œπ‘š π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘Žπ‘›π‘”π‘™π‘’π‘  π‘“π‘œπ‘Ÿπ‘š π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’, π‘‘β„Žπ‘’π‘› 𝑀𝑒 π‘˜π‘›π‘œπ‘€ π‘‘β„Žπ‘Žπ‘‘ π‘‘β„Žπ‘’ π‘‘π‘€π‘œ π‘Žπ‘›π‘”π‘™π‘’π‘  π‘Žπ‘Ÿπ‘’ π‘π‘œπ‘šπ‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦. ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ πΆπ‘œπ‘šπ‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ (π‘ π‘’π‘š π‘œπ‘“ π‘šβˆ 1 π‘Žπ‘›π‘‘ π‘šβˆ 2 π‘’π‘žπ‘’π‘Žπ‘™π‘  90Β°) (π‘π‘’π‘Ÿπ‘π‘™π‘’ π‘π‘œπ‘₯ π‘ π‘¦π‘šπ‘π‘œπ‘™ π‘šπ‘’π‘Žπ‘›π‘  π‘π‘™π‘Žπ‘π‘˜ 𝑙𝑖𝑛𝑒𝑠 π‘Žπ‘Ÿπ‘’ π‘Ž π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒 π‘œπ‘Ÿ 90Β°)
Congruent Supplement Theorem (3 angles) Congruent Supplement Theorem (4 angles) 𝐼𝑓 π‘‘π‘€π‘œ π‘Žπ‘›π‘”π‘™π‘’π‘  π‘Žπ‘Ÿπ‘’ π‘ π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ π‘Žπ‘›π‘‘ π‘Žπ‘›π‘”π‘™π‘’ π‘œπ‘›π‘’ 𝑖𝑠 π‘ π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ π‘€π‘–π‘‘β„Ž π‘Ž π‘‘β„Žπ‘–π‘Ÿπ‘‘ π‘Žπ‘›π‘”π‘™π‘’, π‘‘β„Žπ‘’π‘› π‘‘β„Žπ‘’ π‘‘β„Žπ‘–π‘Ÿπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ 𝑖𝑠 π‘π‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ π‘‘π‘œ π‘‘β„Žπ‘’ π‘ π‘’π‘π‘œπ‘›π‘‘ π‘Žπ‘›π‘”π‘™π‘’. 𝐼𝑓 π‘‘π‘€π‘œ 𝑠𝑒𝑑𝑠 π‘œπ‘“ π‘Žπ‘›π‘”π‘™π‘’π‘  π‘π‘Ÿπ‘’π‘Žπ‘‘π‘’ π‘‘π‘€π‘œ π‘ π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ π‘π‘Žπ‘–π‘Ÿπ‘ , π‘Žπ‘›π‘‘ π‘œπ‘›π‘’ π‘Žπ‘›π‘”π‘™π‘’ π‘“π‘Ÿπ‘œπ‘š π‘’π‘Žπ‘β„Ž 𝑠𝑒𝑑 𝑖𝑠 π‘π‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ π‘‘π‘œ π‘Žπ‘› π‘Žπ‘›π‘”π‘™π‘’ π‘œπ‘“ π‘‘β„Žπ‘’ π‘œπ‘‘β„Žπ‘’π‘Ÿ 𝑠𝑒𝑑, π‘‘β„Žπ‘’π‘› π‘‘β„Žπ‘’ π‘Žπ‘›π‘”π‘™π‘’π‘  π‘›π‘œπ‘‘ π‘™π‘Žπ‘π‘’π‘™π‘’π‘‘ π‘Žπ‘  π‘π‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ π‘€π‘œπ‘’π‘™π‘‘ 𝑏𝑒 π‘π‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ π‘Žπ‘™π‘ π‘œ. ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ ∠1 π‘Žπ‘›π‘‘ ∠3 π‘Žπ‘Ÿπ‘’ π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ π‘‡β„Žπ‘’π‘Ÿπ‘’π‘“π‘œπ‘Ÿπ‘’ ∠2 β‰… ∠3 ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ ∠3 π‘Žπ‘›π‘‘ ∠4 π‘Žπ‘Ÿπ‘’ π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ ∠2 β‰… ∠3 π‘‡β„Žπ‘’π‘Ÿπ‘’π‘“π‘œπ‘Ÿπ‘’, ∠1 β‰… ∠4 Congruent Complement Theorem 𝐼𝑓 π‘‘π‘€π‘œ π‘Žπ‘›π‘”π‘™π‘’π‘  π‘Žπ‘Ÿπ‘’ π‘π‘œπ‘šπ‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ π‘Žπ‘›π‘‘ π‘Žπ‘›π‘”π‘™π‘’ π‘œπ‘›π‘’ 𝑖𝑠 π‘π‘œπ‘šπ‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ π‘€π‘–π‘‘β„Ž π‘Ž π‘‘β„Žπ‘–π‘Ÿπ‘‘ π‘Žπ‘›π‘”π‘™π‘’, π‘‘β„Žπ‘’π‘› π‘‘β„Žπ‘’ π‘‘β„Žπ‘–π‘Ÿπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ 𝑖𝑠 π‘π‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ π‘‘π‘œ π‘‘β„Žπ‘’ π‘ π‘’π‘π‘œπ‘›π‘‘ π‘Žπ‘›π‘”π‘™π‘’. ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ πΆπ‘œπ‘šπ‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ ∠2 π‘Žπ‘›π‘‘ ∠3 π‘Žπ‘Ÿπ‘’ πΆπ‘œπ‘šπ‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ π‘‡β„Žπ‘’π‘Ÿπ‘’π‘“π‘œπ‘Ÿπ‘’, ∠1 β‰… ∠3 Vertical Angles Theorem 𝐼𝑓 π‘‘π‘€π‘œ π‘Žπ‘›π‘”π‘™π‘’π‘  π‘Žπ‘Ÿπ‘’ π‘£π‘’π‘Ÿπ‘‘π‘–π‘π‘Žπ‘™ π‘Žπ‘›π‘”π‘™π‘’π‘ , π‘‘β„Žπ‘’π‘› 𝑀𝑒 π‘˜π‘›π‘œπ‘€ π‘‘β„Žπ‘Žπ‘‘ π‘‘β„Žπ‘’π‘¦ π‘Žπ‘Ÿπ‘’ π‘π‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘. ∠1 β‰… ∠3 ∠2 β‰… ∠4
Right Angle Theorems Perpendicular Lines Intersect to Form Four Right Angles All Right Angles are Congruent Perpendicular Lines Will Form Four Congruent Adjacent Angles If Two Angles are Both Congruent and Supplementary, Then Each Angle Will be a Right Angle If Two Angles of a Linear Pair are Congruent, Then They Are Right Angles 𝐼𝑓 π‘‘π‘€π‘œ 𝑙𝑖𝑛𝑒𝑠 π‘Žπ‘Ÿπ‘’ π‘π‘’π‘Ÿπ‘π‘’π‘›π‘‘π‘–π‘π‘’π‘™π‘Žπ‘Ÿ, π‘‘β„Žπ‘’π‘› π‘“π‘œπ‘’π‘Ÿ π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’π‘  𝑀𝑖𝑙𝑙 𝑏𝑒 π‘™π‘œπ‘π‘Žπ‘‘π‘’π‘‘ π‘Žπ‘‘ π‘‘β„Žπ‘’ π‘–π‘›π‘‘π‘’π‘Ÿπ‘ π‘’π‘π‘‘π‘–π‘œπ‘› π‘π‘œπ‘–π‘›π‘‘. 𝐼𝑓 π‘‘π‘€π‘œ π‘œπ‘Ÿ π‘šπ‘œπ‘Ÿπ‘’ π‘Žπ‘›π‘”π‘™π‘’π‘  π‘Žπ‘Ÿπ‘’ π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’π‘ , π‘‘β„Žπ‘’π‘› π‘Žπ‘™π‘™ π‘œπ‘“ π‘‘β„Žπ‘’ π‘Žπ‘›π‘”π‘™π‘’π‘  π‘Žπ‘Ÿπ‘’ π‘π‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘. 𝐼𝑓 π‘‘π‘€π‘œ 𝑙𝑖𝑛𝑒𝑠 π‘–π‘›π‘‘π‘’π‘Ÿπ‘ π‘’π‘π‘‘ π‘‘π‘œ π‘Žπ‘‘ π‘Ž 90Β° π‘Žπ‘›π‘”π‘™π‘’, π‘‘β„Žπ‘’π‘› π‘π‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘Žπ‘›π‘”π‘™π‘’π‘  𝑀𝑖𝑙𝑙 𝑏𝑒 π‘“π‘œπ‘Ÿπ‘šπ‘’π‘‘. 𝐼𝑓 π‘‘π‘€π‘œ π‘Žπ‘›π‘”π‘™π‘’π‘  π‘Žπ‘Ÿπ‘’ π‘π‘œπ‘‘β„Ž π‘π‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ 𝑖𝑛 π‘šπ‘’π‘Žπ‘ π‘’π‘Ÿπ‘’ π‘Žπ‘›π‘‘ π‘‘β„Žπ‘’ π‘ π‘’π‘š π‘œπ‘“ π‘‘β„Žπ‘’ π‘Žπ‘›π‘”π‘™π‘’π‘  𝑖𝑠 180Β°, π‘‘β„Žπ‘’π‘› 𝑀𝑒 π‘˜π‘›π‘œπ‘€ π‘‘β„Žπ‘Žπ‘‘ π‘’π‘Žπ‘β„Ž π‘Žπ‘›π‘”π‘™π‘’ 𝑖𝑠 90Β° 𝐼𝑓 π‘Ž π‘™π‘–π‘›π‘’π‘Žπ‘Ÿ π‘π‘Žπ‘–π‘Ÿ 𝑖𝑠 π‘“π‘œπ‘Ÿπ‘šπ‘’π‘‘ π‘“π‘Ÿπ‘œπ‘š π‘‘π‘€π‘œ π‘π‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ π‘Žπ‘›π‘”π‘™π‘’π‘ , π‘‘β„Žπ‘’π‘› 𝑀𝑒 π‘˜π‘›π‘œπ‘€ π‘‘β„Žπ‘Žπ‘‘ π‘π‘œπ‘‘β„Ž π‘Žπ‘›π‘”π‘™π‘’π‘  π‘Žπ‘Ÿπ‘’ 90Β° πΉπ‘œπ‘’π‘Ÿ π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠 π‘Žπ‘Ÿπ‘’ π‘π‘Ÿπ‘’π‘Žπ‘‘π‘’π‘‘ π‘Žπ‘‘ π‘‘β„Žπ‘’ πΌπ‘›π‘‘π‘’π‘Ÿπ‘ π‘’π‘π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘‘β„Žπ‘’ π‘ƒπ‘’π‘Ÿπ‘π‘’π‘›π‘‘π‘–π‘π‘’π‘™π‘Žπ‘Ÿ 𝐿𝑖𝑛𝑒𝑠 𝐴𝑙𝑙 π‘‘β„Žπ‘’ π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠 π‘’π‘žπ‘’π‘Žπ‘™ 90Β° ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π΄π‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ ∠2 π‘Žπ‘›π‘‘ ∠3 π‘Žπ‘Ÿπ‘’ π΄π‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ ∠3 π‘Žπ‘›π‘‘ ∠4 π‘Žπ‘Ÿπ‘’ π΄π‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ ∠4 π‘Žπ‘›π‘‘ ∠1 π‘Žπ‘Ÿπ‘’ π΄π‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ 𝐴𝑙𝑙 π‘‘β„Žπ‘’ π‘Žπ‘›π‘”π‘™π‘’π‘  π‘œπ‘“ π‘‘β„Žπ‘’ π΄π‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ 𝐴𝑛𝑔𝑙𝑒 π‘ƒπ‘Žπ‘–π‘Ÿπ‘  π‘Žπ‘Ÿπ‘’ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ π‘Žπ‘›π‘‘ π‘Žπ‘™π‘ π‘œ π‘‘β„Žπ‘’ π‘π‘Žπ‘–π‘Ÿπ‘  π‘Žπ‘Ÿπ‘’ π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦, π‘€β„Žπ‘–π‘β„Ž π‘šπ‘’π‘Žπ‘›π‘  π‘’π‘Žπ‘β„Ž π‘Žπ‘›π‘”π‘™π‘’ 𝑖𝑠 π‘Ž π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒. 𝐴𝑙𝑙 π‘‘β„Žπ‘’ π‘Žπ‘›π‘”π‘™π‘’π‘  π‘œπ‘“ π‘‘β„Žπ‘’ π΄π‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ 𝐴𝑛𝑔𝑙𝑒 π‘ƒπ‘Žπ‘–π‘Ÿπ‘  π‘Žπ‘Ÿπ‘’ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ π‘Žπ‘›π‘‘ π‘Žπ‘™π‘ π‘œ π‘π‘Ÿπ‘’π‘Žπ‘‘π‘’ πΏπ‘–π‘›π‘’π‘Žπ‘Ÿ π‘ƒπ‘Žπ‘–π‘Ÿπ‘  π‘€β„Žπ‘–π‘β„Ž π‘šπ‘’π‘Žπ‘›π‘  π‘’π‘Žπ‘β„Ž π‘Žπ‘›π‘”π‘™π‘’ 𝑖𝑠 π‘Ž π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒. Theorem Proofs Midpoint Theorem: π‘ƒπ‘Ÿπ‘œπ‘£π‘’ π‘‘β„Žπ‘Žπ‘‘: 𝐴𝐡 Μ…Μ…Μ…Μ… β‰… 𝐡𝐢 Μ…Μ…Μ…Μ… 𝐡 𝑖𝑠 π‘€π‘–π‘‘π‘π‘œπ‘–π‘›π‘‘ π‘œπ‘“ 𝐴𝐢 Μ…Μ…Μ…Μ… 𝑔𝑖𝑣𝑒𝑛 𝐴𝐡 = 𝐡𝐢 π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘€π‘–π‘‘π‘π‘œπ‘–π‘›π‘‘ π‘œπ‘“ π‘Ž π‘†π‘’π‘”π‘šπ‘’π‘›π‘‘ 𝐴𝐡 Μ…Μ…Μ…Μ… β‰… 𝐡𝐢 Μ…Μ…Μ…Μ… π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘†π‘’π‘”π‘šπ‘’π‘›π‘‘ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘π‘’
Segment Congruence Theorems: Reflexive Property: π‘ƒπ‘Ÿπ‘œπ‘£π‘’ π‘‘β„Žπ‘Žπ‘‘: 𝐴𝐡 Μ…Μ…Μ…Μ…Μ… β‰… 𝐴𝐡 Μ…Μ…Μ…Μ… 𝐴𝐡 Μ…Μ…Μ…Μ… β‰… 𝐴𝐡 Μ…Μ…Μ…Μ… 𝑔𝑖𝑣𝑒𝑛 𝐴𝐡 = 𝐴𝐡 π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘†π‘’π‘”π‘šπ‘’π‘›π‘‘ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘π‘’ 𝐴𝐡 = 𝐴𝐡 𝑅𝑒𝑓𝑙𝑒π‘₯𝑖𝑣𝑒 𝐴𝐡 Μ…Μ…Μ…Μ… β‰… 𝐴𝐡 Μ…Μ…Μ…Μ… π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘†π‘’π‘”π‘šπ‘’π‘›π‘‘ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘π‘’ Symmetric Property: π‘ƒπ‘Ÿπ‘œπ‘£π‘’ π‘‘β„Žπ‘Žπ‘‘: 𝐢𝐷 Μ…Μ…Μ…Μ… β‰… 𝐴𝐡 Μ…Μ…Μ…Μ… 𝐴𝐡 Μ…Μ…Μ…Μ… β‰… 𝐢𝐷 Μ…Μ…Μ…Μ… 𝑔𝑖𝑣𝑒𝑛 𝐴𝐡 = 𝐢𝐷 π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘†π‘’π‘”π‘šπ‘’π‘›π‘‘ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘π‘’ 𝐢𝐷 = 𝐴𝐡 π‘†π‘¦π‘šπ‘šπ‘’π‘‘π‘Ÿπ‘–π‘ π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦ 𝐢𝐷 Μ…Μ…Μ…Μ… β‰… 𝐴𝐡 Μ…Μ…Μ…Μ… π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘†π‘’π‘”π‘šπ‘’π‘›π‘‘ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘π‘’ Transitive Property: π‘ƒπ‘Ÿπ‘œπ‘£π‘’ π‘‘β„Žπ‘Žπ‘‘: 𝐴𝐡 Μ…Μ…Μ…Μ… β‰… 𝐸𝐹 Μ…Μ…Μ…Μ… 𝐴𝐡 Μ…Μ…Μ…Μ… β‰… 𝐢𝐷 Μ…Μ…Μ…Μ… π‘Žπ‘›π‘‘ 𝐢𝐷 Μ…Μ…Μ…Μ… β‰… 𝐸𝐹 Μ…Μ…Μ…Μ… 𝑔𝑖𝑣𝑒𝑛 𝐴𝐡 = 𝐢𝐷 π‘Žπ‘›π‘‘ 𝐢𝐷 = 𝐸𝐹 π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘†π‘’π‘”π‘šπ‘’π‘›π‘‘ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘π‘’ 𝐴𝐡 = 𝐸𝐹 𝐴𝐡 = 𝐸𝐹 π‘‡π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘π‘–π‘£π‘’ π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦ π‘œπ‘Ÿ, π‘†π‘’π‘π‘ π‘‘π‘–π‘‘π‘’π‘‘π‘–π‘œπ‘› π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦ (𝑠𝑒𝑏 𝐴𝐡 π‘“π‘œπ‘Ÿ 𝐢𝐷) 𝐴𝐡 Μ…Μ…Μ…Μ… β‰… 𝐸𝐹 Μ…Μ…Μ…Μ… π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘†π‘’π‘”π‘šπ‘’π‘›π‘‘ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘π‘’ Supplement Theorem: π‘ƒπ‘Ÿπ‘œπ‘£π‘’ π‘‘β„Žπ‘Žπ‘‘ ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ ∠1 π‘Žπ‘›π‘‘ ∠2 π‘“π‘œπ‘Ÿπ‘š π‘Ž π‘™π‘–π‘›π‘’π‘Žπ‘Ÿ π‘π‘Žπ‘–π‘Ÿ 𝑔𝑖𝑣𝑒𝑛 π‘π‘œπ‘‘π‘’: π»π‘œπ‘€ π‘‘π‘œ π‘‘π‘’π‘‘π‘’π‘Ÿπ‘šπ‘–π‘›π‘’ π‘Ž πΏπ‘–π‘›π‘’π‘Žπ‘Ÿ π‘ƒπ‘Žπ‘–π‘Ÿ? π‘‡π‘€π‘œ 𝐴𝑛𝑔𝑙𝑒𝑠 π‘‘β„Žπ‘Žπ‘‘: π΄π‘Ÿπ‘’ π΄π‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠 π‘ˆπ‘›π‘ β„Žπ‘Žπ‘Ÿπ‘’π‘‘ 𝑠𝑖𝑑𝑒𝑠 π‘“π‘œπ‘Ÿπ‘š π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ π‘Ÿπ‘Žπ‘¦π‘  (180Β°) π‘šβˆ 1 + π‘šβˆ 2 = 180Β° π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ πΏπ‘–π‘›π‘’π‘Žπ‘Ÿ π‘ƒπ‘Žπ‘–π‘Ÿ ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ 𝐴𝑛𝑔𝑙𝑒𝑠 Complement Theorem: π‘ƒπ‘Ÿπ‘œπ‘£π‘’ π‘‘β„Žπ‘Žπ‘‘ ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ πΆπ‘œπ‘šπ‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘Žπ‘›π‘”π‘™π‘’π‘  ∠1 π‘Žπ‘›π‘‘ ∠2 π‘“π‘œπ‘Ÿπ‘š π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ 𝑔𝑖𝑣𝑒𝑛 π‘π‘œπ‘‘π‘’: π»π‘œπ‘€ π‘‘π‘œ π‘‘π‘’π‘‘π‘’π‘Ÿπ‘šπ‘–π‘›π‘’ π΄π‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠? π‘ƒπ‘Žπ‘–π‘Ÿ π‘œπ‘“ 𝐴𝑛𝑔𝑙𝑒𝑠 π‘‘β„Žπ‘Žπ‘‘ π‘†β„Žπ‘Žπ‘Ÿπ‘’ π‘Ž π‘‰π‘’π‘Ÿπ‘‘π‘’π‘₯, π‘†β„Žπ‘Žπ‘Ÿπ‘’ π‘Ž 𝑆𝑖𝑑𝑒, π‘Žπ‘›π‘‘ π‘Žπ‘Ÿπ‘’ π‘œπ‘› π‘‘β„Žπ‘’ π‘†π‘Žπ‘šπ‘’ π‘ƒπ‘™π‘Žπ‘›π‘’ π»π‘œπ‘€ π‘‘π‘œ π‘‘π‘’π‘‘π‘’π‘Ÿπ‘šπ‘–π‘›π‘’ π‘Ž π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒? π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒 = 90Β° π‘Žπ‘›π‘”π‘™π‘’ π‘šβˆ 1 + π‘šβˆ 2 = 90Β° π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒 ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘π‘œπ‘šπ‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ πΆπ‘œπ‘šπ‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ 𝐴𝑛𝑔𝑙𝑒𝑠 Alternative Complement Theorem: ∠1 π‘Žπ‘›π‘‘ ∠2 π‘π‘Ÿπ‘’π‘Žπ‘‘π‘’ ∠𝐴𝐡𝐢, π‘€β„Žπ‘–π‘β„Ž 𝑖𝑠 π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ π‘ƒπ‘Ÿπ‘œπ‘£π‘’ π‘‘β„Žπ‘Žπ‘‘ ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ πΆπ‘œπ‘šπ‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦
