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Unit 23
CONGRUENT AND SIMILAR
FIGURES
CONGRUENT FIGURES
Congruent figures have exactly the same
size and shape. The symbol  means
congruent
 Corresponding parts of congruent triangles
are equal



The sides that lie opposite equal angles are
corresponding sides
The angles that lie opposite equal sides are
corresponding angles
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CONGRUENT FIGURES
B
5"
4.75"
A
• BCA and CAD are corresponding
angles because they are both opposite 5"
long sides
C
4.75"
5"
• BAC and ACD are corresponding
angles because they are both opposite
4.75" long sides
D
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SIMILAR FIGURES
Similar figures mean figures that are alike
in shape but different in size
 Similar polygons have the same number
of sides, equal corresponding angles, and
proportional corresponding sides
 The symbol ~ means similar

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SIMILAR FIGURE EXAMPLE

Given that the two polygons shown below are
similar and A = A', B = B', C = C', and
D = D’, then
D 15.75"
3"
A
3.25"
B
16"
C'
D'
6"
A'
C
The corresponding sides are
proportional as follows:
AD
DC
CB
AB



A ' D'
D'C' C' B' A ' B'
B'
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SIMILAR FIGURE EXAMPLE (Cont)

Given that the two polygons shown below are
similar and A = A', B = B', C = C',
and D = D’,
D 15.75"
3"
A
3.25"
B
16"
(a )
C'
D'
3"
16"
 ' ' so A ' B'  32" Ans
6" A B
3" 3.25"
( b)
 ' ' so B'C '  6.5" Ans
6"
BC
6"
A'
C
Determine the lengths of sides (a) A'B',
(b) B'C', and (c) C'D'
B'
3" 15.75"
'
'
(c)

so
C
D
 31.5" Ans
'
'
6"
CD
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SIMILAR TRIANGLES
If two angles of a triangle are equal to two
angles of another triangle, the triangles
are similar
 If the corresponding sides of two triangles
are proportional, the triangles are similar
 If two sides of a triangle are proportional
to two sides of another triangle and if the
included angles are equal, the triangles
are similar

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SIMILAR TRIANGLES (Cont)
Within a triangle, if a line parallel to one
side intersects the other two sides, the
triangle formed and the given triangle are
similar
 If the altitude is drawn to the hypotenuse
of a right triangle, the two triangles
formed are similar to each other and to
the given triangle

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SIMILAR TRIANGLES
EXAMPLE

Determine AD and DC in the right triangle
shown below:
– BD  AC so ABC ~ ABD ~ BDC
A
20 mm
B
D
15 mm
C
AB
AD

AC
AB
20 mm
AD

so
25 mm
20 mm
AD  16 mm Ans
DC = AC – AD = 25 mm – 16 mm = 9 mm Ans
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PRACTICE PROBLEMS
1.
Identify the pairs of corresponding angles
in the figure below:
B
4"
A
C
5"
3"
5"
E
4"
3"
D
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PRACTICE PROBLEMS (Cont)
2.
The two polygons below are similar. A =
A, B = B, C = C, D = D, E
= E, F = F. Find each of the
following:
a)
b)
c)
d)
Side
Side
Side
Side
BC
CD
DE
AF
A
8"
A 10"
B
C
F
D
E
11"
B
13.5"
F
9"
C
12.5"
E
11.5"
D
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PRACTICE PROBLEMS (Cont)
3.
CDE ~ ABE in the figure below. Given
that AE = 70 ft, BE = 87.5 ft, EC = 25 ft,
ED = 20 ft, and CD = 40 ft, find AB.
A
B
E
C
D
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PRACTICE PROBLEMS (Cont)
4.
Determine length F in the figure below
given that AB = 32 m and AC = 10 m.
F
A
C
B
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PROBLEM ANSWER KEY
1.
2.
3.
4.
ABE and CED
AEB and ECD
BAE and EDC
a) 10.8 inches
b) 10 inches
c) 9.2 inches
d) 8.8 inches
140 feet
15.625 m
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