Congruent and Similar Triangles

1
Congruent and Similar Triangles
2
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.
3
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.

4
Examples
These figures are similar and congruent. Theyre
the same shape and size.
These figures are similar but not congruent.
Theyre the same shape, but not the same size.
5
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)

6
Ratios and Similar Figures
A
B
E
F
Example
G
H
C
D
These angles correspond
A and E
B and F
D and H
C and G
These sides correspond
AB and EF
BD and FH
CD and GH
AC and EG
7
Ratios and Similar Figures
Example
These rectangles are similar, because the ratios
of these corresponding sides are equal


8
Proportions and Similar Figures

• A proportion is an equation that states that two
  ratios are equal.

• Examples

• n 5 m 4

9
Proportions and Similar Figures
You can use proportions of corresponding sides to
figure out unknown lengths of sides of polygons.

• Solve for n

10/16 5/n so n 8 m
10
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

Example
Side B 6 cm
Angle 1 90o
11
Similar Triangles

• Ways to Prove Triangles Are Similar

12
Similar triangles have corresponding angles that
are CONGRUENT and their corresponding sides are
PROPORTIONAL.
10
5
6
3
8
4
13
But you dont need ALL that information to be
able to tell that two triangles are similar.
14
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
15
SSS Similarity

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

21
14
18
8
12
12
16
Video Clip

• Medical Triangles!!