∠𝐴𝐡𝐢 𝑖𝑠 π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ ∠1 π‘Žπ‘›π‘‘ ∠2 π‘“π‘œπ‘Ÿπ‘š ∠𝐴𝐡𝐢 𝑔𝑖𝑣𝑒𝑛 π‘šβˆ π΄π΅πΆ = 90Β° π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ π‘šβˆ 1 + π‘šβˆ 2 = π‘šβˆ π΄π΅πΆ 𝐴𝑛𝑔𝑙𝑒 π΄π‘‘π‘‘π‘–π‘‘π‘–π‘œπ‘› π‘ƒπ‘œπ‘ π‘‘π‘’π‘™π‘Žπ‘‘π‘’ π‘šβˆ 1 + π‘šβˆ 2 = 90Β° π‘†π‘’π‘π‘ π‘‘π‘–π‘‘π‘’π‘‘π‘–π‘œπ‘› π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦ 𝑠𝑒𝑏 90Β° 𝑖𝑛 π‘“π‘œπ‘Ÿ ∠𝐴𝐡𝐢 ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘π‘œπ‘šπ‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ πΆπ‘œπ‘šπ‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ 𝐴𝑛𝑔𝑙𝑒𝑠 Congruent Supplement Theorems: Congruent Supplement Theorem (3 angles): π‘ƒπ‘Ÿπ‘œπ‘£π‘’ ∠2 β‰… ∠3 ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘ π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ ∠1 π‘Žπ‘›π‘‘ ∠3 π‘Žπ‘Ÿπ‘’ π‘ π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ 𝑔𝑖𝑣𝑒𝑛 π‘šβˆ 1 + π‘šβˆ 2 = 180Β° π‘šβˆ 1 + π‘šβˆ 3 = 180Β° π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ 𝐴𝑛𝑔𝑙𝑒𝑠 180Β° = π‘šβˆ 1 + π‘šβˆ 3 π‘†π‘¦π‘šπ‘šπ‘’π‘‘π‘Ÿπ‘–π‘ π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦ π‘šβˆ 1 + π‘šβˆ 2 = π‘šβˆ 1 + π‘šβˆ 3 π‘‡π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘π‘–π‘£π‘’ π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦: 𝐼𝑓 𝑁 = 𝑃 π‘Žπ‘›π‘‘ 𝑃 = 𝑀 π‘‘β„Žπ‘’π‘›, 𝑁 = 𝑀 π‘šβˆ 1 + π‘šβˆ 2 = 180Β° = π‘šβˆ 1 + π‘šβˆ 3 𝑁 = 𝑃 = 𝑀 π‘šβˆ 2 = π‘šβˆ 3 𝑆𝑑𝑒𝑝𝑠: π‘šβˆ 1 + π‘šβˆ 2 = π‘šβˆ 1 + π‘šβˆ 3 π‘šβˆ 1 + π‘šβˆ 2 βˆ’ π‘šβˆ 1 = π‘šβˆ 1 + π‘šβˆ 3 βˆ’ π‘šβˆ 1 π‘†π‘’π‘π‘‘π‘Ÿπ‘Žπ‘π‘‘π‘–π‘œπ‘› π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦: 𝑃 = π‘šβˆ 1 (𝑃 𝑖𝑠 π‘Ÿπ‘’π‘‘π‘’π‘“π‘–π‘›π‘’π‘‘ π‘‘π‘œ π‘’π‘žπ‘’π‘Žπ‘™ π‘šβˆ 1) 𝑁 = 𝑀 𝑁 βˆ’ 𝑃 = 𝑀 βˆ’ 𝑃 ∠2 β‰… ∠3 π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠 Congruent Supplement Theorem (4 angles): π‘ƒπ‘Ÿπ‘œπ‘£π‘’ π‘‘β„Žπ‘Žπ‘‘ ∠2 β‰… ∠3 ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘ π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ π‘Žπ‘›π‘”π‘™π‘’π‘  ∠3 π‘Žπ‘›π‘‘ ∠4 π‘Žπ‘Ÿπ‘’ π‘ π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ π‘Žπ‘›π‘”π‘™π‘’π‘  ∠1 β‰… ∠4 𝑔𝑖𝑣𝑒𝑛 π‘šβˆ 1 + π‘šβˆ 2 = 180Β° π‘šβˆ 3 + π‘šβˆ 4 = 180Β° π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ 𝐴𝑛𝑔𝑙𝑒𝑠 180Β° = π‘šβˆ 3 + π‘šβˆ 4 π‘†π‘¦π‘šπ‘šπ‘’π‘‘π‘Ÿπ‘–π‘ π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦ π‘šβˆ 1 + π‘šβˆ 2 = π‘šβˆ 3 + π‘šβˆ 4 π‘‡π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘π‘–π‘£π‘’ π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦: 𝐼𝑓 𝑁 = 𝑃 π‘Žπ‘›π‘‘ 𝑃 = 𝑀, π‘‘β„Žπ‘’π‘› 𝑁 = 𝑀 π‘šβˆ 1 + π‘šβˆ 2 = 180Β° = π‘šβˆ 3 + π‘šβˆ 4 𝑁 = 𝑃 = 𝑀 ∠1 β‰… ∠4 𝑔𝑖𝑣𝑒𝑛 π‘šβˆ 1 = π‘šβˆ 4 π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠 𝑅𝑒𝑓𝑙𝑒π‘₯𝑖𝑣𝑒 π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦: πΉπ‘œπ‘Ÿ π‘Žπ‘›π‘¦ π‘π‘’π‘šπ‘π‘’π‘Ÿ 𝑃, 𝑃 = 𝑃 𝑃 𝑖𝑠 π‘Ÿπ‘’π‘‘π‘’π‘“π‘–π‘›π‘’π‘‘ π‘‘π‘œ π‘’π‘žπ‘’π‘Žπ‘™ π‘‘β„Žπ‘’ π‘šπ‘’π‘Žπ‘ π‘’π‘Ÿπ‘’ π‘œπ‘“ π‘šβˆ 1 π‘Žπ‘›π‘‘ π‘šβˆ 4 π‘šβˆ 2 = π‘šβˆ 3 𝑆𝑑𝑒𝑝𝑠: π‘šβˆ 1 + π‘šβˆ 2 = π‘šβˆ 3 + π‘šβˆ 4 π‘šβˆ 1 + π‘šβˆ 2 βˆ’ π‘šβˆ 1 = π‘šβˆ 3 + π‘šβˆ 4 βˆ’ π‘šβˆ 4 π‘†π‘’π‘π‘‘π‘Ÿπ‘Žπ‘π‘‘π‘–π‘œπ‘› π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦: 𝑁 = 𝑀 𝑁 βˆ’ 𝑃 = 𝑀 βˆ’ 𝑃
∠2 β‰… ∠3 π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠 Congruent Complementary Theorem: π‘ƒπ‘Ÿπ‘œπ‘£π‘’ ∠2 β‰… ∠3 ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘π‘œπ‘šπ‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ ∠1 π‘Žπ‘›π‘‘ ∠3 π‘Žπ‘Ÿπ‘’ π‘π‘œπ‘šπ‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ 𝑔𝑖𝑣𝑒𝑛 π‘šβˆ 1 + π‘šβˆ 2 = 90Β° π‘šβˆ 1 + π‘šβˆ 3 = 90Β° π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ πΆπ‘œπ‘šπ‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ 𝐴𝑛𝑔𝑙𝑒𝑠 90Β° = π‘šβˆ 1 + π‘šβˆ 3 π‘†π‘¦π‘šπ‘šπ‘’π‘‘π‘Ÿπ‘–π‘ π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦ π‘šβˆ 1 + π‘šβˆ 2 = π‘šβˆ 1 + π‘šβˆ 3 π‘‡π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘π‘–π‘£π‘’ π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦: 𝐼𝑓 𝑁 = 𝑃 π‘Žπ‘›π‘‘ 𝑃 = 𝑀 π‘‘β„Žπ‘’π‘›, 𝑁 = 𝑀 π‘šβˆ 1 + π‘šβˆ 2 = 90Β° = π‘šβˆ 1 + π‘šβˆ 3 𝑁 = 𝑃 = 𝑀 π‘šβˆ 2 = π‘šβˆ 3 𝑆𝑑𝑒𝑝𝑠: π‘šβˆ 1 + π‘šβˆ 2 = π‘šβˆ 1 + π‘šβˆ 3 π‘šβˆ 1 + π‘šβˆ 2 βˆ’ π‘šβˆ 1 = π‘šβˆ 1 + π‘šβˆ 3 βˆ’ π‘šβˆ 1 π‘†π‘’π‘π‘‘π‘Ÿπ‘Žπ‘π‘‘π‘–π‘œπ‘› π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦: 𝑃 = π‘šβˆ 1 (𝑃 𝑖𝑠 π‘Ÿπ‘’π‘‘π‘’π‘“π‘–π‘›π‘’π‘‘ π‘‘π‘œ π‘’π‘žπ‘’π‘Žπ‘™ π‘šβˆ 1) 𝑁 = 𝑀 𝑁 βˆ’ 𝑃 = 𝑀 βˆ’ 𝑃 ∠2 β‰… ∠3 π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠 Vertical Angles Theorem: π‘ƒπ‘Ÿπ‘œπ‘£π‘’ ∠1 β‰… ∠3 ∠1 π‘Žπ‘›π‘‘ ∠3 π‘Žπ‘Ÿπ‘’ π‘£π‘’π‘Ÿπ‘‘π‘–π‘π‘Žπ‘™ π‘Žπ‘›π‘”π‘™π‘’π‘  ∠2 π‘Žπ‘›π‘‘ ∠3 π‘Žπ‘Ÿπ‘’ π‘£π‘’π‘Ÿπ‘‘π‘–π‘π‘Žπ‘™ π‘Žπ‘›π‘”π‘™π‘’π‘  𝑔𝑖𝑣𝑒𝑛 π‘π‘œπ‘‘π‘’: π»π‘œπ‘€ π‘‘π‘œ π‘‘π‘’π‘‘π‘’π‘Ÿπ‘šπ‘–π‘›π‘’ π‘‰π‘’π‘Ÿπ‘‘π‘–π‘π‘Žπ‘™ 𝐴𝑛𝑔𝑙𝑒𝑠? π‘ƒπ‘Žπ‘–π‘Ÿ π‘œπ‘“ 𝐴𝑛𝑔𝑙𝑒𝑠 π‘‘β„Žπ‘Žπ‘‘ π‘†β„Žπ‘Žπ‘Ÿπ‘’ π‘Ž π‘‰π‘’π‘Ÿπ‘‘π‘’π‘₯, π‘Žπ‘Ÿπ‘’ π‘π‘œπ‘‘ π΄π‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠, π‘Žπ‘›π‘‘ π‘Žπ‘Ÿπ‘’ π‘“π‘œπ‘Ÿπ‘šπ‘’π‘‘ π‘“π‘Ÿπ‘œπ‘š πΌπ‘›π‘‘π‘’π‘Ÿπ‘ π‘’π‘π‘‘π‘–π‘›π‘” 𝐿𝑖𝑛𝑒𝑠 ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘ π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ ∠2 π‘Žπ‘›π‘‘ ∠3 π‘Žπ‘Ÿπ‘’ π‘ π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ πΏπ‘–π‘›π‘’π‘Žπ‘Ÿ π‘ƒπ‘Žπ‘–π‘Ÿ π‘šβˆ 1 + π‘šβˆ 2 = 180Β° π‘šβˆ 2 + π‘šβˆ 3 = 180Β° π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ 𝐴𝑛𝑔𝑙𝑒𝑠 180Β° = π‘šβˆ 2 + ∠3 π‘†π‘¦π‘šπ‘šπ‘’π‘‘π‘Ÿπ‘–π‘ π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦ π‘šβˆ 1 + π‘šβˆ 2 = π‘šβˆ 2 + π‘šβˆ 3 π‘‡π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘π‘–π‘£π‘’ π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘¦ 𝐼𝑓 𝑁 = 𝑃 π‘Žπ‘›π‘‘ 𝑃 = 𝑀 π‘‘β„Žπ‘’π‘›, 𝑁 = 𝑀 π‘šβˆ 1 + π‘šβˆ 2 = 180Β° = π‘šβˆ 2 + π‘šβˆ 3 𝑁 = 𝑃 = 𝑀 π‘šβˆ 1 = π‘šβˆ 3 𝑆𝑑𝑒𝑝𝑠: π‘šβˆ 1 + π‘šβˆ 2 = π‘šβˆ 2 + π‘šβˆ 3 π‘šβˆ 1 + π‘šβˆ 2 βˆ’ π‘šβˆ 2 = π‘šβˆ 2 + π‘šβˆ 3 βˆ’ π‘šβˆ 2 π‘†π‘’π‘π‘‘π‘Ÿπ‘Žπ‘π‘‘π‘–π‘œπ‘› π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦: 𝑃 = π‘šβˆ 2 (𝑃 𝑖𝑠 π‘Ÿπ‘’π‘‘π‘’π‘“π‘–π‘›π‘’π‘‘ π‘‘π‘œ π‘’π‘žπ‘’π‘Žπ‘™ π‘šβˆ 2) 𝑁 = 𝑀 𝑁 βˆ’ 𝑃 = 𝑀 βˆ’ 𝑃 ∠1 β‰… ∠3 π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠
Perpendicular Lines Theorems: Perpendicular Lines Intersect to Form Four Right Angles Theorem π‘ƒπ‘Ÿπ‘œπ‘£π‘’ π‘‘β„Žπ‘Žπ‘‘ ∠1, ∠2, ∠3, π‘Žπ‘›π‘‘ ∠4 π‘Žπ‘Ÿπ‘’ π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠 𝐿𝑖𝑛𝑒𝑠 𝑑 π‘Žπ‘›π‘‘ 𝑠 π‘Žπ‘Ÿπ‘’ π‘π‘’π‘Ÿπ‘π‘’π‘›π‘‘π‘–π‘π‘’π‘™π‘Žπ‘Ÿ (𝑑 βŠ₯ 𝑠) 𝑔𝑖𝑣𝑒𝑛 ∠1 𝑖𝑠 π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘ƒπ‘’π‘Ÿπ‘π‘’π‘›π‘‘π‘–π‘π‘’π‘™π‘Žπ‘Ÿ 𝐿𝑖𝑛𝑒𝑠 π‘šβˆ 1 = 90Β° π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒 π‘šβˆ 1 + π‘šβˆ 2 = 180Β° π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘ π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘š π‘π‘œπ‘‘π‘’: 𝑁𝑒𝑒𝑑 π‘‘π‘œ 𝐼𝑑𝑒𝑛𝑑𝑖𝑓𝑦 ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘  πΏπ‘–π‘›π‘’π‘Žπ‘Ÿ π‘ƒπ‘Žπ‘–π‘Ÿ 90Β° + π‘šβˆ 2 = 180Β° π‘†π‘’π‘π‘ π‘‘π‘–π‘‘π‘’π‘‘π‘–π‘œπ‘› (𝑠𝑒𝑏 90Β° π‘“π‘œπ‘Ÿ π‘šβˆ 1) π‘šβˆ 2 = 90Β° π‘†π‘’π‘π‘‘π‘Ÿπ‘Žπ‘π‘‘π‘–π‘œπ‘› π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦ (π‘†π‘’π‘π‘ π‘‘π‘Ÿπ‘Žπ‘π‘‘ 90Β° π‘“π‘œπ‘Ÿ π‘’π‘Žπ‘β„Ž 𝑠𝑖𝑑𝑒 π‘œπ‘“ π‘‘β„Žπ‘’ πΈπ‘žπ‘’π‘Žπ‘‘π‘–π‘œπ‘›) ∠2 𝑖𝑠 π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒 ∠1 β‰… ∠3 π‘‰π‘’π‘Ÿπ‘‘π‘–π‘π‘Žπ‘™ 𝐴𝑛𝑔𝑙𝑒 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘š π‘π‘œπ‘‘π‘’: 𝑁𝑒𝑒𝑑 π‘‘π‘œ 𝐼𝑑𝑒𝑛𝑑𝑖𝑓𝑦 ∠1 π‘Žπ‘›π‘‘ ∠3 π‘Žπ‘  π‘‰π‘’π‘Ÿπ‘‘π‘–π‘π‘Žπ‘™ 𝐴𝑛𝑔𝑙𝑒𝑠, π‘‘β„Žπ‘’π‘Ÿπ‘’π‘“π‘œπ‘Ÿπ‘’ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ π‘šβˆ 1 = π‘šβˆ 3 π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠 90Β° = π‘šβˆ 3 π‘†π‘’π‘π‘ π‘‘π‘–π‘‘π‘’π‘‘π‘–π‘œπ‘› (𝑠𝑒𝑏 90Β° π‘“π‘œπ‘Ÿ π‘šβˆ 1) ∠3 𝑖𝑠 π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒 π‘šβˆ 1 + π‘šβˆ 4 = 180Β° π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘ π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘š (πΏπ‘–π‘›π‘’π‘Žπ‘Ÿ π‘ƒπ‘Žπ‘–π‘Ÿ) 90Β° + π‘šβˆ 4 = 180Β° π‘†π‘’π‘π‘ π‘‘π‘–π‘‘π‘’π‘‘π‘–π‘œπ‘› (𝑠𝑒𝑏 90Β° π‘“π‘œπ‘Ÿ π‘šβˆ 1) π‘šβˆ 4 = 90Β° π‘†π‘’π‘π‘‘π‘Ÿπ‘Žπ‘π‘‘π‘–π‘œπ‘› π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦ (π‘†π‘’π‘π‘ π‘‘π‘Ÿπ‘Žπ‘π‘‘ 90Β° π‘“π‘œπ‘Ÿ π‘’π‘Žπ‘β„Ž 𝑠𝑖𝑑𝑒 π‘œπ‘“ π‘‘β„Žπ‘’ πΈπ‘žπ‘’π‘Žπ‘‘π‘–π‘œπ‘›) ∠4 𝑖𝑠 π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒 All Right Angles are Congruent Theorem π‘ƒπ‘Ÿπ‘œπ‘£π‘’ ∠1 β‰… ∠2 β‰… ∠3 β‰… ∠4 𝑑 βŠ₯ 𝑠 𝑔𝑖𝑣𝑒𝑛 ∠1 𝑖𝑠 π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ ∠2 𝑖𝑠 π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ ∠3 𝑖𝑠 π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ ∠4 𝑖𝑠 π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ π‘ƒπ‘’π‘Ÿπ‘π‘’π‘›π‘‘π‘–π‘π‘’π‘™π‘Žπ‘Ÿ 𝐿𝑖𝑛𝑒𝑠 πΌπ‘›π‘‘π‘’π‘Ÿπ‘ π‘’π‘π‘‘ π‘‘π‘œ πΉπ‘œπ‘Ÿπ‘š πΉπ‘œπ‘’π‘Ÿ π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘š π‘šβˆ 1 = 90Β° π‘šβˆ 2 = 90Β° π‘šβˆ 3 = 90Β° π‘šβˆ 4 = 90Β° π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒 π‘šβˆ 1 = π‘šβˆ 2 = π‘šβˆ 3 = π‘šβˆ 4 π‘†π‘’π‘π‘ π‘‘π‘–π‘‘π‘’π‘‘π‘–π‘œπ‘› ∠1 β‰… ∠2 β‰… ∠3 β‰… ∠4 π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠
Perpendicular Lines Will Form Four Congruent Adjacent Angles Theorem π‘ƒπ‘Ÿπ‘œπ‘£π‘’ π‘‘β„Žπ‘Žπ‘‘ ∠1 β‰… ∠2 𝑑 βŠ₯ 𝑠 𝑔𝑖𝑣𝑒𝑛 ∠1 𝑖𝑠 π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ ∠2 𝑖𝑠 π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ ∠3 𝑖𝑠 π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ ∠4 𝑖𝑠 π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ π‘ƒπ‘’π‘Ÿπ‘π‘’π‘›π‘‘π‘–π‘π‘’π‘™π‘Žπ‘Ÿ 𝐿𝑖𝑛𝑒𝑠 πΌπ‘›π‘‘π‘’π‘Ÿπ‘ π‘’π‘π‘‘ π‘‘π‘œ πΉπ‘œπ‘Ÿπ‘š πΉπ‘œπ‘’π‘Ÿ π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘š ∠1 β‰… ∠2 β‰… ∠3 β‰… ∠4 𝐴𝑙𝑙 π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠 π‘Žπ‘Ÿπ‘’ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘š ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ ∠2 π‘Žπ‘›π‘‘ ∠3 π‘Žπ‘Ÿπ‘’ π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ ∠3 π‘Žπ‘›π‘‘ ∠4 π‘Žπ‘Ÿπ‘’ π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ ∠4 π‘Žπ‘›π‘‘ ∠1 π‘Žπ‘Ÿπ‘’ π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π΄π‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠 π‘π‘œπ‘‘π‘’: π»π‘œπ‘€ π‘‘π‘œ 𝐼𝑑𝑒𝑛𝑑𝑖𝑓𝑦 π΄π‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠? π‘†β„Žπ‘Žπ‘Ÿπ‘’ π‘Ž π‘‰π‘’π‘Ÿπ‘‘π‘’π‘₯, π‘†β„Žπ‘Žπ‘Ÿπ‘’ π‘Ž 𝑆𝑖𝑑𝑒, π‘Žπ‘›π‘‘ π‘Žπ‘Ÿπ‘’ π‘œπ‘› π‘‘β„Žπ‘’ π‘†π‘Žπ‘šπ‘’ π‘ƒπ‘™π‘Žπ‘›π‘’ If Two Angles are Both Congruent and Supplementary, Then Each Angle Will be a Right Angle Theorem π‘ƒπ‘Ÿπ‘œπ‘£π‘’ π‘‘β„Žπ‘Žπ‘‘ ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠 ∠1 β‰… ∠2 ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘ π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ 𝑔𝑖𝑣𝑒𝑛 π‘šβˆ 1 = π‘šβˆ 2 π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠 π‘šβˆ 1 + π‘šβˆ 2 = 180Β° π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ 𝐴𝑛𝑔𝑙𝑒𝑠 π‘šβˆ 1 + π‘šβˆ 1 = 180Β° π‘†π‘’π‘π‘ π‘‘π‘–π‘‘π‘’π‘‘π‘–π‘œπ‘› π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦ (𝑠𝑒𝑏 π‘šβˆ 1 π‘“π‘œπ‘Ÿ π‘šβˆ 2 2 βˆ— π‘šβˆ 1 = 180Β° π‘†π‘’π‘π‘ π‘‘π‘–π‘‘π‘’π‘‘π‘–π‘œπ‘› π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦ (𝑠𝑒𝑏 2 βˆ— π‘šβˆ 1 π‘“π‘œπ‘Ÿ π‘šβˆ 1 + π‘šβˆ 1) π‘šβˆ 1 2 = 180Β° 2 π‘šβˆ 1 = 90Β° π·π‘–π‘£π‘–π‘ π‘–π‘œπ‘› π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦ ∠1 𝑖𝑠 π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒 π‘šβˆ 2 = 90Β° π‘†π‘’π‘π‘ π‘‘π‘–π‘‘π‘’π‘‘π‘–π‘œπ‘› π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦ (𝑠𝑒𝑏 π‘šβˆ 2 π‘“π‘œπ‘Ÿ π‘šβˆ 1) ∠2 𝑖𝑠 π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒 If Two Angles of a Linear Pair are Congruent, Then They Are Right Angles Theorem π‘ƒπ‘Ÿπ‘œπ‘£π‘’ π‘‘β„Žπ‘Žπ‘‘ ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠 ∠1 β‰… ∠2 ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘Ž π‘™π‘–π‘›π‘’π‘Žπ‘Ÿ π‘π‘Žπ‘–π‘Ÿ 𝑔𝑖𝑣𝑒𝑛 ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘ π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘ π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘š ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’π‘  𝐼𝑓 π‘‡π‘€π‘œ 𝐴𝑛𝑔𝑙𝑒𝑠 π‘Žπ‘Ÿπ‘’ π΅π‘œπ‘‘β„Ž πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ π‘Žπ‘›π‘‘ π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦, π‘‡β„Žπ‘’π‘› πΈπ‘Žπ‘β„Ž 𝐴𝑛𝑔𝑙𝑒 π‘Šπ‘–π‘™π‘™ 𝑏𝑒 π‘Ž π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘š

