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.
Similar and Congruent Figures Note: Two figures can be similar but not congruent, but they can’t be congruent but not similar. Think about why!
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 G H 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 These rectangles are similar, because the ratios of these corresponding sides are equal: 7 m 3 m 6 m 14 m
Proportions and Similar Figures
A proportion is an equation that states that two ratios are equal.
n = 5 m = 4
Proportions and Similar Figures You can use proportions of corresponding sides to figure out unknown lengths of sides of polygons. 10/16 = 5/n so n = 8 m
Solve for n:
16 m 10 m n 5 m
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
Example Angle 1 = 90 o Side B = 6 cm 65 o 25 o A 4 cm 2cm 12cm B
Similar Triangles Ways to Prove Triangles Are Similar
Similar triangles have corresponding angles that are CONGRUENT and their corresponding sides are PROPORTIONAL. 6 10 8 3 4 5
But you don’t need ALL that information to be able to tell that two triangles are similar….
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 25 degrees
If all three sides of a triangle are proportional to the corresponding sides of another triangle, then the two triangles are similar.