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6.4 and 6.5 Congruent and Similar Triangles. Similar and Congruent Figures. Congruent polygons have all sides congruent and all angles congruent. Similar polygons have the same shape; they may or may not have the same size. Tests for Congruency. Ways to prove triangles congruent :
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Similar and Congruent Figures • Congruent polygons have all sides congruent and all angles congruent. • Similar polygons have the same shape; they may or may not have the same size.
Tests for Congruency Ways to prove triangles congruent : • SSS ( Side – Side – Side ) • SAS ( Side – Angle – Side ) • ASA ( Angle – Side – Angle ) or AAS ( Angle –Angle – Side ) • RHS ( Right angle – Hypotenuse – Side )
Thinking Time ????? • If 3 angles on A are equal to the 3 corresponding angles on the other B, are the two triangles congruent ?
? 2cm 4 cm 65o 25o ? 12cm Similar triangles For two similar triangles, • Similar triangles are triangles with the same shape • corresponding angles have the same measure • length of corresponding sides have the same ratio Example Side = 6 cm Angle = 90o
Similar Triangles 3 Ways to Prove Triangles Similar
Similar triangles are like similar polygons. Their corresponding angles are CONGRUENT and their corresponding sides are PROPORTIONAL. 10 5 6 3 8 4
But you don’t need ALL that information to be able to tell that two triangles are similar….
AA Similarity • If two angles of a triangle are congruent to the two corresponding angles of another triangle, then the triangles are similar. 25 degrees 25 degrees
SSS Similarity • If all three sides of a triangle are proportional to the corresponding sides of another triangle, then the two triangles are similar. 21 14 18 8 12 12
SSS Similarity Theorem If the sides of two triangles are in proportion, then the triangles are similar. D A C B F E
SAS Similarity • If two sides of a triangle are proportional to two corresponding sides of another triangle AND the angles between those sides are congruent, then the triangles are similar. 14 21 18 12
SAS Similarity Theorem D A C B F E 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.
D A C B F E SAS Similarity Theorem Idea for proof
A 80 D E 80 B C ABC ~ ADE by AA ~ Postulate
C 6 10 D E 5 3 A B CDE~ CAB by SAS ~ Theorem
L 5 3 M 6 6 K N 6 10 O KLM~ KON by SSS ~ Theorem
A 20 D 30 24 16 B C 36 ACB~ DCA by SSS ~ Theorem
L 15 P A 25 9 N LNP~ ANL by SAS ~ Theorem