Recommended

Engineering Mechanics made simple
Engineering Mechanics made simpleEngineering Mechanics made simple
Engineering Mechanics made simpleProf. S.Rajendiran
Β 
Experimental strain analysis
Experimental strain analysisExperimental strain analysis
Experimental strain analysis
Kanok Phoocam
Β 
MOTION OF A RIGID BODY IN SPACE
MOTION OF A RIGID BODY IN SPACEMOTION OF A RIGID BODY IN SPACE
MOTION OF A RIGID BODY IN SPACE
AmenahGondal1
Β 
Momento en estructuras
Momento en estructurasMomento en estructuras
Momento en estructuras
Rol D
Β 
Engineering Mechanice Lecture 06
Engineering Mechanice Lecture 06Engineering Mechanice Lecture 06
Engineering Mechanice Lecture 06
Self-employed
Β 
Work and Energy
Work and EnergyWork and Energy
Work and Energy
Sumit Basu
Β 
Spherical harmonics
Spherical harmonicsSpherical harmonics
Spherical harmonics
Mahmoud Solomon
Β 
General system of forces unit 4 bce & engg mechanics
General system of forces unit 4 bce & engg mechanicsGeneral system of forces unit 4 bce & engg mechanics
General system of forces unit 4 bce & engg mechanics
Parimal Jha
Β 

Recommended

Engineering Mechanics made simple
Engineering Mechanics made simpleEngineering Mechanics made simple
Engineering Mechanics made simpleProf. S.Rajendiran
Β 
Experimental strain analysis
Experimental strain analysisExperimental strain analysis
Experimental strain analysis
Kanok Phoocam
Β 
MOTION OF A RIGID BODY IN SPACE
MOTION OF A RIGID BODY IN SPACEMOTION OF A RIGID BODY IN SPACE
MOTION OF A RIGID BODY IN SPACE
AmenahGondal1
Β 
Momento en estructuras
Momento en estructurasMomento en estructuras
Momento en estructuras
Rol D
Β 
Engineering Mechanice Lecture 06
Engineering Mechanice Lecture 06Engineering Mechanice Lecture 06
Engineering Mechanice Lecture 06
Self-employed
Β 
Work and Energy
Work and EnergyWork and Energy
Work and Energy
Sumit Basu
Β 
Spherical harmonics
Spherical harmonicsSpherical harmonics
Spherical harmonics
Mahmoud Solomon
Β 
General system of forces unit 4 bce & engg mechanics
General system of forces unit 4 bce & engg mechanicsGeneral system of forces unit 4 bce & engg mechanics
General system of forces unit 4 bce & engg mechanics
Parimal Jha
Β 
Center of mass_&_rocket_propulsion
Center of mass_&_rocket_propulsionCenter of mass_&_rocket_propulsion
Center of mass_&_rocket_propulsion
Tharika Weerakoon
Β 
Mechanics Lec 1
Mechanics Lec 1Mechanics Lec 1
Mechanics Lec 1
asad ali
Β 
Reflection refraction
Reflection refractionReflection refraction
Reflection refraction
UCP
Β 
Simple harmonic oscillator - Classical Mechanics
Simple harmonic oscillator - Classical MechanicsSimple harmonic oscillator - Classical Mechanics
Simple harmonic oscillator - Classical Mechanics
Debashis Baidya
Β 
Civil Engineering structure
Civil Engineering structureCivil Engineering structure
Civil Engineering structure
Ayaz Pirkhan
Β 
Ce 255 handout
Ce 255 handoutCe 255 handout
Ce 255 handout
DanielAkorful
Β 
engineering statics: distributed forces-1
engineering statics: distributed forces-1engineering statics: distributed forces-1
engineering statics: distributed forces-1
musadoto
Β 
Teaching Slide1
Teaching Slide1Teaching Slide1
Teaching Slide1
Solomon Ojiemudia
Β 
Review of Seiberg Witten duality.pptx
Review of Seiberg Witten duality.pptxReview of Seiberg Witten duality.pptx
Review of Seiberg Witten duality.pptx
Hassaan Saleem
Β 
Handbook to ssc je mechanical
Handbook to ssc je mechanical Handbook to ssc je mechanical
Handbook to ssc je mechanical
mechanical Singh
Β 
HSC Physics - Torque
HSC Physics - TorqueHSC Physics - Torque
HSC Physics - Torque
Megaminds Brainworks
Β 
1 equilbrium
1 equilbrium1 equilbrium
1 equilbriumMr Patrick NIYISHAKA
Β 
Wk 1 p7 wk 3-p8_13.1-13.3 & 14.6_oscillations & ultrasound
Wk 1 p7 wk 3-p8_13.1-13.3 & 14.6_oscillations & ultrasoundWk 1 p7 wk 3-p8_13.1-13.3 & 14.6_oscillations & ultrasound
Wk 1 p7 wk 3-p8_13.1-13.3 & 14.6_oscillations & ultrasound
chris lembalemba
Β 
Rigid body solutions
Rigid body solutionsRigid body solutions
Rigid body solutions
Ejiro Makamaks
Β 
Balancing of rotating masses
Balancing of rotating massesBalancing of rotating masses
Balancing of rotating masses
yamini champaneri
Β 
engineering statics :force systems
engineering statics :force systems engineering statics :force systems
engineering statics :force systems
musadoto
Β 
MECHANICS OF SOLIDS(coplanar concurrent forces)
MECHANICS OF SOLIDS(coplanar concurrent forces)MECHANICS OF SOLIDS(coplanar concurrent forces)
MECHANICS OF SOLIDS(coplanar concurrent forces)
Parthivpal17
Β 
Solving trigonometric equations 1
Solving trigonometric equations 1Solving trigonometric equations 1
Solving trigonometric equations 1
MalekBashaireh
Β 
Centroid & Centre of Gravity
Centroid & Centre of GravityCentroid & Centre of Gravity
Centroid & Centre of Gravity
Akash Patel
Β 
Differential Geometry for Machine Learning
Differential Geometry for Machine LearningDifferential Geometry for Machine Learning
Differential Geometry for Machine Learning
SEMINARGROOT
Β 
SUEC 高中 Adv Maths (Degree-Radian, Arc length)
SUEC 高中 Adv Maths (Degree-Radian, Arc length)SUEC 高中 Adv Maths (Degree-Radian, Arc length)
SUEC 高中 Adv Maths (Degree-Radian, Arc length)
tungwc
Β 
07.mdsd_modelado_termicos_liquidos
07.mdsd_modelado_termicos_liquidos07.mdsd_modelado_termicos_liquidos
07.mdsd_modelado_termicos_liquidos
HipΓ³lito Aguilar
Β 

