IntroductionRecognizing and using congruent andsimilar shapes can make calculations anddesign work easier. For instance, in thedesign at the corner, only two differentshapes were actually drawn. The designwas put together by copying andmanipulating these shapes to produceversions of them of different sizes and indifferent 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 FiguresNote: Two figures can be similar but not congruent,but they can’t be congruent but not similar. Thinkabout 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 A B E FExample G H C D These angles correspond:These sides correspond: A and EAB and EF B and FBD and FH D and HCD and GH C and GAC and EG
Ratios and Similar Figures 7m 14 m Example 3m 6mThese rectangles 7 14 3 6are similar, becausethe ratios of these 3 6 7 14corresponding sides 7 3 14 6are equal: 14 6 7 3
Proportions and Similar Figures •A proportion is an equation that states that two ratios are equal.•Examples: 4 8 6 m n 10 3 2 n=5 m=4
Proportions and Similar FiguresYou can use proportions of correspondingsides to figure out unknown lengths ofsides of polygons. 16 m n 10 m 5m –Solve for n: 10/16 = 5/n so n = 8 m
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 sameratio Example 4 cm A 2cm 65o 25o B 12cm Angle 1 = 90o Side B = 6 cm
Similar TrianglesWays to Prove Triangles Are Similar
Similar triangles have corresponding angles that are CONGRUENT and their corresponding sides are PROPORTIONAL. 10 6 5 3 8 4
But you don’t need ALLthat information to be able to tell that twotriangles 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 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. 12 3 8 2 21 18 318 14 8 12 2 21 3 12 12 14 2