Learn to use properties of congruent figures to solve problems. 56. Congruence. PreAlgebra. 56. Congruence. PreAlgebra. A correspondence is a way of matching up two sets of objects. If two polygons are congruent, ALL of their corresponding sides and angles are congruent.
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Congruence
PreAlgebra
A correspondence is a way of matching up two sets of objects.
If two polygons are congruent, ALL of their corresponding sides and angles are congruent.
In a congruence statement, the vertices in the second polygon are written in order of correspondence with the first polygon.
Congruence
55
55
PreAlgebra
Example 1A: Writing Congruent Statements
Write a congruence statement for the pair of polygons.
The first triangle can be named triangle ABC. To complete the congruence statement, the vertices in the second triangle have to be written in order of the correspondence.
∠A≅∠Q, so ∠A corresponds to ∠Q.
∠B≅∠R, so ∠B corresponds to ∠R.
∠C≅∠P, so ∠C corresponds to ∠P.
The congruence statement is triangle ABC≅ triangle QRP.
Congruence
PreAlgebra
Example 1B: Writing Congruent Statements
Write a congruence statement for the pair of polygons.
The vertices in the first pentagon are written in order around the pentagon starting at any vertex.
∠D≅∠ M, so ∠D corresponds to ∠M.
∠E≅∠ N, so ∠E corresponds to ∠N.
∠F≅∠ O, so ∠F corresponds to ∠O.
∠G≅∠ P, so ∠G corresponds to ∠P.
∠H≅∠Q, so ∠H corresponds to ∠Q.
The congruence statement is pentagon DEFGH≅ pentagon MNOPQ.
WX ≅ KL
a + 8 = 24
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Congruence
a = 16
PreAlgebra
Example 2A: Using Congruence Relationships to Find Unknown Values
In the figure, quadrilateral VWXY≅ quadrilateral JKLM.
A. Find a.
Subtract 8 from both sides.
6b = 30
ML ≅ YX
6b = 30
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Congruence
6 6
PreAlgebra
Example 2B: Using Congruence Relationships to Find Unknown Values
In the figure, quadrilateral VWXY≅ quadrilateral JKLM.
B. Find b.
Divide both sides by 6.
b = 5
5c = 85
∠J ≅∠V
5c = 85
56
Congruence
5 5
PreAlgebra
Example 2C: Using Congruence Relationships to Find Unknown Values
In the figure, quadrilateral VWXY≅ quadrilateral JKLM.
C. Find c.
Divide both sides by 5.
c = 17
Congruence
PreAlgebra
A correspondence is a way of matching up two sets of objects.
If two polygons are congruent, all of their corresponding sides and angles are congruent.
In a congruence statement, the vertices in the second polygon are written in order of correspondence with the first polygon.
Congruence
PreAlgebra
Try This: Example 1A
Write a congruence statement for the pair of polygons.
The first trapezoid can be named trapezoid ABCD. To complete the congruence statement, the vertices in the second trapezoid have to be written in order of the correspondence.
A
B

60°
60°


120°
120°

D
C
∠A≅∠S, so ∠A corresponds to ∠S.
Q
R

120°
120°
∠B≅∠T, so ∠B corresponds to ∠T.


∠C≅∠Q, so ∠C corresponds to ∠Q.
60°
60°

∠D≅∠R, so ∠D corresponds to ∠R.
T
S
The congruence statement is trapezoid ABCD≅ trapezoid STQR.
Congruence
PreAlgebra
Try This: Example 1B
Write a congruence statement for the pair of polygons.
The vertices in the first pentagon are written in order around the pentagon starting at any vertex.
110°
A
B
∠A≅∠M, so ∠A corresponds to ∠M.
110°
140°
140°
F
∠B≅∠N, so ∠B corresponds to ∠N.
C
110°
∠C≅∠O, so ∠C corresponds to ∠O.
E
110°
D
N
∠D≅∠P, so ∠D corresponds to ∠P.
110°
O
M
∠E≅∠Q, so ∠E corresponds to ∠Q.
140°
110°
110°
∠F≅∠L, so ∠F corresponds to ∠L.
P
140°
L
The congruence statement is hexagon ABCDEF≅ hexagon MNOPQL.
110°
Q
3a = 6
IH ≅ RS
3a = 6
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Congruence
3 3
PreAlgebra
Try This: Example 2A
In the figure, quadrilateral JIHK≅ quadrilateral QRST.
A. Find a.
Divide both sides by 3.
3a
I
H
a = 2
6
4b°
S
R
120°
J
30°
Q
K
c + 10°
T
4b = 120
∠H ≅∠S
4b = 120
56
Congruence
4 4
PreAlgebra
Try This: Example 2B
In the figure, quadrilateral JIHK≅ quadrilateral QRST.
B. Find b.
Divide both sides by 4.
3a
I
H
b = 30°
6
4b°
S
R
120°
J
30°
Q
K
c + 10°
T
c + 10 = 30
∠K ≅∠T
c + 10 = 30
56
Congruence
PreAlgebra
Try This: Example 2C
In the figure, quadrilateral JIHK≅ quadrilateral QRST.
C. Find c.
Subtract 10 from both sides.
3a
I
H
c = 20°
6
90°
4b°
S
R
120°
90°
J
30°
c + 10°
Q
K
T