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Name Class Date 4-1

Determining If Figures are Congruent

Use the definition of congruence in terms of rigid motions to determine whether the

two figures are congruent and explain your answer.

A �ABC and �DEF have different sizes.

Since rigid motions preserve distance, there is no

sequence of rigid motions that will map �ABC to

�DEF.

Therefore,

B You can map JKLM to PQRS by the translation

that has the following coordinate notation:

A translation is a rigid motion.

Therefore,

REFLECT

1a. Why does the fact that �ABC and �DEF have different sizes lead to the conclusion

that there is no sequence of rigid motions that maps �ABC to �DEF?

E X AM P L E1

Congruence and TransformationsGoing DeeperEssential question: How can you use transformations to determine whether figures are congruent?

Two figures are congruent if they have the same size and shape.

A more formal mathematical definition of congruence depends

on the notion of rigid motions.

Two plane figures are congruent if and only if one can be obtained

from the other by rigid motions (that is, by a sequence of reflections,

translations, and/or rotations.)

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The definition of congruence tells you that when two figures are known to be congruent,

there must be some sequence of rigid motions that maps one to the other. You will

investigate this idea in the next example.

Finding a Sequence of Rigid Motions

For each pair of congruent figures, find a sequence of rigid motions that maps one figure

to the other.

A You can map �ABC to �RST by a reflection followed by

a translation. Provide the coordinate notation for each.

Reflection:

Followed by...

Translation:

B You can map �DFG to �HJK by a rotation followed by

a translation. Provide the coordinate notation for each.

Rotation:

Followed by...

Translation:

REFLECT

2a. Explain how you could use tracing paper to help you find a sequence of rigid

motions that maps one figure to another congruent figure.

2b. Given two congruent figures, is there a unique sequence of rigid motions that

maps one figure to the other? Use one or more of the above examples to explain

your answer.

E X AM P L E2G-CO.1.5

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Investigating Congruent Segments and Angles

A Use a straightedge to trace ___

AB on a piece of tracing paper. Then slide, flip, and/or turn

the tracing paper to determine if there is a sequence of rigid motions that maps ___

AB to

one of the other line segments.

B Repeat the process with the other line segments and the angles in order to determine

which pairs of line segments and which pairs of angles, if any, are congruent.

Congruent line segments:

Congruent angles:

C Use a ruler to measure the congruent line segments. Use a protractor to measure the

congruent angles.

REFLECT

3a. Make a conjecture about congruent line segments.

3b. Make a conjecture about congruent angles.

The symbol of congruence is �. You read the statement ___

UV � ___

XY as “Line segment UV is

congruent to line segment XY.”

Congruent line segments have the same length, so ___

UV � ___

XY implies UV = XY and vice

versa. Congruent angles have the same measure, so ∠C � ∠D implies m∠C = m∠D and vice versa. Because of this, there are properties of congruence that resemble the

properties of equality, and you can use these properties as reasons in proofs.

E X P L O R E3

Properties of Congruence

Reflexive Property of Congruence ___

AB � ___

AB

Symmetric Property of Congruence If ___

AB � ___

CD , then ___

CD � ___

AB .

Transitive Property of Congruence If ___

AB � ___

CD and ___

CD � ___

EF , then ___

AB � ___

EF .

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P R A C T I C E

Use the definition of congruence in terms of rigid motions to determine whether the two figures are congruent and explain your answer.

1. 2. 3.

For each pair of congruent figures, find a sequence of rigid motions that maps one figure to the other. Give coordinate notation for the transformations you use.

4. 5. 6.

7. �ABC� �DEF and �DEF � �GHJ. Can you conclude �ABC � �GHJ? Explain.

Chapter 4 134 Lesson 1

Apply the transformation to the polygon with the given vertices. Identify and describe the transformation. 1. M: (x, y) → (x − 2, y + 3) A(−1, −3), B(2, −1), C(2, −4)

_____________________________________

2. M: (x, y) → (−x, y) P(−1, 2), Q(−2, −3), R(1, −2)

____________________________________

3. M: (x, y) → (y, −x) G(−4, 3), H(−2, 3), J(−2, −1), K(−4, −1)

_____________________________________

4. M: (x, y) → (2x, 2y) E(−2, 2), F(1, 1), G(2, 2)

____________________________________

Determine whether the polygons with the given vertices are congruent. 5. A(−4, 4), B(−2, 4), C(−2, 2), D(−3, 1), E(−4, 2); P(2, 6), Q(4, 6), R(4, 4), S(3, 3), T(2, 4)

_____________________________________

6. P(4, 4), Q(−4, 2), R(−2, 6); J(2, 2), K(−2, 1), L(−1, 3)

____________________________________

4-1Name Class Date

Additional Practice©

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Chapter 4 135 Lesson 1

Name ________________________________________ Date __________________ Class__________________

Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.

Holt McDougal Geometry

Problem Solving Congruence and Transformations

1. Irena is designing a quilt. She made this diagram to follow when making her quilt. What transformation(s) are used on the triangles to create the pattern in the quilt design?

_____________________________________

2. An architect used this design for a stained glass window. What frieze transformation(s) is used to create the pattern in the window?

_____________________________________

3. A graphic artist incorporated the universal symbol for radiation for one of his designs. Describe the transformation(s) he used.

_____________________________________

4. Richard developed a tessellating shape for floor tile. Describe the series of transformations he used to create the design.

_____________________________________

Choose the best answer. 5. A team flag is made using a fabric with the

design shown. What transformation is used?

6. An art student used transformations in all her art. What transformation did she use for her design shown?

F translation G reflection H rotation J dilation

A translation B reflection C rotation D dilation

105

LESSON

4-1

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Problem Solving

Chapter 4 136 Lesson 1