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Triangle Similarity

Triangle Similarity. Keystone Geometry. F. C. A. B. D. E. Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are proportional. Similar Polygons. ~ means “is similar to”. Congruence vs. Similarity.

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Triangle Similarity

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  1. Triangle Similarity Keystone Geometry

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

  3. 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

  4. 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 AA Similarity Postulate

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

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

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

  8. 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.

  9. * Yes, they are similar by SSS theorem! 16 20 10 32 24 15 Example: Are the two triangles similar? The proportions of the sides are equal! The scale factor of the two triangles is 5:8.

  10. SAS Similarity Theorem Remember! Proportional Side – Congruent Angle – Proportional Side! 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.

  11. M J 6 12 N 9 8 L K Included Angle Example: Are the two triangles similar? Side Side Yes, they are similar by SAS theorem!

  12. 20 16 20 24 32 32 24 24 16 36 36 Example: Are the two triangles similar? NO! They are NOT similar because not all of the sides are proportional!

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