SIMILAR TRIANGLES

1. SIMILAR TRIANGLES • Identify similar triangles • Use similar triangles to solve problems Eiffel Tower

2. POSTULATE Angle-Angle (AA) Similarity If the two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Example:

3. THEOREM Side-Side-Side (SSS) Similarity If the measures of the corresponding sides of two triangles are proportional, then the triangles are similar. Example: S Q a ax b bx c P R cx T U

4. THEOREM Side-Angle-Side (SAS) Similarity If the measures of two sides of a triangle are proportional to the measures of two corresponding sides of another triangle and the included angles are congruent, then the triangles are similar. S Q Example: a ax b bx P R T U

5. Example 1– Determine Whether Triangles are Similar In the figure, , BE = 15, CF = 20, AE = 9, and DF = 12. Determine which triangles in the figure are similar. C B G A F E D

6. Example 1– Determine Whether Triangles are Similar In the figure, , BE = 15, CF = 20, AE = 9, and DF = 12. Determine which triangles in the figure are similar. C Solution: Triangle FGE is an isosceles triangle, so B G A F E D

7. Example 1– Determine Whether Triangles are Similar In the figure, , BE = 15, CF = 20, AE = 9, and DF = 12. Determine which triangles in the figure are similar. C If the measures of the corresponding sides that include the angles are proportional, then the triangles are similar. B G A F E D By SAS similarity, ∆ABE ~ ∆DCF

8. THEOREM Similarity of triangles is reflexive, symmetric, and transitive. Examples: Reflexive: ∆ABC ~ ∆ABC Symmetric: If ∆ABC ~ ∆DEF, then ∆DEF ~ ∆ABC Transitive: If ∆ABC ~ ∆DEF and ∆DEF ~ ∆GHI, then ∆ABC ~ ∆GHI

9. Example 2– Parts of Similar Triangles Algebra: Find AE and DE C 5 X - 1 A E 2 X + 5 D B