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Similar triangles

Similar Triangles - one of the topics in Plane Geometry

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Similar triangles

  1. 1. Similar Triangles
  2. 2. The AAA Similarity Postulate If three angles of one triangle are congruent to three angle of another triangle, then the two triangles are similar.
  3. 3. The AAA Similarity Postulate If ∠𝐴 β‰… ∠𝐷, π‘Žπ‘›π‘‘βˆ π΅ β‰… ∠𝐸, ∠𝐢 β‰… ∠𝐹. Then βˆ†π΄π΅πΆ~βˆ†π·πΈπΉ.
  4. 4. The AA Similarity Theorem If ∠𝐴 β‰… ∠𝐷, π‘Žπ‘›π‘‘βˆ π΅ β‰… ∠𝐸. Then βˆ†π΄π΅πΆ~βˆ†π·πΈπΉ.
  5. 5. Example 1 RI II NO, RI =8, RB=3x+4,ON=16, and OB=x+18 Find a. RB b. OB Ans. x=2 RB=10 OB=20
  6. 6. The SAS Similarity Theorem If two sides of one triangle are proportional to the corresponding two sides of another triangle and their respective included angles are congruent, then the triangles are similar.
  7. 7. The SAS Similarity Theorem If 𝐴𝐡 𝐷𝐸 = 𝐴𝐢 𝐷𝐹 π‘Žπ‘›π‘‘ ∠𝐴 β‰… ∠𝐷, π‘‡β„Žπ‘’π‘› βˆ†π΄π΅πΆ~βˆ†π·πΈπΉ
  8. 8. Example 2 Are the two triangles similar? Justify your answer.
  9. 9. The SSS Similarity Theorem If the sides of one triangle are proportional to the corresponding sides of a second triangle, then the triangles are similar.
  10. 10. Similar right triangles The L-L Similarity Theorem If the legs of a right triangle are proportional to the corresponding legs of another right triangle, the right triangles are similar.
  11. 11. The L-L Similarity Theorem If ∠𝐢 π‘Žπ‘›π‘‘βˆ πΉ π‘Žπ‘Ÿπ‘’ π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’π‘  π‘Žπ‘›π‘‘ 𝐴𝐢 𝐡𝐢 = 𝐷𝐹 𝐸𝐹 π‘‡β„Žπ‘’π‘› βˆ†π΄π΅πΆ~βˆ†π·πΈπΉ
  12. 12. Similar right triangles The H-L Similarity Theorem If the hypotenuse and a leg of a right triangle are proportional to the corresponding hypotenuse and leg of another right triangle, then the right triangles are similar.
  13. 13. The H-L Similarity Theorem If ∠𝐢 π‘Žπ‘›π‘‘βˆ πΉ π‘Žπ‘Ÿπ‘’ π‘Ÿπ‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘”π‘™π‘’π‘  π‘Žπ‘›π‘‘ 𝐴𝐡 𝐷𝐸 = 𝐴𝐢 𝐷𝐹 π‘‡β„Žπ‘’π‘› βˆ†π΄π΅πΆ~βˆ†π·πΈπΉ
  14. 14. Example 3 In the figure UA βŠ₯ 𝐴𝑀, 𝑀𝐸 βŠ₯ 𝐸𝑅, π‘ˆπ΄ = 24, 𝐴𝑀 = 10, 𝑅𝐸 = 5π‘₯ + 2, π‘Žπ‘›π‘‘ 𝐸𝑀π‘₯ + 3. π·π‘’π‘‘π‘’π‘Ÿπ‘šπ‘–π‘›π‘’ π‘₯ π‘ π‘œ π‘‘β„Žπ‘Žπ‘‘ βˆ†π‘ˆπ΄π‘€~βˆ†π‘…πΈπ‘€.
  15. 15. Proportional Segments The Proportional Segments Theorem If a line intersects two sides of a triangle at distinct points and is parallel to the third side, the line divides the two sides in two proportional segments.
  16. 16. The Proportional Segments Theorem 𝑙𝑄𝑃 𝐼𝑓 𝑙 || BC, then 𝐡𝑃 𝐴𝑃 = 𝐢𝑄 𝐴𝑄
  17. 17. Example 4 In βˆ†π‘ƒπ‘„π‘…, AB||QR. If OA=5, PA=2, and BR=10, find PB.
  18. 18. Example 5 A flagpole 8m high casts a shadow of 12m, while a nearby building casts a shadow of 60m. How high is a building?
  19. 19. Proportional Segments The Bisector of an angle of a triangle divides the opposite side into segments which are proportional to the adjacent sides.
  20. 20. Proportional Segments If βˆ†π΄π΅πΆ with AD an angle bisector, then 𝐴𝐡 𝐴𝐢 = 𝐡𝐷 𝐢𝐷
  21. 21. Example 6 Find the value of x.

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