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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
Inside Cover
Congruence and Rigid Motions
At a Glance
Binder SectionTriangles & Transformations
DifficultyBeginner
Estimated Time15–20 minutes
PrerequisitesNone
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.
2
The triangles are congruent.
Example 2
Two right triangles have equal hypotenuse and one equal leg
1
Apply HL.
2
The triangles are congruent.
Ask and explain
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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Warm-Up
Guided
Independent
Challenge
1
Quick Start
Warm-Up
Are two figures related by a reflection congruent?
2
We Do
Guided Practice
If triangle ABC is congruent to triangle DEF, which angle matches B? Answer: E.
Do sides 3,4,5 and 3,4,5 prove congruence? Answer: yes, SSS.
Two sides and the included angle are congruent. Answer: SAS.
Is AAA a congruence theorem? Answer: no.
After proving triangles congruent, justify matching sides. Answer: CPCTC.
3
You Do
Independent Practice
Decide whether SAS data proves congruence.
Explain why SSA is generally insufficient.
Write a correct correspondence statement.
4
Stretch Your Thinking
Challenge & Real-World Practice
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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Open resourceDeltaMath: Triangle Congruence
Open resourceDeltaMath: CPCTC
Open resource
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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.
1
Question 1
Which statement shows the key idea in this lesson?
2
Question 2
Which habit best supports accuracy?
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Question 3
What should a student do after solving?
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Question 4
Which explanation is strongest?
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Question 5
How can an error be found?
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