More Related Content

What's hot

Center of mass_&_rocket_propulsion
Center of mass_&_rocket_propulsionCenter of mass_&_rocket_propulsion
Center of mass_&_rocket_propulsion
Tharika Weerakoon
Β 
Mechanics Lec 1
Mechanics Lec 1Mechanics Lec 1
Mechanics Lec 1
asad ali
Β 
Reflection refraction
Reflection refractionReflection refraction
Reflection refraction
UCP
Β 
Simple harmonic oscillator - Classical Mechanics
Simple harmonic oscillator - Classical MechanicsSimple harmonic oscillator - Classical Mechanics
Simple harmonic oscillator - Classical Mechanics
Debashis Baidya
Β 
Civil Engineering structure
Civil Engineering structureCivil Engineering structure
Civil Engineering structure
Ayaz Pirkhan
Β 
Ce 255 handout
Ce 255 handoutCe 255 handout
Ce 255 handout
DanielAkorful
Β 
engineering statics: distributed forces-1
engineering statics: distributed forces-1engineering statics: distributed forces-1
engineering statics: distributed forces-1
musadoto
Β 
Teaching Slide1
Teaching Slide1Teaching Slide1
Teaching Slide1
Solomon Ojiemudia
Β 
Review of Seiberg Witten duality.pptx
Review of Seiberg Witten duality.pptxReview of Seiberg Witten duality.pptx
Review of Seiberg Witten duality.pptx
Hassaan Saleem
Β 
Handbook to ssc je mechanical
Handbook to ssc je mechanical Handbook to ssc je mechanical
Handbook to ssc je mechanical
mechanical Singh
Β 
HSC Physics - Torque
HSC Physics - TorqueHSC Physics - Torque
HSC Physics - Torque
Megaminds Brainworks
Β 
Wk 1 p7 wk 3-p8_13.1-13.3 & 14.6_oscillations & ultrasound
Wk 1 p7 wk 3-p8_13.1-13.3 & 14.6_oscillations & ultrasoundWk 1 p7 wk 3-p8_13.1-13.3 & 14.6_oscillations & ultrasound
Wk 1 p7 wk 3-p8_13.1-13.3 & 14.6_oscillations & ultrasound
chris lembalemba
Β 
Rigid body solutions
Rigid body solutionsRigid body solutions
Rigid body solutions
Ejiro Makamaks
Β 
Balancing of rotating masses
Balancing of rotating massesBalancing of rotating masses
Balancing of rotating masses
yamini champaneri
Β 
engineering statics :force systems
engineering statics :force systems engineering statics :force systems
engineering statics :force systems
musadoto
Β 
MECHANICS OF SOLIDS(coplanar concurrent forces)
MECHANICS OF SOLIDS(coplanar concurrent forces)MECHANICS OF SOLIDS(coplanar concurrent forces)
MECHANICS OF SOLIDS(coplanar concurrent forces)
Parthivpal17
Β 
Solving trigonometric equations 1
Solving trigonometric equations 1Solving trigonometric equations 1
Solving trigonometric equations 1
MalekBashaireh
Β 
Centroid & Centre of Gravity
Centroid & Centre of GravityCentroid & Centre of Gravity
Centroid & Centre of Gravity
Akash Patel
Β 

What's hot (19)

Center of mass_&_rocket_propulsion
Center of mass_&_rocket_propulsionCenter of mass_&_rocket_propulsion
Center of mass_&_rocket_propulsion
Β 
Mechanics Lec 1
Mechanics Lec 1Mechanics Lec 1
Mechanics Lec 1
Β 
Reflection refraction
Reflection refractionReflection refraction
Reflection refraction
Β 
Simple harmonic oscillator - Classical Mechanics
Simple harmonic oscillator - Classical MechanicsSimple harmonic oscillator - Classical Mechanics
Simple harmonic oscillator - Classical Mechanics
Β 
Civil Engineering structure
Civil Engineering structureCivil Engineering structure
Civil Engineering structure
Β 
Ce 255 handout
Ce 255 handoutCe 255 handout
Ce 255 handout
Β 
engineering statics: distributed forces-1
engineering statics: distributed forces-1engineering statics: distributed forces-1
engineering statics: distributed forces-1
Β 
Teaching Slide1
Teaching Slide1Teaching Slide1
Teaching Slide1
Β 
Review of Seiberg Witten duality.pptx
Review of Seiberg Witten duality.pptxReview of Seiberg Witten duality.pptx
Review of Seiberg Witten duality.pptx
Β 
Handbook to ssc je mechanical
Handbook to ssc je mechanical Handbook to ssc je mechanical
Handbook to ssc je mechanical
Β 
HSC Physics - Torque
HSC Physics - TorqueHSC Physics - Torque
HSC Physics - Torque
Β 
1 equilbrium
1 equilbrium1 equilbrium
1 equilbrium
Β 
Wk 1 p7 wk 3-p8_13.1-13.3 & 14.6_oscillations & ultrasound
Wk 1 p7 wk 3-p8_13.1-13.3 & 14.6_oscillations & ultrasoundWk 1 p7 wk 3-p8_13.1-13.3 & 14.6_oscillations & ultrasound
Wk 1 p7 wk 3-p8_13.1-13.3 & 14.6_oscillations & ultrasound
Β 
Rigid body solutions
Rigid body solutionsRigid body solutions
Rigid body solutions
Β 
Balancing of rotating masses
Balancing of rotating massesBalancing of rotating masses
Balancing of rotating masses
Β 
engineering statics :force systems
engineering statics :force systems engineering statics :force systems
engineering statics :force systems
Β 
MECHANICS OF SOLIDS(coplanar concurrent forces)
MECHANICS OF SOLIDS(coplanar concurrent forces)MECHANICS OF SOLIDS(coplanar concurrent forces)
MECHANICS OF SOLIDS(coplanar concurrent forces)
Β 
Solving trigonometric equations 1
Solving trigonometric equations 1Solving trigonometric equations 1
Solving trigonometric equations 1
Β 
Centroid & Centre of Gravity
Centroid & Centre of GravityCentroid & Centre of Gravity
Centroid & Centre of Gravity
Β 

Similar to Geometry Theorems 1 REMC Tutoring

Differential Geometry for Machine Learning
Differential Geometry for Machine LearningDifferential Geometry for Machine Learning
Differential Geometry for Machine Learning
SEMINARGROOT
Β 
SUEC 高中 Adv Maths (Degree-Radian, Arc length)
SUEC 高中 Adv Maths (Degree-Radian, Arc length)SUEC 高中 Adv Maths (Degree-Radian, Arc length)
SUEC 高中 Adv Maths (Degree-Radian, Arc length)
tungwc
Β 
07.mdsd_modelado_termicos_liquidos
07.mdsd_modelado_termicos_liquidos07.mdsd_modelado_termicos_liquidos
07.mdsd_modelado_termicos_liquidos
HipΓ³lito Aguilar
Β 
HEAT CONDUCTION DEMYSTIFIED.pdf
HEAT CONDUCTION DEMYSTIFIED.pdfHEAT CONDUCTION DEMYSTIFIED.pdf
HEAT CONDUCTION DEMYSTIFIED.pdf
Wasswaderrick3
Β 
FUNDAMENTALS OF HEAT TRANSFER .pdf
FUNDAMENTALS OF HEAT TRANSFER .pdfFUNDAMENTALS OF HEAT TRANSFER .pdf
FUNDAMENTALS OF HEAT TRANSFER .pdf
Wasswaderrick3
Β 
RADIAL HEAT CONDUCTION SOLVED USING THE INTEGRAL EQUATION .pdf
RADIAL HEAT CONDUCTION SOLVED USING THE INTEGRAL EQUATION .pdfRADIAL HEAT CONDUCTION SOLVED USING THE INTEGRAL EQUATION .pdf
RADIAL HEAT CONDUCTION SOLVED USING THE INTEGRAL EQUATION .pdf
Wasswaderrick3
Β 
K to 12 math
K to 12 mathK to 12 math
K to 12 math
GrantWSmith
Β 
SUEC 高中 Adv Maths (Trigo Equation) (Part 3)
SUEC 高中 Adv Maths (Trigo Equation) (Part 3)SUEC 高中 Adv Maths (Trigo Equation) (Part 3)
SUEC 高中 Adv Maths (Trigo Equation) (Part 3)
tungwc
Β 
Integral method of the Analytic solutions to the heat equation With Experimen...
Integral method of the Analytic solutions to the heat equation With Experimen...Integral method of the Analytic solutions to the heat equation With Experimen...
Integral method of the Analytic solutions to the heat equation With Experimen...
Wasswaderrick3
Β 
Mathematical analysis of non-uniform polyhedra having 2 congruent regular n-g...
Mathematical analysis of non-uniform polyhedra having 2 congruent regular n-g...Mathematical analysis of non-uniform polyhedra having 2 congruent regular n-g...
Mathematical analysis of non-uniform polyhedra having 2 congruent regular n-g...
Harish Chandra Rajpoot
Β 
Parallelogram Law Force | Civil Engineering
Parallelogram Law Force | Civil EngineeringParallelogram Law Force | Civil Engineering
Parallelogram Law Force | Civil Engineering
Transweb Global Inc
Β 
7). mechanical waves (finished)
7). mechanical waves (finished)7). mechanical waves (finished)
7). mechanical waves (finished)PhysicsLover
Β 
SUEC 高中 Adv Maths (Trigo Function Part 3)
SUEC 高中 Adv Maths (Trigo Function Part 3)SUEC 高中 Adv Maths (Trigo Function Part 3)
SUEC 高中 Adv Maths (Trigo Function Part 3)
tungwc
Β 
TRANSIENT AND STEADY STATE HEAT CONDUCTION WITH NO LATERAL CONVECTION SOLVED ...
TRANSIENT AND STEADY STATE HEAT CONDUCTION WITH NO LATERAL CONVECTION SOLVED ...TRANSIENT AND STEADY STATE HEAT CONDUCTION WITH NO LATERAL CONVECTION SOLVED ...
TRANSIENT AND STEADY STATE HEAT CONDUCTION WITH NO LATERAL CONVECTION SOLVED ...
Wasswaderrick3
Β 
09.sdcd_lugar_geometrico_raices
09.sdcd_lugar_geometrico_raices09.sdcd_lugar_geometrico_raices
09.sdcd_lugar_geometrico_raices
HipΓ³lito Aguilar
Β 
SEMI-INFINITE ROD SOLUTION FOR TRANSIENT AND STEADY STATE.pdf
SEMI-INFINITE ROD SOLUTION FOR TRANSIENT AND STEADY STATE.pdfSEMI-INFINITE ROD SOLUTION FOR TRANSIENT AND STEADY STATE.pdf
SEMI-INFINITE ROD SOLUTION FOR TRANSIENT AND STEADY STATE.pdf
Wasswaderrick3
Β 
Equations_3_Industrial Instrumentation - Temperature & Level Measurement Impo...
Equations_3_Industrial Instrumentation - Temperature & Level Measurement Impo...Equations_3_Industrial Instrumentation - Temperature & Level Measurement Impo...
Equations_3_Industrial Instrumentation - Temperature & Level Measurement Impo...
Burdwan University
Β 
Teoria NumΓ©rica (Palestra 01)
Teoria NumΓ©rica (Palestra 01)Teoria NumΓ©rica (Palestra 01)
Teoria NumΓ©rica (Palestra 01)
Eugenio Souza
Β 
Jordan Higher (𝜎, 𝜏)-Centralizer on Prime Ring
Jordan Higher (𝜎, 𝜏)-Centralizer on Prime RingJordan Higher (𝜎, 𝜏)-Centralizer on Prime Ring
Jordan Higher (𝜎, 𝜏)-Centralizer on Prime Ring
IOSR Journals
Β 
ANALTICAL SOLUTIONS TO THE HEAT EQUATION USING THE INTEGRAL METHODS.pdf
ANALTICAL SOLUTIONS TO THE HEAT EQUATION USING THE INTEGRAL METHODS.pdfANALTICAL SOLUTIONS TO THE HEAT EQUATION USING THE INTEGRAL METHODS.pdf
ANALTICAL SOLUTIONS TO THE HEAT EQUATION USING THE INTEGRAL METHODS.pdf
Wasswaderrick3
Β 

Similar to Geometry Theorems 1 REMC Tutoring (20)

