Name———————————————————————— Lesson
4.3
Date —————————————
Study Guide
For use with the lesson “Relate Transformations and Congruence”
goal
Use transformations to show congruence.
Vocabulary
A rigid motion is a transformation that preserves length, angle
measure, and area. A rigid motion is also called an isometry.
Translations, reflections, and rotations are rigid motions.
example 1
Describe rigid motions to show congruence
Describe the transformation(s) you can use to move figure A onto
figure B.
a.
b.
A
B
A
B
a.
example
example 2
1
translation
b.
rotation and then reflection
Show figures are congruent or not congruent
Tell whether a rigid motion can move figure A onto figure B. If so,
describe the transformation(s) that can be used. If not, explain why
the figures are not congruent.
a.
y
b.
y
B
1
1
A
Lesson 4.3
B
4-40
1
x
1
A
Solution
a. No; a translation maps one side to a congruent side, but the other
sides are not congruent.
b. Yes; a reflection in the x-axis maps figure A onto figure B.
Geometry
Chapter Resource Book
x
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
Solution
Name———————————————————————— Lesson
4.3
Date —————————————
Study Guide continued
For use with the lesson “Relate Transformations and Congruence”
Exercises for Examples 1 and 2
Describe the transformation(s) you can use to move figure A onto
figure B.
1.
2.
A
B
A
B
Tell whether a rigid motion can move figure A onto figure
B. If so, describe the transformation(s) that can be used. If
not, explain why the figures are not congruent.
3.
y
y
4.
B
A
1
1
x
1
B
Move one figure in a pattern onto another
Designs for floor tiles are shown below. Describe the rigid
motion(s) that can be used to move figure A onto figure B.
a.
b.
A
A
B
B
rotation 90° clockwise
translation and then reflection
Exercises for Example 3
Lesson 4.3
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
A
example 3
x
1
Describe the rigid motion(s) that can be used to move
figure A onto figure B.
5.
6.
A
A
B
B
Geometry
Chapter Resource Book
4-41
Lesson 4.3 Relate Transformations and Congruence, continued
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
Study Guide
1. translation and then reflection 2. rotation 908
clockwise and then translation 3. Yes; a
rotation of 1808 maps figure A onto figure B. 4. No; a 908 clockwise rotation maps two sides to
two corresponding congruent sides, but the other
sides are not congruent. 5. Sample answer:
translation down and to the right 6. Sample answer: 908 rotation clockwise, followed by reflection
Real-Life Application
1. reflection 2. translation 3. rotation 4. Yes, the lengths of the lines and the angle
­ easures formed by the lines were preserved in
m
each transformation, so the figures are congruent. 5. Yes, in Question 1, four congruent triangles
are formed on each side. In Question 2 there are
congruent trapezoids. In Question 3 there are
­congruent triangles in each corner. 6. More
­patterns would emerge, with more congruent
figures.
Challenge Practice
1
answers
8. Sample answer: n ABD > n CBD; n ABD
is not congruent to n AED; a reflection maps
n ABD onto n CBD and since reflection is a
rigid motion, the triangles are congruent. When
you reflect n ABD onto n AED, they share only
one side and one angle. So the figures are not
­congruent.
9. Sample answer: translation 4 units right and
6 units down or a reflection in the diagonal and a
1808 rotation.
10. Ayo is correct. Sample answer: A combination
of a translation, reflection, and rotation was used
to move figure A onto figure C. Since one or more
rigid motions were used, the figures are congruent.
A rigid motion is also called an isometry.
A translation 4 units right and 2 units up maps one
side to a congruent side, but the other sides are not
congruent.
2. Sample answer:
y
x
1
A reflection in the y-axis maps two sides to
corresponding congruent sides, but the other two
sides are not congruent.
3. Sample answer:
y
1
x
1
A rotation 908 maps one side to a congruent side,
but the other sides are not congruent.
4. reflection, reflection, reflection, and so on
5. reflection, translation, reflection, translation, and so on 6. Sample answer: Translate the
preimage triangle horizontally to the right twice
the distance between the parallel lines to move it
onto the final image triangle. 7. Sample answer:
Reflect the quadrilateral in lines that intersect at
the center of rotation such that the angle between
the reflecting lines is half the measure of the angle
of rotation.
8. 1. Sample answer:
y
9. 1
A
1
x
B
Geometry
Chapter Resource Book
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