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Congruent and Similar Triangles
A PERESENTATION BYRITIK
XF
GLT Saraswati Bal Mandir Sr. Sec. SchoolNehru Nagar ND=65
IntroductionRecognizing 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
ExamplesThese 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 EFBD and FHCD and GHAC and EG
These angles correspond:
A and EB and FD and HC 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:
7 143 6
3 67 14
7 314 6
14 67 3
• A proportion is an equation that states that two ratios are equal.
• Examples:4 8
10n
63 2
m
n = 5 m = 4
Proportions and Similar Figures
Proportions and Similar FiguresYou 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 shapeFor two similar triangles, • corresponding angles have the same measure• length of corresponding sides have the same
ratio
65o
25o
A4 cm 2cm
12cmB
Example
Angle 1 = 90o 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.
610
8
3
4
5
But you don’t need ALL that information to be able to tell that 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 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.
18
12
8
12
1421
23
1421
23
1218
23
812
THE END