Differential Geometry for Machine Learning
Differential Geometry for Machine LearningDifferential Geometry for Machine Learning
Differential Geometry for Machine Learning
Β 
SUEC 高中 Adv Maths (Degree-Radian, Arc length)
SUEC 高中 Adv Maths (Degree-Radian, Arc length)SUEC 高中 Adv Maths (Degree-Radian, Arc length)
SUEC 高中 Adv Maths (Degree-Radian, Arc length)
Β 
07.mdsd_modelado_termicos_liquidos
07.mdsd_modelado_termicos_liquidos07.mdsd_modelado_termicos_liquidos
07.mdsd_modelado_termicos_liquidos
Β 
HEAT CONDUCTION DEMYSTIFIED.pdf
HEAT CONDUCTION DEMYSTIFIED.pdfHEAT CONDUCTION DEMYSTIFIED.pdf
HEAT CONDUCTION DEMYSTIFIED.pdf
Β 
FUNDAMENTALS OF HEAT TRANSFER .pdf
FUNDAMENTALS OF HEAT TRANSFER .pdfFUNDAMENTALS OF HEAT TRANSFER .pdf
FUNDAMENTALS OF HEAT TRANSFER .pdf
Β 
RADIAL HEAT CONDUCTION SOLVED USING THE INTEGRAL EQUATION .pdf
RADIAL HEAT CONDUCTION SOLVED USING THE INTEGRAL EQUATION .pdfRADIAL HEAT CONDUCTION SOLVED USING THE INTEGRAL EQUATION .pdf
RADIAL HEAT CONDUCTION SOLVED USING THE INTEGRAL EQUATION .pdf
Β 
K to 12 math
K to 12 mathK to 12 math
K to 12 math
Β 
SUEC 高中 Adv Maths (Trigo Equation) (Part 3)
SUEC 高中 Adv Maths (Trigo Equation) (Part 3)SUEC 高中 Adv Maths (Trigo Equation) (Part 3)
SUEC 高中 Adv Maths (Trigo Equation) (Part 3)
Β 
Integral method of the Analytic solutions to the heat equation With Experimen...
Integral method of the Analytic solutions to the heat equation With Experimen...Integral method of the Analytic solutions to the heat equation With Experimen...
Integral method of the Analytic solutions to the heat equation With Experimen...
Β 
Mathematical analysis of non-uniform polyhedra having 2 congruent regular n-g...
Mathematical analysis of non-uniform polyhedra having 2 congruent regular n-g...Mathematical analysis of non-uniform polyhedra having 2 congruent regular n-g...
Mathematical analysis of non-uniform polyhedra having 2 congruent regular n-g...
Β 
Parallelogram Law Force | Civil Engineering
Parallelogram Law Force | Civil EngineeringParallelogram Law Force | Civil Engineering
Parallelogram Law Force | Civil Engineering
Β 
7). mechanical waves (finished)
7). mechanical waves (finished)7). mechanical waves (finished)
7). mechanical waves (finished)
Β 
SUEC 高中 Adv Maths (Trigo Function Part 3)
SUEC 高中 Adv Maths (Trigo Function Part 3)SUEC 高中 Adv Maths (Trigo Function Part 3)
SUEC 高中 Adv Maths (Trigo Function Part 3)
Β 
TRANSIENT AND STEADY STATE HEAT CONDUCTION WITH NO LATERAL CONVECTION SOLVED ...
TRANSIENT AND STEADY STATE HEAT CONDUCTION WITH NO LATERAL CONVECTION SOLVED ...TRANSIENT AND STEADY STATE HEAT CONDUCTION WITH NO LATERAL CONVECTION SOLVED ...
TRANSIENT AND STEADY STATE HEAT CONDUCTION WITH NO LATERAL CONVECTION SOLVED ...
Β 
09.sdcd_lugar_geometrico_raices
09.sdcd_lugar_geometrico_raices09.sdcd_lugar_geometrico_raices
09.sdcd_lugar_geometrico_raices
Β 
SEMI-INFINITE ROD SOLUTION FOR TRANSIENT AND STEADY STATE.pdf
SEMI-INFINITE ROD SOLUTION FOR TRANSIENT AND STEADY STATE.pdfSEMI-INFINITE ROD SOLUTION FOR TRANSIENT AND STEADY STATE.pdf
SEMI-INFINITE ROD SOLUTION FOR TRANSIENT AND STEADY STATE.pdf
Β 
Equations_3_Industrial Instrumentation - Temperature & Level Measurement Impo...
Equations_3_Industrial Instrumentation - Temperature & Level Measurement Impo...Equations_3_Industrial Instrumentation - Temperature & Level Measurement Impo...
Equations_3_Industrial Instrumentation - Temperature & Level Measurement Impo...
Β 
Teoria NumΓ©rica (Palestra 01)
Teoria NumΓ©rica (Palestra 01)Teoria NumΓ©rica (Palestra 01)
Teoria NumΓ©rica (Palestra 01)
Β 
Jordan Higher (𝜎, 𝜏)-Centralizer on Prime Ring
Jordan Higher (𝜎, 𝜏)-Centralizer on Prime RingJordan Higher (𝜎, 𝜏)-Centralizer on Prime Ring
Jordan Higher (𝜎, 𝜏)-Centralizer on Prime Ring
Β 
ANALTICAL SOLUTIONS TO THE HEAT EQUATION USING THE INTEGRAL METHODS.pdf
ANALTICAL SOLUTIONS TO THE HEAT EQUATION USING THE INTEGRAL METHODS.pdfANALTICAL SOLUTIONS TO THE HEAT EQUATION USING THE INTEGRAL METHODS.pdf
ANALTICAL SOLUTIONS TO THE HEAT EQUATION USING THE INTEGRAL METHODS.pdf
Β 

Recently uploaded

Acetabularia Information For Class 9 .docx
Acetabularia Information For Class 9 .docxAcetabularia Information For Class 9 .docx
Acetabularia Information For Class 9 .docx
vaibhavrinwa19
Β 
The basics of sentences session 5pptx.pptx
The basics of sentences session 5pptx.pptxThe basics of sentences session 5pptx.pptx
The basics of sentences session 5pptx.pptx
heathfieldcps1
Β 
Chapter 3 - Islamic Banking Products and Services.pptx
Chapter 3 - Islamic Banking Products and Services.pptxChapter 3 - Islamic Banking Products and Services.pptx
Chapter 3 - Islamic Banking Products and Services.pptx
Mohd Adib Abd Muin, Senior Lecturer at Universiti Utara Malaysia
Β 
Thesis Statement for students diagnonsed withADHD.ppt
Thesis Statement for students diagnonsed withADHD.pptThesis Statement for students diagnonsed withADHD.ppt
Thesis Statement for students diagnonsed withADHD.ppt
EverAndrsGuerraGuerr
Β 
Honest Reviews of Tim Han LMA Course Program.pptx
Honest Reviews of Tim Han LMA Course Program.pptxHonest Reviews of Tim Han LMA Course Program.pptx
Honest Reviews of Tim Han LMA Course Program.pptx
timhan337
Β 
Unit 2- Research Aptitude (UGC NET Paper I).pdf
Unit 2- Research Aptitude (UGC NET Paper I).pdfUnit 2- Research Aptitude (UGC NET Paper I).pdf
Unit 2- Research Aptitude (UGC NET Paper I).pdf
Thiyagu K
Β 
How libraries can support authors with open access requirements for UKRI fund...
How libraries can support authors with open access requirements for UKRI fund...How libraries can support authors with open access requirements for UKRI fund...
How libraries can support authors with open access requirements for UKRI fund...
Jisc
Β 
2024.06.01 Introducing a competency framework for languag learning materials ...
2024.06.01 Introducing a competency framework for languag learning materials ...2024.06.01 Introducing a competency framework for languag learning materials ...
2024.06.01 Introducing a competency framework for languag learning materials ...
Sandy Millin
Β 
Mule 4.6 & Java 17 Upgrade | MuleSoft Mysore Meetup #46
Mule 4.6 & Java 17 Upgrade | MuleSoft Mysore Meetup #46Mule 4.6 & Java 17 Upgrade | MuleSoft Mysore Meetup #46
Mule 4.6 & Java 17 Upgrade | MuleSoft Mysore Meetup #46
MysoreMuleSoftMeetup
Β 
BΓ€I TαΊ¬P Bα»” TRα»’ TIαΊΎNG ANH GLOBAL SUCCESS LỚP 3 - CαΊ’ NΔ‚M (CΓ“ FILE NGHE VΓ€ ĐÁP Á...
BΓ€I TαΊ¬P Bα»” TRα»’ TIαΊΎNG ANH GLOBAL SUCCESS LỚP 3 - CαΊ’ NΔ‚M (CΓ“ FILE NGHE VΓ€ ĐÁP Á...BΓ€I TαΊ¬P Bα»” TRα»’ TIαΊΎNG ANH GLOBAL SUCCESS LỚP 3 - CαΊ’ NΔ‚M (CΓ“ FILE NGHE VΓ€ ĐÁP Á...
BΓ€I TαΊ¬P Bα»” TRα»’ TIαΊΎNG ANH GLOBAL SUCCESS LỚP 3 - CαΊ’ NΔ‚M (CΓ“ FILE NGHE VΓ€ ĐÁP Á...
Nguyen Thanh Tu Collection
Β 
Adversarial Attention Modeling for Multi-dimensional Emotion Regression.pdf
Adversarial Attention Modeling for Multi-dimensional Emotion Regression.pdfAdversarial Attention Modeling for Multi-dimensional Emotion Regression.pdf
Adversarial Attention Modeling for Multi-dimensional Emotion Regression.pdf
Po-Chuan Chen
Β 
Polish students' mobility in the Czech Republic
Polish students' mobility in the Czech RepublicPolish students' mobility in the Czech Republic
Polish students' mobility in the Czech Republic
Anna Sz.
Β 
Embracing GenAI - A Strategic Imperative
Embracing GenAI - A Strategic ImperativeEmbracing GenAI - A Strategic Imperative
Embracing GenAI - A Strategic Imperative
Peter Windle
Β 
The Challenger.pdf DNHS Official Publication
The Challenger.pdf DNHS Official PublicationThe Challenger.pdf DNHS Official Publication
The Challenger.pdf DNHS Official Publication
Delapenabediema
Β 
The French Revolution Class 9 Study Material pdf free download
The French Revolution Class 9 Study Material pdf free downloadThe French Revolution Class 9 Study Material pdf free download
The French Revolution Class 9 Study Material pdf free download
Vivekanand Anglo Vedic Academy
Β 
Home assignment II on Spectroscopy 2024 Answers.pdf
Home assignment II on Spectroscopy 2024 Answers.pdfHome assignment II on Spectroscopy 2024 Answers.pdf
Home assignment II on Spectroscopy 2024 Answers.pdf
Tamralipta Mahavidyalaya
Β 
Biological Screening of Herbal Drugs in detailed.
Biological Screening of Herbal Drugs in detailed.Biological Screening of Herbal Drugs in detailed.
Biological Screening of Herbal Drugs in detailed.
Ashokrao Mane college of Pharmacy Peth-Vadgaon
Β 
Phrasal Verbs.XXXXXXXXXXXXXXXXXXXXXXXXXX
Phrasal Verbs.XXXXXXXXXXXXXXXXXXXXXXXXXXPhrasal Verbs.XXXXXXXXXXXXXXXXXXXXXXXXXX
Phrasal Verbs.XXXXXXXXXXXXXXXXXXXXXXXXXX
MIRIAMSALINAS13
Β 
Sha'Carri Richardson Presentation 202345
Sha'Carri Richardson Presentation 202345Sha'Carri Richardson Presentation 202345
Sha'Carri Richardson Presentation 202345
beazzy04
Β 
CLASS 11 CBSE B.St Project AIDS TO TRADE - INSURANCE
CLASS 11 CBSE B.St Project AIDS TO TRADE - INSURANCECLASS 11 CBSE B.St Project AIDS TO TRADE - INSURANCE
CLASS 11 CBSE B.St Project AIDS TO TRADE - INSURANCE
BhavyaRajput3
Β 

Recently uploaded (20)

Acetabularia Information For Class 9 .docx
Acetabularia Information For Class 9 .docxAcetabularia Information For Class 9 .docx
Acetabularia Information For Class 9 .docx
Β 
The basics of sentences session 5pptx.pptx
The basics of sentences session 5pptx.pptxThe basics of sentences session 5pptx.pptx
The basics of sentences session 5pptx.pptx
Β 
Chapter 3 - Islamic Banking Products and Services.pptx
Chapter 3 - Islamic Banking Products and Services.pptxChapter 3 - Islamic Banking Products and Services.pptx
Chapter 3 - Islamic Banking Products and Services.pptx
Β 
Thesis Statement for students diagnonsed withADHD.ppt
Thesis Statement for students diagnonsed withADHD.pptThesis Statement for students diagnonsed withADHD.ppt
Thesis Statement for students diagnonsed withADHD.ppt
Β 
Honest Reviews of Tim Han LMA Course Program.pptx
Honest Reviews of Tim Han LMA Course Program.pptxHonest Reviews of Tim Han LMA Course Program.pptx
Honest Reviews of Tim Han LMA Course Program.pptx
Β 
Unit 2- Research Aptitude (UGC NET Paper I).pdf
Unit 2- Research Aptitude (UGC NET Paper I).pdfUnit 2- Research Aptitude (UGC NET Paper I).pdf
Unit 2- Research Aptitude (UGC NET Paper I).pdf
Β 
How libraries can support authors with open access requirements for UKRI fund...
How libraries can support authors with open access requirements for UKRI fund...How libraries can support authors with open access requirements for UKRI fund...
How libraries can support authors with open access requirements for UKRI fund...
Β 
2024.06.01 Introducing a competency framework for languag learning materials ...
2024.06.01 Introducing a competency framework for languag learning materials ...2024.06.01 Introducing a competency framework for languag learning materials ...
2024.06.01 Introducing a competency framework for languag learning materials ...
Β 
Mule 4.6 & Java 17 Upgrade | MuleSoft Mysore Meetup #46
Mule 4.6 & Java 17 Upgrade | MuleSoft Mysore Meetup #46Mule 4.6 & Java 17 Upgrade | MuleSoft Mysore Meetup #46
Mule 4.6 & Java 17 Upgrade | MuleSoft Mysore Meetup #46
Β 
BΓ€I TαΊ¬P Bα»” TRα»’ TIαΊΎNG ANH GLOBAL SUCCESS LỚP 3 - CαΊ’ NΔ‚M (CΓ“ FILE NGHE VΓ€ ĐÁP Á...
BΓ€I TαΊ¬P Bα»” TRα»’ TIαΊΎNG ANH GLOBAL SUCCESS LỚP 3 - CαΊ’ NΔ‚M (CΓ“ FILE NGHE VΓ€ ĐÁP Á...BΓ€I TαΊ¬P Bα»” TRα»’ TIαΊΎNG ANH GLOBAL SUCCESS LỚP 3 - CαΊ’ NΔ‚M (CΓ“ FILE NGHE VΓ€ ĐÁP Á...
BΓ€I TαΊ¬P Bα»” TRα»’ TIαΊΎNG ANH GLOBAL SUCCESS LỚP 3 - CαΊ’ NΔ‚M (CΓ“ FILE NGHE VΓ€ ĐÁP Á...
Β 
Adversarial Attention Modeling for Multi-dimensional Emotion Regression.pdf
Adversarial Attention Modeling for Multi-dimensional Emotion Regression.pdfAdversarial Attention Modeling for Multi-dimensional Emotion Regression.pdf
Adversarial Attention Modeling for Multi-dimensional Emotion Regression.pdf
Β 
Polish students' mobility in the Czech Republic
Polish students' mobility in the Czech RepublicPolish students' mobility in the Czech Republic
Polish students' mobility in the Czech Republic
Β 
Embracing GenAI - A Strategic Imperative
Embracing GenAI - A Strategic ImperativeEmbracing GenAI - A Strategic Imperative
Embracing GenAI - A Strategic Imperative
Β 
The Challenger.pdf DNHS Official Publication
The Challenger.pdf DNHS Official PublicationThe Challenger.pdf DNHS Official Publication
The Challenger.pdf DNHS Official Publication
Β 
The French Revolution Class 9 Study Material pdf free download
The French Revolution Class 9 Study Material pdf free downloadThe French Revolution Class 9 Study Material pdf free download
The French Revolution Class 9 Study Material pdf free download
Β 
Home assignment II on Spectroscopy 2024 Answers.pdf
Home assignment II on Spectroscopy 2024 Answers.pdfHome assignment II on Spectroscopy 2024 Answers.pdf
Home assignment II on Spectroscopy 2024 Answers.pdf
Β 
Biological Screening of Herbal Drugs in detailed.
Biological Screening of Herbal Drugs in detailed.Biological Screening of Herbal Drugs in detailed.
Biological Screening of Herbal Drugs in detailed.
Β 
Phrasal Verbs.XXXXXXXXXXXXXXXXXXXXXXXXXX
Phrasal Verbs.XXXXXXXXXXXXXXXXXXXXXXXXXXPhrasal Verbs.XXXXXXXXXXXXXXXXXXXXXXXXXX
Phrasal Verbs.XXXXXXXXXXXXXXXXXXXXXXXXXX
Β 
Sha'Carri Richardson Presentation 202345
Sha'Carri Richardson Presentation 202345Sha'Carri Richardson Presentation 202345
Sha'Carri Richardson Presentation 202345
Β 
CLASS 11 CBSE B.St Project AIDS TO TRADE - INSURANCE
CLASS 11 CBSE B.St Project AIDS TO TRADE - INSURANCECLASS 11 CBSE B.St Project AIDS TO TRADE - INSURANCE
CLASS 11 CBSE B.St Project AIDS TO TRADE - INSURANCE
Β 

