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Congruent and Similar Triangles (MASMTS408).notebook February 27, 2014 Congruent Triangles Two triangles are congruent when all three corresponding sides and all three corresponding angles have the same measurements. D A B E C F To prove that two triangles are congruent, it is not necessary to show all six conditions. (Note: Isometric and congruent mean the same thing.) 1 Congruent and Similar Triangles (MASMTS408).notebook February 27, 2014 There are minimum conditions for showing that triangles are congruent. These are known as Theorems of Congruence. T Q R V P U 1) If the three sides of one triangle are congruent to the three corresponding sides of another triangle, then the triangles are congruent. This is called the Side Side Side theorem (SSS). 2 Congruent and Similar Triangles (MASMTS408).notebook February 27, 2014 = = 2) If two sides and the contained angle of one triangle are congruent to two corresponding sides and contained angle of another triangle, then the triangles are congruent. This is called the Side Angle Side theorem (SAS). 3 Congruent and Similar Triangles (MASMTS408).notebook February 27, 2014 _ _ 3) If two angles and the contained side of one triangle are congruent to two corresponding angles and contained side of another triangle, then the triangles are congruent. This is called the Angle Side Angle theorem (ASA). 4 Congruent and Similar Triangles (MASMTS408).notebook February 27, 2014 Similar Triangles _ _ Two figures are similar when ... All corresponding angles are congruent. All corresponding sides are proportional. Therefore, similar figures have the same shape, but are not necessarily the same size. 5 Congruent and Similar Triangles (MASMTS408).notebook February 27, 2014 There are also minimum conditions to prove that two triangles are similar. 1) Side Side Side (SSS) If the corresponding sides of two triangles are proportional in length, then the triangles are similar. 2) Angle Angle (AA) If two corresponding angles of two triangles are congruent, then the triangles are similar. _ _ 3) Side Angle Side (SAS) If two triangles have one congruent angle contained between corresponding sides of proportional length, then the triangles are similar. 6 Congruent and Similar Triangles (MASMTS408).notebook February 27, 2014 Name the similar triangles and why they are similar. A a) 6 M 2 B 5 C N 1.66... P A b) D E B C 7 Congruent and Similar Triangles (MASMTS408).notebook February 27, 2014 Similar Triangles Knowing that triangles are similar allows us to solve some geometric problems. Example: Given that the triangles below are similar, solve triangle DEF. (To solve a triangle is to find all its measures.) A D F E C B Set up a proportion: 8 Congruent and Similar Triangles (MASMTS408).notebook February 27, 2014 Example: Determine the value of x. A M P C N B 9 Congruent and Similar Triangles (MASMTS408).notebook February 27, 2014 Example:Determine the value of x. Triangles are similar by AA Separate the triangles and make their orientations the same. Set up and solve the proportion 10