Triangle Similarity Keystone Geometry. 2 Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding.

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Presentation transcript:

Triangle Similarity Keystone Geometry

2 Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are proportional. AB C DE F Similar Polygons ~ means “is similar to”

Congruence vs. Similarity If two triangles are congruent, then they are exactly the same. ◦Methods: SAS, SSS, ASA, AAS, and HL If two triangle are similar, then they will have congruent angles and their sides will be proportional. ◦Methods: AA, SSS, SAS

AA Similarity Postulate If two angles of one triangle are congruent (equal) to two angles of another triangle, then the triangles are similar. Note: If you know two angles, the third angle is not negotiable

Examples: Tell whether the triangles are similar or not. Example 1: Example 2: C H E G D F 40º 50º YES- 2 angles of triangle CDE are congruent to 2 angles of triangle FHG 61º 60º 58º 59º 62º NO- only 1 angle is congruent in both triangles *You need two angles to be congruent to prove the triangles are similar!

Example: Find the values of x and y. 3 4 y 2 4 x 6 y 6 x y+34+2=6 *If two triangles share an angle, then they share a congruent angle The triangle on the inside is similar to the larger triangle on the outside because of AA similarity.

Example: Complete the following statement: ΔJKN ~ _______ J L M N K ΔLMN *Make sure you match up corresponding angles *It matters how you name your triangle, just like with congruence!!

SSS Similarity Theorem If the sides of two triangles are in proportion, then the triangles are similar. All proportions will be equal to the scale factor of the two triangles.

Example: Are the two triangles similar? * Yes, they are similar by SSS theorem! The proportions of the sides are equal! The scale factor of the two triangles is 5:8.

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. Remember! Proportional Side – Congruent Angle – Proportional Side!

Example: Are the two triangles similar? J L M N K Yes, they are similar by SAS theorem! Side Included Angle

Example: Are the two triangles similar? NO! They are NOT similar because not all of the sides are proportional!