56. Congruence. Warm Up. Problem of the Day. Lesson Presentation. PreAlgebra. Learning Goal Assignment Learn to use properties of congruent figures to solve problems. Vocabulary. correspondence. A correspondence is a way of matching up two sets of objects.
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Learn to use properties of congruent figures to solve problems.
correspondence
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.
Additional 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.
[email protected]Q, so A corresponds to Q.
[email protected]R, so B corresponds to R.
[email protected]P, so C corresponds to P.
The congruence statement is triangle [email protected] triangle QRP.
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
[email protected]S, so A corresponds to S.
Q
R

120°
120°
[email protected]T, so B corresponds to T.


[email protected]Q, so C corresponds to Q.
60°
60°

[email protected]R, so D corresponds to R.
T
S
The congruence statement is trapezoid [email protected] trapezoid STQR.
Additional 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.
[email protected]M, so D corresponds to M.
[email protected]N, so E corresponds to N.
[email protected]O, so F corresponds to O.
[email protected]P, so G corresponds to P.
[email protected]Q, so H corresponds to Q.
The congruence statement is pentagon [email protected] pentagon MNOPQ.
Write a congruence statement for the pair of polygons.
The vertices in the first hexagon are written in order around the hexagon starting at any vertex.
110°
A
B
[email protected]M, so A corresponds to M.
110°
140°
140°
F
[email protected]N, so B corresponds to N.
C
110°
[email protected]O, so C corresponds to O.
E
110°
D
N
[email protected]P, so D corresponds to P.
110°
O
M
[email protected]Q, so E corresponds to Q.
140°
110°
110°
[email protected]L, so F corresponds to L.
P
140°
L
The congruence statement is hexagon [email protected] hexagon MNOPQL.
110°
Q
a + 8 = 24
–8 –8
a = 16
Additional Example 2A: Using Congruence Relationships to Find Unknown Values
In the figure, quadrilateral [email protected] quadrilateral JKLM.
A. Find a.
Subtract 8 from both sides.
3a = 6
3a = 6
3 3
Try This: Example 2A
In the figure, quadrilateral [email protected] 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
6b = 30
6b = 30
6 6
Additional Example 2B: Using Congruence Relationships to Find Unknown Values
In the figure, quadrilateral [email protected] quadrilateral JKLM.
B. Find b.
Divide both sides by 6.
b = 5
4b = 120
4b = 120
4 4
Try This: Example 2B
In the figure, quadrilateral [email protected] 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
5c = 85
5c = 85
5 5
Additional Example 2C: Using Congruence Relationships to Find Unknown Values
In the figure, quadrilateral [email protected] quadrilateral JKLM.
C. Find c.
Divide both sides by 5.
c = 17
c + 10 = 30
c + 10 = 30
–10 –10
Try This: Example 2C
In the figure, quadrilateral [email protected] 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
Learning Goal Assignment
Learn to transform plane figures using translations, rotations, and reflections.
When you are on an amusement park ride,
you are undergoing a transformation. Ferris wheels and merrygorounds are rotations. Free fall rides and water slides are translations. Translations, rotations, and reflectionsare type of transformations.
The resulting figure or image, of a translation, rotation or reflection is congruent to the original figure.
Additional Example 1A & 1B: Identifying Transformations
Identify each as a translation, rotation, reflection, or none of these.
B.
A.
rotation
reflection
C’
D’
A’
B’
B’
C’
Try This: Example 1A & 1B
Identify each as a translation, rotation, reflection, or none of these.
A.
B.
B
A
A
C
D
C
B
reflection
translation
Additional Example 1C & 1D: Identifying Transformations
Identify each as a translation, rotation, reflection, or none of these.
C.
D.
none of the these
translation
Identify each as a translation, rotation, reflection, or none of these.
E’
C.
D.
A’
F’
D’
A
B’
B
C’
F
C
D
none of these
rotation
E
B’
C’
Additional Example 2A: Drawing Transformations
Draw the image of the triangle after the transformation.
A. Translation along AB so that A’ coincides with B
A
B
C
A. Translation along DE so that E’ coincides with D
C’
F’
A’
D’
E’
Try This: Example 2A
Draw the image of the polygon after the transformation.
B
C
A
F
D
E
C’
A’
Additional Example 2B: Drawing Transformations
Draw the image of the triangle after the transformation.
B. Reflection across BC.
A
B
C
B’
C’
A’
F’
D’
E’
Try This: Example 2B
Draw the image of the polygon after the transformation.
B
C
A
D
F
E
A’
B’
Additional Example 2C: Drawing Transformations
Draw the image of the triangle after the transformation.
C. 90° clockwise rotation around point B
A
B
C
C’
B’
F’
E’
A’
Try This: Example 2C
Draw the image of the polygon after the transformation.
C. 90° counterclockwise rotation around point C
B
C
A
F
D
E
Additional Example 3A: Graphing Transformations
Draw the image of a triangle with vertices of (1, 1), (2, –2 ), and (5, 0) after each transformation.
A. 180° counterclockwise rotation around (0, 0)
Draw the image of a shape with vertices of (1, –2), (3, 2), (7, 3), and (6, –1) after each transformation.
A. 180° clockwise rotation around (0, 0)
y
2
x
–2
Additional Example 3B: Graphing Transformations
Draw the image of a triangle with vertices of (1, 1), (2, –2 ), and (5, 0) after each transformation.
B. Translation 5 units left
Draw the image of a shape with vertices of (1, –2), (3, 2), (7, 3), and (6, –1) after each transformation.
B. Translation 10 units left
y
2
x
–2
Additional Example 3C: Graphing Transformations
Draw the image of a triangle with vertices of (1, 1), (2, –2 ), and (5, 0) after each transformation.
C. Reflection across the xaxis
Draw the image of a shape with vertices of (1, –2), (3, 2), (7, 3), and (6, –1) after each transformation.
C. Reflection across the xaxis
y
2
x
–2