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. 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!
5. 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.
6. 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)
7. Ratios and Similar Figures
A B E F
Example
G H
C D These angles correspond:
These sides correspond:
A and E
AB and EF
B and F
BD and FH
D and H
CD and GH
C and G
AC and EG
8. Ratios and Similar Figures
7m 14 m
Example
3m 6m
These rectangles
are similar, because
7 14 3 6
the ratios of these 3 6 7 14
corresponding sides 7 3 14 6
are equal: 14 6 7 3
9. 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
10. Proportions and Similar Figures
You can use proportions of corresponding
sides to figure out unknown lengths of
sides of polygons.
16 m
n
10 m 5m
–Solve for n:
10/16 = 5/n so n = 8 m
11. 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 4 cm A 2cm
65o
25o B
12cm
Angle 1 = 90o Side B = 6 cm
13. Similar triangles have corresponding
angles that are CONGRUENT and
their corresponding sides are
PROPORTIONAL.
10
6 5
3
8 4
14. But you don’t need ALL
that information to be
able to tell that two
triangles are similar….
15. 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
16. 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 3
18 14 8
12 2
21 3
12
12 14 2