Geometry Theorems 1 REMC Tutoring

  • 1. Theorems Midpoint Theorem 𝐼𝑓 𝐡 𝑖𝑠 π‘‘β„Žπ‘’ π‘€π‘–π‘‘π‘π‘œπ‘–π‘›π‘‘ π‘œπ‘“ 𝐴𝐢 Μ…Μ…Μ…Μ…, π‘‘β„Žπ‘’π‘› 𝐴𝐡 Μ…Μ…Μ…Μ… β‰… 𝐡𝐢 Μ…Μ…Μ…Μ… Segment Congruence Theorems Reflexive Symmetric Transitive πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘π‘’ π‘“π‘œπ‘™π‘™π‘œπ‘€π‘  π‘‘β„Žπ‘’ π΄π‘™π‘”π‘’π‘π‘Ÿπ‘Žπ‘–π‘ π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘–π‘’π‘  π‘œπ‘“ 𝑅𝑒𝑓𝑙𝑒π‘₯𝑖𝑣𝑒, π‘†π‘¦π‘šπ‘šπ‘’π‘‘π‘Ÿπ‘–π‘, π‘Žπ‘›π‘‘ π‘‡π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘π‘–π‘£π‘’ 𝐴𝐡 Μ…Μ…Μ…Μ…Μ… β‰… 𝐴𝐡 Μ…Μ…Μ…Μ… 𝐼𝑓 𝐴𝐡 Μ…Μ…Μ…Μ… β‰… 𝐢𝐷 Μ…Μ…Μ…Μ…, π‘‘β„Žπ‘’π‘› 𝐢𝐷 Μ…Μ…Μ…Μ… β‰… 𝐴𝐡 Μ…Μ…Μ…Μ… 𝐼𝑓 𝐴𝐡 Μ…Μ…Μ…Μ… β‰… 𝐢𝐷 Μ…Μ…Μ…Μ… π‘Žπ‘›π‘‘ 𝐢𝐷 Μ…Μ…Μ…Μ… β‰… 𝐸𝐹 Μ…Μ…Μ…Μ…, π‘‘β„Žπ‘’π‘› 𝐴𝐡 Μ…Μ…Μ…Μ… β‰… 𝐸𝐹 Μ…Μ…Μ…Μ… 𝐴𝐡 Μ…Μ…Μ…Μ… β‰… 𝐢𝐷 Μ…Μ…Μ…Μ… π‘Žπ‘›π‘‘ 𝐢𝐷 Μ…Μ…Μ…Μ… β‰… 𝐸𝐹 Μ…Μ…Μ…Μ…, π‘‘β„Žπ‘’π‘Ÿπ‘’π‘“π‘œπ‘Ÿπ‘’ 𝐴𝐡 Μ…Μ…Μ…Μ… β‰… 𝐸𝐹 Μ…Μ…Μ…Μ… (π‘‘π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘π‘–π‘£π‘’ π‘π‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦ 𝑖𝑠 𝑒𝑠𝑒𝑑 π‘€β„Žπ‘’π‘› π‘œπ‘π‘—π‘’π‘π‘‘π‘  π‘Žπ‘Ÿπ‘’ π‘π‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ π‘‘π‘œ π‘Žπ‘›π‘œπ‘‘β„Žπ‘’π‘Ÿ) Supplement Theorem 𝐼𝑓 π‘Ž π‘™π‘–π‘›π‘’π‘Žπ‘Ÿ π‘π‘Žπ‘–π‘Ÿ 𝑖𝑠 π‘“π‘œπ‘Ÿπ‘šπ‘’π‘‘ 𝑏𝑦 π‘‘π‘€π‘œ π‘Žπ‘›π‘”π‘™π‘’π‘ , π‘‘β„Žπ‘’π‘› 𝑀𝑒 π‘˜π‘›π‘œπ‘€ π‘‘β„Žπ‘Žπ‘‘ π‘‘β„Žπ‘’ π‘‘π‘€π‘œ π‘Žπ‘›π‘”π‘™π‘’π‘  π‘Žπ‘Ÿπ‘’ π‘ π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦. ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ (π‘ π‘’π‘š π‘œπ‘“ π‘šβˆ 1 π‘Žπ‘›π‘‘ π‘šβˆ 2 π‘’π‘žπ‘’π‘Žπ‘™π‘  180Β°) Complement Theorem 𝐼𝑓 π‘‘β„Žπ‘’ π‘’π‘›π‘π‘œπ‘šπ‘šπ‘œπ‘› 𝑠𝑖𝑑𝑒𝑠 π‘“π‘Ÿπ‘œπ‘š π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘Žπ‘›π‘”π‘™π‘’π‘  π‘“π‘œπ‘Ÿπ‘š π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’, π‘‘β„Žπ‘’π‘› 𝑀𝑒 π‘˜π‘›π‘œπ‘€ π‘‘β„Žπ‘Žπ‘‘ π‘‘β„Žπ‘’ π‘‘π‘€π‘œ π‘Žπ‘›π‘”π‘™π‘’π‘  π‘Žπ‘Ÿπ‘’ π‘π‘œπ‘šπ‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦. ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ πΆπ‘œπ‘šπ‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ (π‘ π‘’π‘š π‘œπ‘“ π‘šβˆ 1 π‘Žπ‘›π‘‘ π‘šβˆ 2 π‘’π‘žπ‘’π‘Žπ‘™π‘  90Β°) (π‘π‘’π‘Ÿπ‘π‘™π‘’ π‘π‘œπ‘₯ π‘ π‘¦π‘šπ‘π‘œπ‘™ π‘šπ‘’π‘Žπ‘›π‘  π‘π‘™π‘Žπ‘π‘˜ 𝑙𝑖𝑛𝑒𝑠 π‘Žπ‘Ÿπ‘’ π‘Ž π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒 π‘œπ‘Ÿ 90Β°)
  • 2. Congruent Supplement Theorem (3 angles) Congruent Supplement Theorem (4 angles) 𝐼𝑓 π‘‘π‘€π‘œ π‘Žπ‘›π‘”π‘™π‘’π‘  π‘Žπ‘Ÿπ‘’ π‘ π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ π‘Žπ‘›π‘‘ π‘Žπ‘›π‘”π‘™π‘’ π‘œπ‘›π‘’ 𝑖𝑠 π‘ π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ π‘€π‘–π‘‘β„Ž π‘Ž π‘‘β„Žπ‘–π‘Ÿπ‘‘ π‘Žπ‘›π‘”π‘™π‘’, π‘‘β„Žπ‘’π‘› π‘‘β„Žπ‘’ π‘‘β„Žπ‘–π‘Ÿπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ 𝑖𝑠 π‘π‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ π‘‘π‘œ π‘‘β„Žπ‘’ π‘ π‘’π‘π‘œπ‘›π‘‘ π‘Žπ‘›π‘”π‘™π‘’. 𝐼𝑓 π‘‘π‘€π‘œ 𝑠𝑒𝑑𝑠 π‘œπ‘“ π‘Žπ‘›π‘”π‘™π‘’π‘  π‘π‘Ÿπ‘’π‘Žπ‘‘π‘’ π‘‘π‘€π‘œ π‘ π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ π‘π‘Žπ‘–π‘Ÿπ‘ , π‘Žπ‘›π‘‘ π‘œπ‘›π‘’ π‘Žπ‘›π‘”π‘™π‘’ π‘“π‘Ÿπ‘œπ‘š π‘’π‘Žπ‘β„Ž 𝑠𝑒𝑑 𝑖𝑠 π‘π‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ π‘‘π‘œ π‘Žπ‘› π‘Žπ‘›π‘”π‘™π‘’ π‘œπ‘“ π‘‘β„Žπ‘’ π‘œπ‘‘β„Žπ‘’π‘Ÿ 𝑠𝑒𝑑, π‘‘β„Žπ‘’π‘› π‘‘β„Žπ‘’ π‘Žπ‘›π‘”π‘™π‘’π‘  π‘›π‘œπ‘‘ π‘™π‘Žπ‘π‘’π‘™π‘’π‘‘ π‘Žπ‘  π‘π‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ π‘€π‘œπ‘’π‘™π‘‘ 𝑏𝑒 π‘π‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ π‘Žπ‘™π‘ π‘œ. ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ ∠1 π‘Žπ‘›π‘‘ ∠3 π‘Žπ‘Ÿπ‘’ π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ π‘‡β„Žπ‘’π‘Ÿπ‘’π‘“π‘œπ‘Ÿπ‘’ ∠2 β‰… ∠3 ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ ∠3 π‘Žπ‘›π‘‘ ∠4 π‘Žπ‘Ÿπ‘’ π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ ∠2 β‰… ∠3 π‘‡β„Žπ‘’π‘Ÿπ‘’π‘“π‘œπ‘Ÿπ‘’, ∠1 β‰… ∠4 Congruent Complement Theorem 𝐼𝑓 π‘‘π‘€π‘œ π‘Žπ‘›π‘”π‘™π‘’π‘  π‘Žπ‘Ÿπ‘’ π‘π‘œπ‘šπ‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ π‘Žπ‘›π‘‘ π‘Žπ‘›π‘”π‘™π‘’ π‘œπ‘›π‘’ 𝑖𝑠 π‘π‘œπ‘šπ‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ π‘€π‘–π‘‘β„Ž π‘Ž π‘‘β„Žπ‘–π‘Ÿπ‘‘ π‘Žπ‘›π‘”π‘™π‘’, π‘‘β„Žπ‘’π‘› π‘‘β„Žπ‘’ π‘‘β„Žπ‘–π‘Ÿπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ 𝑖𝑠 π‘π‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ π‘‘π‘œ π‘‘β„Žπ‘’ π‘ π‘’π‘π‘œπ‘›π‘‘ π‘Žπ‘›π‘”π‘™π‘’. ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ πΆπ‘œπ‘šπ‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ ∠2 π‘Žπ‘›π‘‘ ∠3 π‘Žπ‘Ÿπ‘’ πΆπ‘œπ‘šπ‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ π‘‡β„Žπ‘’π‘Ÿπ‘’π‘“π‘œπ‘Ÿπ‘’, ∠1 β‰… ∠3 Vertical Angles Theorem 𝐼𝑓 π‘‘π‘€π‘œ π‘Žπ‘›π‘”π‘™π‘’π‘  π‘Žπ‘Ÿπ‘’ π‘£π‘’π‘Ÿπ‘‘π‘–π‘π‘Žπ‘™ π‘Žπ‘›π‘”π‘™π‘’π‘ , π‘‘β„Žπ‘’π‘› 𝑀𝑒 π‘˜π‘›π‘œπ‘€ π‘‘β„Žπ‘Žπ‘‘ π‘‘β„Žπ‘’π‘¦ π‘Žπ‘Ÿπ‘’ π‘π‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘. ∠1 β‰… ∠3 ∠2 β‰… ∠4
  • 3. Right Angle Theorems Perpendicular Lines Intersect to Form Four Right Angles All Right Angles are Congruent Perpendicular Lines Will Form Four Congruent Adjacent Angles If Two Angles are Both Congruent and Supplementary, Then Each Angle Will be a Right Angle If Two Angles of a Linear Pair are Congruent, Then They Are Right Angles 𝐼𝑓 π‘‘π‘€π‘œ 𝑙𝑖𝑛𝑒𝑠 π‘Žπ‘Ÿπ‘’ π‘π‘’π‘Ÿπ‘π‘’π‘›π‘‘π‘–π‘π‘’π‘™π‘Žπ‘Ÿ, π‘‘β„Žπ‘’π‘› π‘“π‘œπ‘’π‘Ÿ π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’π‘  𝑀𝑖𝑙𝑙 𝑏𝑒 π‘™π‘œπ‘π‘Žπ‘‘π‘’π‘‘ π‘Žπ‘‘ π‘‘β„Žπ‘’ π‘–π‘›π‘‘π‘’π‘Ÿπ‘ π‘’π‘π‘‘π‘–π‘œπ‘› π‘π‘œπ‘–π‘›π‘‘. 𝐼𝑓 π‘‘π‘€π‘œ π‘œπ‘Ÿ π‘šπ‘œπ‘Ÿπ‘’ π‘Žπ‘›π‘”π‘™π‘’π‘  π‘Žπ‘Ÿπ‘’ π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’π‘ , π‘‘β„Žπ‘’π‘› π‘Žπ‘™π‘™ π‘œπ‘“ π‘‘β„Žπ‘’ π‘Žπ‘›π‘”π‘™π‘’π‘  π‘Žπ‘Ÿπ‘’ π‘π‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘. 𝐼𝑓 π‘‘π‘€π‘œ 𝑙𝑖𝑛𝑒𝑠 π‘–π‘›π‘‘π‘’π‘Ÿπ‘ π‘’π‘π‘‘ π‘‘π‘œ π‘Žπ‘‘ π‘Ž 90Β° π‘Žπ‘›π‘”π‘™π‘’, π‘‘β„Žπ‘’π‘› π‘π‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘Žπ‘›π‘”π‘™π‘’π‘  𝑀𝑖𝑙𝑙 𝑏𝑒 π‘“π‘œπ‘Ÿπ‘šπ‘’π‘‘. 𝐼𝑓 π‘‘π‘€π‘œ π‘Žπ‘›π‘”π‘™π‘’π‘  π‘Žπ‘Ÿπ‘’ π‘π‘œπ‘‘β„Ž π‘π‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ 𝑖𝑛 π‘šπ‘’π‘Žπ‘ π‘’π‘Ÿπ‘’ π‘Žπ‘›π‘‘ π‘‘β„Žπ‘’ π‘ π‘’π‘š π‘œπ‘“ π‘‘β„Žπ‘’ π‘Žπ‘›π‘”π‘™π‘’π‘  𝑖𝑠 180Β°, π‘‘β„Žπ‘’π‘› 𝑀𝑒 π‘˜π‘›π‘œπ‘€ π‘‘β„Žπ‘Žπ‘‘ π‘’π‘Žπ‘β„Ž π‘Žπ‘›π‘”π‘™π‘’ 𝑖𝑠 90Β° 𝐼𝑓 π‘Ž π‘™π‘–π‘›π‘’π‘Žπ‘Ÿ π‘π‘Žπ‘–π‘Ÿ 𝑖𝑠 π‘“π‘œπ‘Ÿπ‘šπ‘’π‘‘ π‘“π‘Ÿπ‘œπ‘š π‘‘π‘€π‘œ π‘π‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ π‘Žπ‘›π‘”π‘™π‘’π‘ , π‘‘β„Žπ‘’π‘› 𝑀𝑒 π‘˜π‘›π‘œπ‘€ π‘‘β„Žπ‘Žπ‘‘ π‘π‘œπ‘‘β„Ž π‘Žπ‘›π‘”π‘™π‘’π‘  π‘Žπ‘Ÿπ‘’ 90Β° πΉπ‘œπ‘’π‘Ÿ π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠 π‘Žπ‘Ÿπ‘’ π‘π‘Ÿπ‘’π‘Žπ‘‘π‘’π‘‘ π‘Žπ‘‘ π‘‘β„Žπ‘’ πΌπ‘›π‘‘π‘’π‘Ÿπ‘ π‘’π‘π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘‘β„Žπ‘’ π‘ƒπ‘’π‘Ÿπ‘π‘’π‘›π‘‘π‘–π‘π‘’π‘™π‘Žπ‘Ÿ 𝐿𝑖𝑛𝑒𝑠 𝐴𝑙𝑙 π‘‘β„Žπ‘’ π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠 π‘’π‘žπ‘’π‘Žπ‘™ 90Β° ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π΄π‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ ∠2 π‘Žπ‘›π‘‘ ∠3 π‘Žπ‘Ÿπ‘’ π΄π‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ ∠3 π‘Žπ‘›π‘‘ ∠4 π‘Žπ‘Ÿπ‘’ π΄π‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ ∠4 π‘Žπ‘›π‘‘ ∠1 π‘Žπ‘Ÿπ‘’ π΄π‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ 𝐴𝑙𝑙 π‘‘β„Žπ‘’ π‘Žπ‘›π‘”π‘™π‘’π‘  π‘œπ‘“ π‘‘β„Žπ‘’ π΄π‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ 𝐴𝑛𝑔𝑙𝑒 π‘ƒπ‘Žπ‘–π‘Ÿπ‘  π‘Žπ‘Ÿπ‘’ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ π‘Žπ‘›π‘‘ π‘Žπ‘™π‘ π‘œ π‘‘β„Žπ‘’ π‘π‘Žπ‘–π‘Ÿπ‘  π‘Žπ‘Ÿπ‘’ π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦, π‘€β„Žπ‘–π‘β„Ž π‘šπ‘’π‘Žπ‘›π‘  π‘’π‘Žπ‘β„Ž π‘Žπ‘›π‘”π‘™π‘’ 𝑖𝑠 π‘Ž π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒. 𝐴𝑙𝑙 π‘‘β„Žπ‘’ π‘Žπ‘›π‘”π‘™π‘’π‘  π‘œπ‘“ π‘‘β„Žπ‘’ π΄π‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ 𝐴𝑛𝑔𝑙𝑒 π‘ƒπ‘Žπ‘–π‘Ÿπ‘  π‘Žπ‘Ÿπ‘’ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ π‘Žπ‘›π‘‘ π‘Žπ‘™π‘ π‘œ π‘π‘Ÿπ‘’π‘Žπ‘‘π‘’ πΏπ‘–π‘›π‘’π‘Žπ‘Ÿ π‘ƒπ‘Žπ‘–π‘Ÿπ‘  π‘€β„Žπ‘–π‘β„Ž π‘šπ‘’π‘Žπ‘›π‘  π‘’π‘Žπ‘β„Ž π‘Žπ‘›π‘”π‘™π‘’ 𝑖𝑠 π‘Ž π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒. Theorem Proofs Midpoint Theorem: π‘ƒπ‘Ÿπ‘œπ‘£π‘’ π‘‘β„Žπ‘Žπ‘‘: 𝐴𝐡 Μ…Μ…Μ…Μ… β‰… 𝐡𝐢 Μ…Μ…Μ…Μ… 𝐡 𝑖𝑠 π‘€π‘–π‘‘π‘π‘œπ‘–π‘›π‘‘ π‘œπ‘“ 𝐴𝐢 Μ…Μ…Μ…Μ… 𝑔𝑖𝑣𝑒𝑛 𝐴𝐡 = 𝐡𝐢 π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘€π‘–π‘‘π‘π‘œπ‘–π‘›π‘‘ π‘œπ‘“ π‘Ž π‘†π‘’π‘”π‘šπ‘’π‘›π‘‘ 𝐴𝐡 Μ…Μ…Μ…Μ… β‰… 𝐡𝐢 Μ…Μ…Μ…Μ… π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘†π‘’π‘”π‘šπ‘’π‘›π‘‘ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘π‘’
