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Congruence and Rigid Motions

Students define congruence through rigid motions and use corresponding parts and triangle congruence criteria to justify conclusions.

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Beginner 15–20 minutes Triangles & Transformations
Triangles & Transformations Page 3 of 8
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At a Glance

Congruence and Rigid Motions
Binder SectionTriangles & Transformations
DifficultyBeginner
Estimated Time15–20 minutes
PrerequisitesNone
Essential Question

Why do translations, rotations, and reflections preserve congruence?

Build Understanding

Figures are congruent when one can be mapped exactly onto the other by translations, rotations, and reflections. These rigid motions preserve side lengths and angle measures.

Corresponding vertices must be listed in matching order. If triangle ABC is congruent to triangle DEF, then A matches D, B matches E, and C matches F.

Triangle congruence shortcuts are SSS, SAS, ASA, AAS, and HL for right triangles. SSA and AAA do not prove congruence. After triangles are proven congruent, corresponding parts are congruent (CPCTC).

A proof names the given facts, uses definitions or theorems, establishes a valid congruence criterion, and then draws a justified conclusion.

Visual Model
4Thousands 3Hundreds 2Tens 7Ones

4,327 = 4,000 + 300 + 20 + 7

Goals

Learning Targets

  • Define congruence using rigid motions.
  • Match corresponding vertices, sides, and angles.
  • Use SSS, SAS, ASA, AAS, and HL.
  • Recognize why SSA and AAA are insufficient.
  • Use CPCTC after proving triangles congruent.
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Interactive Vocabulary

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Worked Examples

Example 1

Triangle ABC maps onto A′B′C′ by a translation

1

Corresponding distances and angles remain equal.

Example 2

Two right triangles have equal hypotenuse and one equal leg

1

Apply HL.

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Common Questions

Does a reflection preserve congruence? Yes.
Does a dilation preserve congruence? Only when the scale factor has absolute value 1.
Why does order matter in a congruence statement? It identifies corresponding parts.
Does AAA prove congruence? No; it proves similarity.
Notice and correct

Common Misconceptions

Using AAA to prove congruence

AAA proves similarity, not equal size.

Matching vertices out of order

Write congruence statements from corresponding positions.

Including a dilation with scale factor not 1

It is not a rigid motion.

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Congruence and Rigid Motions lesson and examples

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Quick Start

Warm-Up

Warm-up 1

Directions: Solve the problem. Show your thinking, then enter only your final answer.

Are two figures related by a reflection congruent?

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Independent Practice

Independent 1

Directions: Solve independently. Show your work, check it, then enter only your final answer.

Decide whether SAS data proves congruence.

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ABC

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Independent 2

Directions: Solve independently. Show your work, check it, then enter only your final answer.

Explain why SSA is generally insufficient.

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Independent 3

Directions: Solve independently. Show your work, check it, then enter only your final answer.

Write a correct correspondence statement.

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Challenge & Real-World Practice

🧩 Challenge

Create two congruent triangles in different orientations. Give a minimum rigid-motion sequence and a second valid sequence mapping one to the other.

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Success Criteria

Before you begin, I can…

  • I match corresponding vertices accurately.
  • I distinguish rigid motions from dilations.
  • I cite an appropriate congruence criterion and supporting evidence.
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Question 1

Which statement shows the key idea in this lesson?

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Question 2

Which habit best supports accuracy?

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Question 4

Which explanation is strongest?

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How can an error be found?

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