7-3 Triangle Similarity: AA, SSS, SAS Geometry

Postulate 7-3-1 Angle – Angle (AA) Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. L Z Y X J K

Writing Proportionality Statements ∠A =∠D = 90° Ex. 1 ) B A C D E a.).Explain why the triangles are similar. b.) Write a similarity statement. ∟ 1a) ∠BCA ≌∠ECD by vertical angles thm; AA 1b) ABC~DEC

THEOREM 7-3-2 Side-Side-Side (SSS) Similarity Theorem If the three sides of one triangle are proportional to the 3 corresponding sides of another triangle, then the triangles are similar. B C E F A D

THEOREM 7-3-3 Side-Angle-Side (SAS) Similarity Theorem If two sides of one triangle are proportional to two sides of another triangle & their included angles are congruent, then the triangles are similar. A D B C E F ∠B ≅∠E

Verifying Triangle Similarity   2 3 4.5 4.5 2 5.8 2.9 70° 1 Use Extra Example 2 picture. 3

Finding lengths in Similar Triangles Ex. 3 Explain why & find CD. D E 5 x A 9 B 3 C

Properties of Similarity Reflexive Property of Similarity ~ Symmetric Property of Similarity If ~ , then ~ . Transitive Property of Similarity If ~ & ~ then ~

Writing Proofs with Similarity Triangles Ex. 4 Given: M is the midpoint of . N is the midpoint of , & P is the midpoint of . Prove: (Hint: Use the Triangle Midsegment Theorem & SSS ). ~

Ex. 4 Proof Statements / Reasons

Assignment

Using Algebra Ex. 3 – Use points G & J in the diagram below to find the slope of the line containing GJ. Name five other segments whose endpoints could be used to find the slope of the line.

Find the value of the variable Ex. 4) Find y. Use picture for #6 & 7

Generalization of Theorem 8.1 If two polygons are similar, then the ratio of any two corresponding lengths (such as altitudes, medians, angle bisector segments, and diagonals) is equal to the scale factor of the similar polygons.

Similar Triangles Ex. 5) The segments in blue are special segments in the similar triangles. Find the value of the variable. Use #45 as example

Slope Slope formula #29?