  • 4. Segment Congruence Theorems: Reflexive Property: π‘ƒπ‘Ÿπ‘œπ‘£π‘’ π‘‘β„Žπ‘Žπ‘‘: 𝐴𝐡 Μ…Μ…Μ…Μ…Μ… β‰… 𝐴𝐡 Μ…Μ…Μ…Μ… 𝐴𝐡 Μ…Μ…Μ…Μ… β‰… 𝐴𝐡 Μ…Μ…Μ…Μ… 𝑔𝑖𝑣𝑒𝑛 𝐴𝐡 = 𝐴𝐡 π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘†π‘’π‘”π‘šπ‘’π‘›π‘‘ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘π‘’ 𝐴𝐡 = 𝐴𝐡 𝑅𝑒𝑓𝑙𝑒π‘₯𝑖𝑣𝑒 𝐴𝐡 Μ…Μ…Μ…Μ… β‰… 𝐴𝐡 Μ…Μ…Μ…Μ… π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘†π‘’π‘”π‘šπ‘’π‘›π‘‘ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘π‘’ Symmetric Property: π‘ƒπ‘Ÿπ‘œπ‘£π‘’ π‘‘β„Žπ‘Žπ‘‘: 𝐢𝐷 Μ…Μ…Μ…Μ… β‰… 𝐴𝐡 Μ…Μ…Μ…Μ… 𝐴𝐡 Μ…Μ…Μ…Μ… β‰… 𝐢𝐷 Μ…Μ…Μ…Μ… 𝑔𝑖𝑣𝑒𝑛 𝐴𝐡 = 𝐢𝐷 π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘†π‘’π‘”π‘šπ‘’π‘›π‘‘ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘π‘’ 𝐢𝐷 = 𝐴𝐡 π‘†π‘¦π‘šπ‘šπ‘’π‘‘π‘Ÿπ‘–π‘ π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦ 𝐢𝐷 Μ…Μ…Μ…Μ… β‰… 𝐴𝐡 Μ…Μ…Μ…Μ… π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘†π‘’π‘”π‘šπ‘’π‘›π‘‘ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘π‘’ Transitive Property: π‘ƒπ‘Ÿπ‘œπ‘£π‘’ π‘‘β„Žπ‘Žπ‘‘: 𝐴𝐡 Μ…Μ…Μ…Μ… β‰… 𝐸𝐹 Μ…Μ…Μ…Μ… 𝐴𝐡 Μ…Μ…Μ…Μ… β‰… 𝐢𝐷 Μ…Μ…Μ…Μ… π‘Žπ‘›π‘‘ 𝐢𝐷 Μ…Μ…Μ…Μ… β‰… 𝐸𝐹 Μ…Μ…Μ…Μ… 𝑔𝑖𝑣𝑒𝑛 𝐴𝐡 = 𝐢𝐷 π‘Žπ‘›π‘‘ 𝐢𝐷 = 𝐸𝐹 π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘†π‘’π‘”π‘šπ‘’π‘›π‘‘ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘π‘’ 𝐴𝐡 = 𝐸𝐹 𝐴𝐡 = 𝐸𝐹 π‘‡π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘π‘–π‘£π‘’ π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦ π‘œπ‘Ÿ, π‘†π‘’π‘π‘ π‘‘π‘–π‘‘π‘’π‘‘π‘–π‘œπ‘› π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦ (𝑠𝑒𝑏 𝐴𝐡 π‘“π‘œπ‘Ÿ 𝐢𝐷) 𝐴𝐡 Μ…Μ…Μ…Μ… β‰… 𝐸𝐹 Μ…Μ…Μ…Μ… π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘†π‘’π‘”π‘šπ‘’π‘›π‘‘ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘π‘’ Supplement Theorem: π‘ƒπ‘Ÿπ‘œπ‘£π‘’ π‘‘β„Žπ‘Žπ‘‘ ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ ∠1 π‘Žπ‘›π‘‘ ∠2 π‘“π‘œπ‘Ÿπ‘š π‘Ž π‘™π‘–π‘›π‘’π‘Žπ‘Ÿ π‘π‘Žπ‘–π‘Ÿ 𝑔𝑖𝑣𝑒𝑛 π‘π‘œπ‘‘π‘’: π»π‘œπ‘€ π‘‘π‘œ π‘‘π‘’π‘‘π‘’π‘Ÿπ‘šπ‘–π‘›π‘’ π‘Ž πΏπ‘–π‘›π‘’π‘Žπ‘Ÿ π‘ƒπ‘Žπ‘–π‘Ÿ? π‘‡π‘€π‘œ 𝐴𝑛𝑔𝑙𝑒𝑠 π‘‘β„Žπ‘Žπ‘‘: π΄π‘Ÿπ‘’ π΄π‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠 π‘ˆπ‘›π‘ β„Žπ‘Žπ‘Ÿπ‘’π‘‘ 𝑠𝑖𝑑𝑒𝑠 π‘“π‘œπ‘Ÿπ‘š π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ π‘Ÿπ‘Žπ‘¦π‘  (180Β°) π‘šβˆ 1 + π‘šβˆ 2 = 180Β° π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ πΏπ‘–π‘›π‘’π‘Žπ‘Ÿ π‘ƒπ‘Žπ‘–π‘Ÿ ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ 𝐴𝑛𝑔𝑙𝑒𝑠 Complement Theorem: π‘ƒπ‘Ÿπ‘œπ‘£π‘’ π‘‘β„Žπ‘Žπ‘‘ ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ πΆπ‘œπ‘šπ‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘Žπ‘›π‘”π‘™π‘’π‘  ∠1 π‘Žπ‘›π‘‘ ∠2 π‘“π‘œπ‘Ÿπ‘š π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ 𝑔𝑖𝑣𝑒𝑛 π‘π‘œπ‘‘π‘’: π»π‘œπ‘€ π‘‘π‘œ π‘‘π‘’π‘‘π‘’π‘Ÿπ‘šπ‘–π‘›π‘’ π΄π‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠? π‘ƒπ‘Žπ‘–π‘Ÿ π‘œπ‘“ 𝐴𝑛𝑔𝑙𝑒𝑠 π‘‘β„Žπ‘Žπ‘‘ π‘†β„Žπ‘Žπ‘Ÿπ‘’ π‘Ž π‘‰π‘’π‘Ÿπ‘‘π‘’π‘₯, π‘†β„Žπ‘Žπ‘Ÿπ‘’ π‘Ž 𝑆𝑖𝑑𝑒, π‘Žπ‘›π‘‘ π‘Žπ‘Ÿπ‘’ π‘œπ‘› π‘‘β„Žπ‘’ π‘†π‘Žπ‘šπ‘’ π‘ƒπ‘™π‘Žπ‘›π‘’ π»π‘œπ‘€ π‘‘π‘œ π‘‘π‘’π‘‘π‘’π‘Ÿπ‘šπ‘–π‘›π‘’ π‘Ž π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒? π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒 = 90Β° π‘Žπ‘›π‘”π‘™π‘’ π‘šβˆ 1 + π‘šβˆ 2 = 90Β° π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒 ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘π‘œπ‘šπ‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ πΆπ‘œπ‘šπ‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ 𝐴𝑛𝑔𝑙𝑒𝑠 Alternative Complement Theorem: ∠1 π‘Žπ‘›π‘‘ ∠2 π‘π‘Ÿπ‘’π‘Žπ‘‘π‘’ ∠𝐴𝐡𝐢, π‘€β„Žπ‘–π‘β„Ž 𝑖𝑠 π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ π‘ƒπ‘Ÿπ‘œπ‘£π‘’ π‘‘β„Žπ‘Žπ‘‘ ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ πΆπ‘œπ‘šπ‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦
  • 5. ∠𝐴𝐡𝐢 𝑖𝑠 π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ ∠1 π‘Žπ‘›π‘‘ ∠2 π‘“π‘œπ‘Ÿπ‘š ∠𝐴𝐡𝐢 𝑔𝑖𝑣𝑒𝑛 π‘šβˆ π΄π΅πΆ = 90Β° π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ π‘šβˆ 1 + π‘šβˆ 2 = π‘šβˆ π΄π΅πΆ 𝐴𝑛𝑔𝑙𝑒 π΄π‘‘π‘‘π‘–π‘‘π‘–π‘œπ‘› π‘ƒπ‘œπ‘ π‘‘π‘’π‘™π‘Žπ‘‘π‘’ π‘šβˆ 1 + π‘šβˆ 2 = 90Β° π‘†π‘’π‘π‘ π‘‘π‘–π‘‘π‘’π‘‘π‘–π‘œπ‘› π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦ 𝑠𝑒𝑏 90Β° 𝑖𝑛 π‘“π‘œπ‘Ÿ ∠𝐴𝐡𝐢 ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘π‘œπ‘šπ‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ πΆπ‘œπ‘šπ‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ 𝐴𝑛𝑔𝑙𝑒𝑠 Congruent Supplement Theorems: Congruent Supplement Theorem (3 angles): π‘ƒπ‘Ÿπ‘œπ‘£π‘’ ∠2 β‰… ∠3 ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘ π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ ∠1 π‘Žπ‘›π‘‘ ∠3 π‘Žπ‘Ÿπ‘’ π‘ π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ 𝑔𝑖𝑣𝑒𝑛 π‘šβˆ 1 + π‘šβˆ 2 = 180Β° π‘šβˆ 1 + π‘šβˆ 3 = 180Β° π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ 𝐴𝑛𝑔𝑙𝑒𝑠 180Β° = π‘šβˆ 1 + π‘šβˆ 3 π‘†π‘¦π‘šπ‘šπ‘’π‘‘π‘Ÿπ‘–π‘ π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦ π‘šβˆ 1 + π‘šβˆ 2 = π‘šβˆ 1 + π‘šβˆ 3 π‘‡π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘π‘–π‘£π‘’ π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦: 𝐼𝑓 𝑁 = 𝑃 π‘Žπ‘›π‘‘ 𝑃 = 𝑀 π‘‘β„Žπ‘’π‘›, 𝑁 = 𝑀 π‘šβˆ 1 + π‘šβˆ 2 = 180Β° = π‘šβˆ 1 + π‘šβˆ 3 𝑁 = 𝑃 = 𝑀 π‘šβˆ 2 = π‘šβˆ 3 𝑆𝑑𝑒𝑝𝑠: π‘šβˆ 1 + π‘šβˆ 2 = π‘šβˆ 1 + π‘šβˆ 3 π‘šβˆ 1 + π‘šβˆ 2 βˆ’ π‘šβˆ 1 = π‘šβˆ 1 + π‘šβˆ 3 βˆ’ π‘šβˆ 1 π‘†π‘’π‘π‘‘π‘Ÿπ‘Žπ‘π‘‘π‘–π‘œπ‘› π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦: 𝑃 = π‘šβˆ 1 (𝑃 𝑖𝑠 π‘Ÿπ‘’π‘‘π‘’π‘“π‘–π‘›π‘’π‘‘ π‘‘π‘œ π‘’π‘žπ‘’π‘Žπ‘™ π‘šβˆ 1) 𝑁 = 𝑀 𝑁 βˆ’ 𝑃 = 𝑀 βˆ’ 𝑃 ∠2 β‰… ∠3 π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠 Congruent Supplement Theorem (4 angles): π‘ƒπ‘Ÿπ‘œπ‘£π‘’ π‘‘β„Žπ‘Žπ‘‘ ∠2 β‰… ∠3 ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘ π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ π‘Žπ‘›π‘”π‘™π‘’π‘  ∠3 π‘Žπ‘›π‘‘ ∠4 π‘Žπ‘Ÿπ‘’ π‘ π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ π‘Žπ‘›π‘”π‘™π‘’π‘  ∠1 β‰… ∠4 𝑔𝑖𝑣𝑒𝑛 π‘šβˆ 1 + π‘šβˆ 2 = 180Β° π‘šβˆ 3 + π‘šβˆ 4 = 180Β° π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ 𝐴𝑛𝑔𝑙𝑒𝑠 180Β° = π‘šβˆ 3 + π‘šβˆ 4 π‘†π‘¦π‘šπ‘šπ‘’π‘‘π‘Ÿπ‘–π‘ π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦ π‘šβˆ 1 + π‘šβˆ 2 = π‘šβˆ 3 + π‘šβˆ 4 π‘‡π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘π‘–π‘£π‘’ π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦: 𝐼𝑓 𝑁 = 𝑃 π‘Žπ‘›π‘‘ 𝑃 = 𝑀, π‘‘β„Žπ‘’π‘› 𝑁 = 𝑀 π‘šβˆ 1 + π‘šβˆ 2 = 180Β° = π‘šβˆ 3 + π‘šβˆ 4 𝑁 = 𝑃 = 𝑀 ∠1 β‰… ∠4 𝑔𝑖𝑣𝑒𝑛 π‘šβˆ 1 = π‘šβˆ 4 π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠 𝑅𝑒𝑓𝑙𝑒π‘₯𝑖𝑣𝑒 π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦: πΉπ‘œπ‘Ÿ π‘Žπ‘›π‘¦ π‘π‘’π‘šπ‘π‘’π‘Ÿ 𝑃, 𝑃 = 𝑃 𝑃 𝑖𝑠 π‘Ÿπ‘’π‘‘π‘’π‘“π‘–π‘›π‘’π‘‘ π‘‘π‘œ π‘’π‘žπ‘’π‘Žπ‘™ π‘‘β„Žπ‘’ π‘šπ‘’π‘Žπ‘ π‘’π‘Ÿπ‘’ π‘œπ‘“ π‘šβˆ 1 π‘Žπ‘›π‘‘ π‘šβˆ 4 π‘šβˆ 2 = π‘šβˆ 3 𝑆𝑑𝑒𝑝𝑠: π‘šβˆ 1 + π‘šβˆ 2 = π‘šβˆ 3 + π‘šβˆ 4 π‘šβˆ 1 + π‘šβˆ 2 βˆ’ π‘šβˆ 1 = π‘šβˆ 3 + π‘šβˆ 4 βˆ’ π‘šβˆ 4 π‘†π‘’π‘π‘‘π‘Ÿπ‘Žπ‘π‘‘π‘–π‘œπ‘› π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦: 𝑁 = 𝑀 𝑁 βˆ’ 𝑃 = 𝑀 βˆ’ 𝑃
