CONGRUENT TRIANGLES Jim Smith JCHS Sections 4-3, 4-5
When we talk about congruent triangles, we mean everything about them is congruent. All 3 pairs of corresponding angles are equal…. And all 3 pairs of corresponding sides are equal
For us to prove that 2 people are identical twins, we don’t need to show that all “2000” body parts are equal. We can take a short cut and show 3 or 4 things are equal such as their face, age and height. If these are the same I think we can agree they are twins. The same is true for triangles. We don’t need to prove all 6 corresponding parts are congruent. We have 5 short cuts or methods.
SSS If we can show all 3 pairs of corr. sides are congruent, the triangles have to be congruent.
SAS Show 2 pairs of sides and the included angles are congruent and the triangles have to be congruent. Included angle Non-included angles
This is called a common side. It is a side for both triangles. We’ll use the reflexive property.
Which method can be used to prove the triangles are congruent
Common side SSS Parallel lines alt int angles Common side Vertical angles SAS SAS
ASA, AAS and HL ASA – 2 angles and the included side A S A AAS – 2 angles and The non-included side A A S
HL ( hypotenuse leg ) is used only with right triangles, BUT, not all right triangles. HL ASA
When Starting A Proof, Make The Marks On The Diagram Indicating The Congruent Parts. Use The Given Info, Properties, Definitions, Etc. We’ll Call Any Given Info That Does Not Specifically State Congruency Or Equality A PREREQUISITE PROOFS
SOME REASONS WE’LL BE USING
DEF OF MIDPOINT
DEF OF A BISECTOR
VERT ANGLES ARE CONGRUENT
DEF OF PERPENDICULAR BISECTOR
REFLEXIVE PROPERTY (COMMON SIDE)
PARALLEL LINES ….. ALT INT ANGLES
A B C D E 1 2 Given: AB = BD EB = BC Prove: ∆ABE ˜ ∆DBC = SAS Our Outline P rerequisites S ides A ngles S ides Triangles ˜ =
A C D Given: AB = BD EB = BC Prove: ∆ABE ˜ ∆DBC = B E 1 2 SAS none AB = BD Given 1 = 2 Vertical angles EB = BC Given ∆ ABE ˜ ∆DBC SAS = STATEMENTS REASONS P S A S ∆’ s
A B C 1 2 Given: CX bisects ACB A ˜ B Prove: ∆ACX ˜ ∆BCX X = = AAS P A A S ∆’ s CX bisects ACB Given 1 = 2 Def of angle bisc A = B Given CX = CX Reflexive Prop ∆ ACX ˜ ∆BCX AAS =
Can you prove these triangles are congruent? A B D C X Given: AB ll DC X is the midpoint of AC Prove: AXB ˜ CXD =
ASA A B D C X Given: AB ll DC X is the midpoint of AC Prove: AXB ˜ CXD =