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Pre-Algebra
5-6 Congruence5-6 Congruence
Pre-Algebra
Warm UpWarm Up
Problem of the DayProblem of the Day
Lesson PresentationLesson Presentation
Pre-Algebra
5-6 Congruence
Learning Goal Assignment
Learn to use properties of congruent figures to solve problems.
Pre-Algebra
5-6 Congruence
correspondence
Vocabulary
Pre-Algebra
5-6 Congruence
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.
Pre-Algebra
5-6 Congruence
Additional Example 1A: Writing Congruent Statements
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.
Write a congruence statement for the pair of polygons.
Pre-Algebra
5-6 Congruence
Try This: Example 1A
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 S, so A corresponds to S.
B T, so B corresponds to T.
C Q, so C corresponds to Q.
The congruence statement is trapezoid ABCD trapezoid STQR.
A B
CD
Q R
STD R, so D corresponds to R.
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Write a congruence statement for the pair of polygons.
60° 60°
120° 120°
60° 60°
120° 120°
Pre-Algebra
5-6 Congruence
Pre-Algebra HW
Page 741
#1-10
Pre-Algebra
5-6 Congruence
Additional Example 1B: Writing Congruent Statements
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.
The congruence statement is pentagon DEFGH pentagon MNOPQ.
G P, so G corresponds to P.
H Q, so H corresponds to Q.
Write a congruence statement for the pair of polygons.
Pre-Algebra
5-6 Congruence
Try This: Example 1B
The vertices in the first hexagon are written in order around the hexagon starting at any vertex.
A M, so A corresponds to M.
B N, so B corresponds to N.
C O, so C corresponds to O.
The congruence statement is hexagon ABCDEF hexagon MNOPQL.
D P, so D corresponds to P.
E Q, so E corresponds to Q.
A B
C
DE
F
N
O
P
QL
M
F L, so F corresponds to L.
Write a congruence statement for the pair of polygons.
140° 140°
110°
110°
110°
110°
140°
140°
110°
110°
110°
110°
Pre-Algebra
5-6 Congruence
Additional Example 2A: Using Congruence Relationships to Find Unknown Values
In the figure, quadrilateral VWXY quadrilateral JKLM.
a = 16
–8 –8 Subtract 8 from both sides.
A. Find a.
a + 8 = 24 WX KL
Pre-Algebra
5-6 Congruence
Try This: Example 2A
In the figure, quadrilateral JIHK quadrilateral QRST.
A. Find a.
3a4b° 6
30°Q
120°R S
H I
JK
3a = 6 3 3
a = 2
c + 10°T
3a = 6 IH RS
Divide both sides by 3.
Pre-Algebra
5-6 Congruence
In the figure, quadrilateral VWXY quadrilateral JKLM.
6 6 6b = 30 Divide both sides by 6.
B. Find b.6b = 30 ML YX
b = 5
Additional Example 2B: Using Congruence Relationships to Find Unknown Values
Pre-Algebra
5-6 Congruence
B. Find b.
Divide both sides by 4. 4 4 4b = 120
b = 30°
4b = 120 H S
Try This: Example 2B
In the figure, quadrilateral JIHK quadrilateral QRST.
3a4b° 6
30°Q
120°R S
H I
JK c + 10°
T
Pre-Algebra
5-6 Congruence
5c = 85 J V
5 5 5c = 85 Divide both sides by 5.
C. Find c.
c = 17
In the figure, quadrilateral VWXY quadrilateral JKLM.
Additional Example 2C: Using Congruence Relationships to Find Unknown Values
Pre-Algebra
5-6 Congruence
C. Find c.
c = 20°
Subtract 10 from both sides.–10 –10
c + 10 = 30
c + 10 = 30 K T
Try This: Example 2C
3a4b° 6
30°
90°
Q
120°90° R S
H I
JK
T
c + 10°
In the figure, quadrilateral JIHK quadrilateral QRST.
Pre-Algebra
5-6 Congruence5-7 Transformations
Pre-Algebra
Warm UpWarm Up
Problem of the DayProblem of the Day
Lesson PresentationLesson Presentation
5-7Learning Goal Assignment
Learn to transform plane figures using translations, rotations, and reflections.
Vocabulary
transformation
translation
rotation
center of rotation
reflection
image
When you are on an amusement park ride,you are undergoing a transformation. Ferris wheels and merry-go-rounds are rotations. Free fall rides and water slides are translations. Translations, rotations, and reflections are 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.
A. B.
reflection rotation
Try This: Example 1A & 1B
Identify each as a translation, rotation, reflection, or none of these.
A
B C
A. B. A B
CD
A’ B’
C’D’
translation reflection
A’
B’ C’
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.
BC
DE
F
C. D.
A
A’
B’C’
D’
F’
E’
rotation none of these
Try This: Example 1C & 1D
Additional Example 2A: Drawing Transformations
Draw the image of the triangle after the transformation.
A
B
C
A. Translation along AB so that A’ coincides with B
A’
B’
C’
Try This: Example 2A
Draw the image of the polygon after the transformation.
A
B C
D
E
F
A’
B’ C’
D’
E’
F’
A. Translation along DE so that E’ coincides with D
Additional Example 2B: Drawing Transformations
Draw the image of the triangle after the transformation.
A
B
C
B. Reflection across BC.
A’
B’
C’
Try This: Example 2B
Draw the image of the polygon after the transformation.
A
BC
D
E
F
B. Reflection across CD.
A’
B’C’
D’
E’
F’
Draw the image of the triangle after the transformation.
A
B
C
C. 90° clockwise rotation around point B A’
B’
C’
Additional Example 2C: Drawing Transformations
Draw the image of the polygon after the transformation.
A
BC
D
E
F
C’
A’
B’
D’
E’F’
C. 90° counterclockwise rotation around point C
Try This: Example 2C
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)
Try This: Example 3A
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)
x
y
–2
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
B. Translation 10 units left
Draw the image of a shape with vertices of (1, –2), (3, 2), (7, 3), and (6, –1) after each transformation.
Try This: Example 3B
x
y
–2
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 x-axis
C. Reflection across the x-axis
Draw the image of a shape with vertices of (1, –2), (3, 2), (7, 3), and (6, –1) after each transformation.
Try This: Example 3C
x
y
–2
2