Congruent and Similar Triangles

Introduction Recognizing and using congruent and similar shapes can make calculations and design work easier. For instance, in the design at the corner, only two different shapes were actually drawn. The design was put together by copying and manipulating these shapes to produce versions of them of different sizes and in different positions.

Similar and Congruent Figures Congruent triangles have all sides congruent and all angles congruent. Similar triangles have the same shape; they may or may not have the same size.

Note: Two figures can be similar but not congruent, but they can’t be congruent but not similar. Think about why! Similar and Congruent Figures

Examples These figures are similar and congruent. They’re the same shape and size. These figures are similar but not congruent. They’re the same shape, but not the same size.

Ratios and Similar Figures Similar figures have corresponding sides and corresponding angles that are located at the same place on the figures. Corresponding sides have to have the same ratios between the two figures. A ratio is a comparison between 2 numbers (usually shown as a fraction)

Ratios and Similar Figures Example A E C F D GH B These sides correspond: AB and EF BD and FH CD and GH AC and EG These angles correspond: A and E B and F D and H C and G

Ratios and Similar Figures Example 7 m 3 m 6 m 14 m These rectangles are similar, because the ratios of these corresponding sides are equal:

A proportion is an equation that states that two ratios are equal. Examples: n = 5 m = 4 Proportions and Similar Figures

You can use proportions of corresponding sides to figure out unknown lengths of sides of polygons. 16 m 10 m n 5 m 10/16 = 5/n so n = 8 m –Solve for n:

Similar triangles Similar triangles are triangles with the same shape For two similar triangles, corresponding angles have the same measure length of corresponding sides have the same ratio 65 o 25 o A 4 cm 2cm 12cm B Example Angle 1 = 90 o Side B = 6 cm

Similar Triangles Ways to Prove Triangles Are Similar

Similar triangles have corresponding angles that are CONGRUENT and their corresponding sides are PROPORTIONAL

But you don’t need ALL that information to be able to tell that two triangles are similar….

SSS Similarity If all three sides of a triangle are proportional to the corresponding sides of another triangle, then the two triangles are similar

AA Similarity If two (or 3) angles of a triangle are congruent to the two corresponding angles of another triangle, then the triangles are similar. 25 degrees