  • 6. ∠2 β‰… ∠3 π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠 Congruent Complementary Theorem: π‘ƒπ‘Ÿπ‘œπ‘£π‘’ ∠2 β‰… ∠3 ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘π‘œπ‘šπ‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ ∠1 π‘Žπ‘›π‘‘ ∠3 π‘Žπ‘Ÿπ‘’ π‘π‘œπ‘šπ‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ 𝑔𝑖𝑣𝑒𝑛 π‘šβˆ 1 + π‘šβˆ 2 = 90Β° π‘šβˆ 1 + π‘šβˆ 3 = 90Β° π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ πΆπ‘œπ‘šπ‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ 𝐴𝑛𝑔𝑙𝑒𝑠 90Β° = π‘šβˆ 1 + π‘šβˆ 3 π‘†π‘¦π‘šπ‘šπ‘’π‘‘π‘Ÿπ‘–π‘ π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦ π‘šβˆ 1 + π‘šβˆ 2 = π‘šβˆ 1 + π‘šβˆ 3 π‘‡π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘π‘–π‘£π‘’ π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦: 𝐼𝑓 𝑁 = 𝑃 π‘Žπ‘›π‘‘ 𝑃 = 𝑀 π‘‘β„Žπ‘’π‘›, 𝑁 = 𝑀 π‘šβˆ 1 + π‘šβˆ 2 = 90Β° = π‘šβˆ 1 + π‘šβˆ 3 𝑁 = 𝑃 = 𝑀 π‘šβˆ 2 = π‘šβˆ 3 𝑆𝑑𝑒𝑝𝑠: π‘šβˆ 1 + π‘šβˆ 2 = π‘šβˆ 1 + π‘šβˆ 3 π‘šβˆ 1 + π‘šβˆ 2 βˆ’ π‘šβˆ 1 = π‘šβˆ 1 + π‘šβˆ 3 βˆ’ π‘šβˆ 1 π‘†π‘’π‘π‘‘π‘Ÿπ‘Žπ‘π‘‘π‘–π‘œπ‘› π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦: 𝑃 = π‘šβˆ 1 (𝑃 𝑖𝑠 π‘Ÿπ‘’π‘‘π‘’π‘“π‘–π‘›π‘’π‘‘ π‘‘π‘œ π‘’π‘žπ‘’π‘Žπ‘™ π‘šβˆ 1) 𝑁 = 𝑀 𝑁 βˆ’ 𝑃 = 𝑀 βˆ’ 𝑃 ∠2 β‰… ∠3 π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠 Vertical Angles Theorem: π‘ƒπ‘Ÿπ‘œπ‘£π‘’ ∠1 β‰… ∠3 ∠1 π‘Žπ‘›π‘‘ ∠3 π‘Žπ‘Ÿπ‘’ π‘£π‘’π‘Ÿπ‘‘π‘–π‘π‘Žπ‘™ π‘Žπ‘›π‘”π‘™π‘’π‘  ∠2 π‘Žπ‘›π‘‘ ∠3 π‘Žπ‘Ÿπ‘’ π‘£π‘’π‘Ÿπ‘‘π‘–π‘π‘Žπ‘™ π‘Žπ‘›π‘”π‘™π‘’π‘  𝑔𝑖𝑣𝑒𝑛 π‘π‘œπ‘‘π‘’: π»π‘œπ‘€ π‘‘π‘œ π‘‘π‘’π‘‘π‘’π‘Ÿπ‘šπ‘–π‘›π‘’ π‘‰π‘’π‘Ÿπ‘‘π‘–π‘π‘Žπ‘™ 𝐴𝑛𝑔𝑙𝑒𝑠? π‘ƒπ‘Žπ‘–π‘Ÿ π‘œπ‘“ 𝐴𝑛𝑔𝑙𝑒𝑠 π‘‘β„Žπ‘Žπ‘‘ π‘†β„Žπ‘Žπ‘Ÿπ‘’ π‘Ž π‘‰π‘’π‘Ÿπ‘‘π‘’π‘₯, π‘Žπ‘Ÿπ‘’ π‘π‘œπ‘‘ π΄π‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠, π‘Žπ‘›π‘‘ π‘Žπ‘Ÿπ‘’ π‘“π‘œπ‘Ÿπ‘šπ‘’π‘‘ π‘“π‘Ÿπ‘œπ‘š πΌπ‘›π‘‘π‘’π‘Ÿπ‘ π‘’π‘π‘‘π‘–π‘›π‘” 𝐿𝑖𝑛𝑒𝑠 ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘ π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ ∠2 π‘Žπ‘›π‘‘ ∠3 π‘Žπ‘Ÿπ‘’ π‘ π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ πΏπ‘–π‘›π‘’π‘Žπ‘Ÿ π‘ƒπ‘Žπ‘–π‘Ÿ π‘šβˆ 1 + π‘šβˆ 2 = 180Β° π‘šβˆ 2 + π‘šβˆ 3 = 180Β° π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ 𝐴𝑛𝑔𝑙𝑒𝑠 180Β° = π‘šβˆ 2 + ∠3 π‘†π‘¦π‘šπ‘šπ‘’π‘‘π‘Ÿπ‘–π‘ π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦ π‘šβˆ 1 + π‘šβˆ 2 = π‘šβˆ 2 + π‘šβˆ 3 π‘‡π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘π‘–π‘£π‘’ π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘¦ 𝐼𝑓 𝑁 = 𝑃 π‘Žπ‘›π‘‘ 𝑃 = 𝑀 π‘‘β„Žπ‘’π‘›, 𝑁 = 𝑀 π‘šβˆ 1 + π‘šβˆ 2 = 180Β° = π‘šβˆ 2 + π‘šβˆ 3 𝑁 = 𝑃 = 𝑀 π‘šβˆ 1 = π‘šβˆ 3 𝑆𝑑𝑒𝑝𝑠: π‘šβˆ 1 + π‘šβˆ 2 = π‘šβˆ 2 + π‘šβˆ 3 π‘šβˆ 1 + π‘šβˆ 2 βˆ’ π‘šβˆ 2 = π‘šβˆ 2 + π‘šβˆ 3 βˆ’ π‘šβˆ 2 π‘†π‘’π‘π‘‘π‘Ÿπ‘Žπ‘π‘‘π‘–π‘œπ‘› π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦: 𝑃 = π‘šβˆ 2 (𝑃 𝑖𝑠 π‘Ÿπ‘’π‘‘π‘’π‘“π‘–π‘›π‘’π‘‘ π‘‘π‘œ π‘’π‘žπ‘’π‘Žπ‘™ π‘šβˆ 2) 𝑁 = 𝑀 𝑁 βˆ’ 𝑃 = 𝑀 βˆ’ 𝑃 ∠1 β‰… ∠3 π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠
  • 7. Perpendicular Lines Theorems: Perpendicular Lines Intersect to Form Four Right Angles Theorem π‘ƒπ‘Ÿπ‘œπ‘£π‘’ π‘‘β„Žπ‘Žπ‘‘ ∠1, ∠2, ∠3, π‘Žπ‘›π‘‘ ∠4 π‘Žπ‘Ÿπ‘’ π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠 𝐿𝑖𝑛𝑒𝑠 𝑑 π‘Žπ‘›π‘‘ 𝑠 π‘Žπ‘Ÿπ‘’ π‘π‘’π‘Ÿπ‘π‘’π‘›π‘‘π‘–π‘π‘’π‘™π‘Žπ‘Ÿ (𝑑 βŠ₯ 𝑠) 𝑔𝑖𝑣𝑒𝑛 ∠1 𝑖𝑠 π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘ƒπ‘’π‘Ÿπ‘π‘’π‘›π‘‘π‘–π‘π‘’π‘™π‘Žπ‘Ÿ 𝐿𝑖𝑛𝑒𝑠 π‘šβˆ 1 = 90Β° π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒 π‘šβˆ 1 + π‘šβˆ 2 = 180Β° π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘ π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘š π‘π‘œπ‘‘π‘’: 𝑁𝑒𝑒𝑑 π‘‘π‘œ 𝐼𝑑𝑒𝑛𝑑𝑖𝑓𝑦 ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘  πΏπ‘–π‘›π‘’π‘Žπ‘Ÿ π‘ƒπ‘Žπ‘–π‘Ÿ 90Β° + π‘šβˆ 2 = 180Β° π‘†π‘’π‘π‘ π‘‘π‘–π‘‘π‘’π‘‘π‘–π‘œπ‘› (𝑠𝑒𝑏 90Β° π‘“π‘œπ‘Ÿ π‘šβˆ 1) π‘šβˆ 2 = 90Β° π‘†π‘’π‘π‘‘π‘Ÿπ‘Žπ‘π‘‘π‘–π‘œπ‘› π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦ (π‘†π‘’π‘π‘ π‘‘π‘Ÿπ‘Žπ‘π‘‘ 90Β° π‘“π‘œπ‘Ÿ π‘’π‘Žπ‘β„Ž 𝑠𝑖𝑑𝑒 π‘œπ‘“ π‘‘β„Žπ‘’ πΈπ‘žπ‘’π‘Žπ‘‘π‘–π‘œπ‘›) ∠2 𝑖𝑠 π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒 ∠1 β‰… ∠3 π‘‰π‘’π‘Ÿπ‘‘π‘–π‘π‘Žπ‘™ 𝐴𝑛𝑔𝑙𝑒 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘š π‘π‘œπ‘‘π‘’: 𝑁𝑒𝑒𝑑 π‘‘π‘œ 𝐼𝑑𝑒𝑛𝑑𝑖𝑓𝑦 ∠1 π‘Žπ‘›π‘‘ ∠3 π‘Žπ‘  π‘‰π‘’π‘Ÿπ‘‘π‘–π‘π‘Žπ‘™ 𝐴𝑛𝑔𝑙𝑒𝑠, π‘‘β„Žπ‘’π‘Ÿπ‘’π‘“π‘œπ‘Ÿπ‘’ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ π‘šβˆ 1 = π‘šβˆ 3 π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠 90Β° = π‘šβˆ 3 π‘†π‘’π‘π‘ π‘‘π‘–π‘‘π‘’π‘‘π‘–π‘œπ‘› (𝑠𝑒𝑏 90Β° π‘“π‘œπ‘Ÿ π‘šβˆ 1) ∠3 𝑖𝑠 π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒 π‘šβˆ 1 + π‘šβˆ 4 = 180Β° π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘ π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘š (πΏπ‘–π‘›π‘’π‘Žπ‘Ÿ π‘ƒπ‘Žπ‘–π‘Ÿ) 90Β° + π‘šβˆ 4 = 180Β° π‘†π‘’π‘π‘ π‘‘π‘–π‘‘π‘’π‘‘π‘–π‘œπ‘› (𝑠𝑒𝑏 90Β° π‘“π‘œπ‘Ÿ π‘šβˆ 1) π‘šβˆ 4 = 90Β° π‘†π‘’π‘π‘‘π‘Ÿπ‘Žπ‘π‘‘π‘–π‘œπ‘› π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦ (π‘†π‘’π‘π‘ π‘‘π‘Ÿπ‘Žπ‘π‘‘ 90Β° π‘“π‘œπ‘Ÿ π‘’π‘Žπ‘β„Ž 𝑠𝑖𝑑𝑒 π‘œπ‘“ π‘‘β„Žπ‘’ πΈπ‘žπ‘’π‘Žπ‘‘π‘–π‘œπ‘›) ∠4 𝑖𝑠 π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒 All Right Angles are Congruent Theorem π‘ƒπ‘Ÿπ‘œπ‘£π‘’ ∠1 β‰… ∠2 β‰… ∠3 β‰… ∠4 𝑑 βŠ₯ 𝑠 𝑔𝑖𝑣𝑒𝑛 ∠1 𝑖𝑠 π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ ∠2 𝑖𝑠 π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ ∠3 𝑖𝑠 π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ ∠4 𝑖𝑠 π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ π‘ƒπ‘’π‘Ÿπ‘π‘’π‘›π‘‘π‘–π‘π‘’π‘™π‘Žπ‘Ÿ 𝐿𝑖𝑛𝑒𝑠 πΌπ‘›π‘‘π‘’π‘Ÿπ‘ π‘’π‘π‘‘ π‘‘π‘œ πΉπ‘œπ‘Ÿπ‘š πΉπ‘œπ‘’π‘Ÿ π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘š π‘šβˆ 1 = 90Β° π‘šβˆ 2 = 90Β° π‘šβˆ 3 = 90Β° π‘šβˆ 4 = 90Β° π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒 π‘šβˆ 1 = π‘šβˆ 2 = π‘šβˆ 3 = π‘šβˆ 4 π‘†π‘’π‘π‘ π‘‘π‘–π‘‘π‘’π‘‘π‘–π‘œπ‘› ∠1 β‰… ∠2 β‰… ∠3 β‰… ∠4 π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠
  • 8. Perpendicular Lines Will Form Four Congruent Adjacent Angles Theorem π‘ƒπ‘Ÿπ‘œπ‘£π‘’ π‘‘β„Žπ‘Žπ‘‘ ∠1 β‰… ∠2 𝑑 βŠ₯ 𝑠 𝑔𝑖𝑣𝑒𝑛 ∠1 𝑖𝑠 π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ ∠2 𝑖𝑠 π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ ∠3 𝑖𝑠 π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ ∠4 𝑖𝑠 π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ π‘ƒπ‘’π‘Ÿπ‘π‘’π‘›π‘‘π‘–π‘π‘’π‘™π‘Žπ‘Ÿ 𝐿𝑖𝑛𝑒𝑠 πΌπ‘›π‘‘π‘’π‘Ÿπ‘ π‘’π‘π‘‘ π‘‘π‘œ πΉπ‘œπ‘Ÿπ‘š πΉπ‘œπ‘’π‘Ÿ π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘š ∠1 β‰… ∠2 β‰… ∠3 β‰… ∠4 𝐴𝑙𝑙 π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠 π‘Žπ‘Ÿπ‘’ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘š ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ ∠2 π‘Žπ‘›π‘‘ ∠3 π‘Žπ‘Ÿπ‘’ π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ ∠3 π‘Žπ‘›π‘‘ ∠4 π‘Žπ‘Ÿπ‘’ π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ ∠4 π‘Žπ‘›π‘‘ ∠1 π‘Žπ‘Ÿπ‘’ π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π΄π‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠 π‘π‘œπ‘‘π‘’: π»π‘œπ‘€ π‘‘π‘œ 𝐼𝑑𝑒𝑛𝑑𝑖𝑓𝑦 π΄π‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠? π‘†β„Žπ‘Žπ‘Ÿπ‘’ π‘Ž π‘‰π‘’π‘Ÿπ‘‘π‘’π‘₯, π‘†β„Žπ‘Žπ‘Ÿπ‘’ π‘Ž 𝑆𝑖𝑑𝑒, π‘Žπ‘›π‘‘ π‘Žπ‘Ÿπ‘’ π‘œπ‘› π‘‘β„Žπ‘’ π‘†π‘Žπ‘šπ‘’ π‘ƒπ‘™π‘Žπ‘›π‘’ If Two Angles are Both Congruent and Supplementary, Then Each Angle Will be a Right Angle Theorem π‘ƒπ‘Ÿπ‘œπ‘£π‘’ π‘‘β„Žπ‘Žπ‘‘ ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠 ∠1 β‰… ∠2 ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘ π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ 𝑔𝑖𝑣𝑒𝑛 π‘šβˆ 1 = π‘šβˆ 2 π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠 π‘šβˆ 1 + π‘šβˆ 2 = 180Β° π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ 𝐴𝑛𝑔𝑙𝑒𝑠 π‘šβˆ 1 + π‘šβˆ 1 = 180Β° π‘†π‘’π‘π‘ π‘‘π‘–π‘‘π‘’π‘‘π‘–π‘œπ‘› π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦ (𝑠𝑒𝑏 π‘šβˆ 1 π‘“π‘œπ‘Ÿ π‘šβˆ 2 2 βˆ— π‘šβˆ 1 = 180Β° π‘†π‘’π‘π‘ π‘‘π‘–π‘‘π‘’π‘‘π‘–π‘œπ‘› π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦ (𝑠𝑒𝑏 2 βˆ— π‘šβˆ 1 π‘“π‘œπ‘Ÿ π‘šβˆ 1 + π‘šβˆ 1) π‘šβˆ 1 2 = 180Β° 2 π‘šβˆ 1 = 90Β° π·π‘–π‘£π‘–π‘ π‘–π‘œπ‘› π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦ ∠1 𝑖𝑠 π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒 π‘šβˆ 2 = 90Β° π‘†π‘’π‘π‘ π‘‘π‘–π‘‘π‘’π‘‘π‘–π‘œπ‘› π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦ (𝑠𝑒𝑏 π‘šβˆ 2 π‘“π‘œπ‘Ÿ π‘šβˆ 1) ∠2 𝑖𝑠 π‘Ž π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’ π·π‘’π‘“π‘–π‘›π‘–π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒 If Two Angles of a Linear Pair are Congruent, Then They Are Right Angles Theorem π‘ƒπ‘Ÿπ‘œπ‘£π‘’ π‘‘β„Žπ‘Žπ‘‘ ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒𝑠 ∠1 β‰… ∠2 ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘Ž π‘™π‘–π‘›π‘’π‘Žπ‘Ÿ π‘π‘Žπ‘–π‘Ÿ 𝑔𝑖𝑣𝑒𝑛 ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘ π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘ π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘š ∠1 π‘Žπ‘›π‘‘ ∠2 π‘Žπ‘Ÿπ‘’ π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’π‘  𝐼𝑓 π‘‡π‘€π‘œ 𝐴𝑛𝑔𝑙𝑒𝑠 π‘Žπ‘Ÿπ‘’ π΅π‘œπ‘‘β„Ž πΆπ‘œπ‘›π‘”π‘Ÿπ‘’π‘’π‘›π‘‘ π‘Žπ‘›π‘‘ π‘†π‘’π‘π‘π‘™π‘’π‘šπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦, π‘‡β„Žπ‘’π‘› πΈπ‘Žπ‘β„Ž 𝐴𝑛𝑔𝑙𝑒 π‘Šπ‘–π‘™π‘™ 𝑏𝑒 π‘Ž π‘…π‘–π‘”β„Žπ‘‘ 𝐴𝑛𝑔𝑙𝑒 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘š