Theorems for Similar Triangles Lesson 7.5

Pre-AP Geometry

Lesson Focus Two theorems for showing that two triangles are similar are presented in this lesson. These theorems, along with the definition of similarity and the AA Similarity Postulate, provide four ways to prove that two triangles are similar.

SAS Similarity Theorem If an angle of one triangle is congruent to an angle of another triangle and the sides including those angles are in proportion, then the triangles are similar.

Triangle ABC is similar to triangle DEF because  A and  D are congruent and AB:DE = AC:DF.

SSS Similarity Theorem If the sides of two triangles are in proportion, then the triangles are similar.

AB:DE = AC:DF = BC:EF The two triangles are similar

SAS/SSS Similarity Theorems Example 1: The measures of the sides of  ABC are 4, 5, and 7 and the measures of the sides of  XYZ are 16, 20, and 28. Are the triangles similar?

Example 2: In  ABC, AB = 2, AC = 5, and BC = 6. In  XYZ, XY = 2.5, YZ = 2, and XZ = 3. Is  ABC  XYZ?

SAS/SSS Similarity Theorems Example 3: Name similar triangles and give the postulate or theorem that justifies your answer.

A D 80  E B 80  C

SAS/SSS Similarity Theorems Example 4: Name similar triangles and give the postulate or theorem that justifies your answer.

C D 6 10 E 3 5 A B

SAS/SSS Similarity Theorems Example 5: Name similar triangles and give the postulate or theorem that justifies your answer.

L 3 5 6 M 6 K N 6 10 O

SAS/SSS Similarity Theorems Example 6: If  ABC   DEF, does correspond to ?

SAS/SSS Similarity Theorems A Example 7: Given:  B   DEC Prove:  ABC   DEC D Proof: Statements B Reasons E C

Similar Triangles Remember, you must match corresponding vertices when naming similar triangles.

Similar Triangles Ways to prove triangles similar: – Definition of Similarity – AA Similarity Postulate – SAS Similarity Theorem – SSS Similarity Theorem

Written Exercises Problem Set 7.5, p.266: # 2 - 12 (even); Handout 7-5