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USA - Geometry Transformations, congruence, and proof

Congruence in terms of rigid motions

20 practice questions 2 video lessons Theory + worked examples

Congruence and Rigid Motions

California Geometry • Standard G-CO.6 • Transformations, Congruence & Proof

Congruence and Rigid Motions is a topic in Transformations, Congruence & Proof in the California Common Core State Standards. It is aligned to Standard G-CO.6, which requires students to use the definition of congruence in terms of rigid motions to decide whether two figures are congruent.

Two figures are congruent exactly when some sequence of rigid motions maps one onto the other.

California Geometry › Transformations, Congruence & Proof › Congruence and Rigid Motions  —  Standard G-CO.6

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Practice questions

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Watch 2 video(s)
  • Lesson 12.6 - Congruent Triangles using Rigid Motions (Concept Development) Watch
  • Geometry Lesson 34 Congruence Via Rigid Motions Watch
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Theory

Two figures are congruent when a sequence of rigid motions (translations, reflections, rotations) maps one exactly onto the other. We write \(\triangle ABC\cong\triangle DEF\).

Because rigid motions preserve distance and angle, corresponding parts of congruent figures are congruent — matching sides and matching angles are equal (abbreviated CPCTC).

Congruence is about a rigid motion existing, not about how the figures happen to be drawn or positioned.
Congruence by a rigid motion Two triangles are congruent because a translation maps one exactly onto the other. ΔABC ΔDEF
A translation maps \(\triangle ABC\) onto \(\triangle DEF\): they are congruent.
Corresponding parts When figures are congruent, matching sides and matching angles are congruent (CPCTC). corresponding parts are congruent
Matching tick marks show congruent corresponding sides.

The definition and CPCTC:

\[\triangle ABC\cong\triangle DEF \iff \text{a rigid motion maps one onto the other}\]
two figures are congruent exactly when a rigid motion maps one onto the other
CPCTC: once triangles are congruent, every pair of corresponding sides and angles is congruent.

How to show two figures are congruent

  1. Find a sequence of rigid motions mapping one onto the other.
  2. Match corresponding vertices in order.
  3. Conclude corresponding sides and angles are congruent (CPCTC).
Example 1 — Name the rigid motion
\(\triangle ABC\) maps to \(\triangle DEF\) by sliding it 4 units right. Are they congruent, and by what motion?
Solution

A slide is a translation, a rigid motion, so the triangles are congruent.

\(\triangle ABC\)\(\cong\)\(\triangle DEF\)
congruent by a translation
Example 2 — Corresponding parts
If \(\triangle ABC\cong\triangle DEF\), which side corresponds to \(\overline{AB}\)?
Solution

Corresponding parts follow the order of the naming: \(A\to D\), \(B\to E\), \(C\to F\).

\(\overline{AB}\)\(\leftrightarrow\)\(\overline{DE}\)
side AB corresponds to side DE
Example 3 — Use CPCTC
Given \(\triangle ABC\cong\triangle DEF\) with \(\angle A=50^\circ\), find \(\angle D\).
Solution

Corresponding parts of congruent triangles are congruent, so \(\angle D=\angle A\).

\(\angle D\)\(=\)\(50^\circ\)
angle D equals 50 degrees
Example 4 — Congruent or not
A figure is dilated to twice its size. Is the image congruent to the original?
Solution

No. A dilation changes size, so it is not a rigid motion; the image is similar but not congruent.

no, a dilation gives a similar but not congruent figure

Common pitfalls

Order the names to match. \(\triangle ABC\cong\triangle DEF\) means \(A\!\leftrightarrow\!D\), \(B\!\leftrightarrow\!E\), \(C\!\leftrightarrow\!F\).
Congruent \(\ne\) similar. A dilation gives similar figures, not congruent ones.
CPCTC comes after proving congruence, never before — it is a conclusion, not a reason to start.

Frequently asked questions

What does congruent mean in terms of rigid motions?

Two figures are congruent if some sequence of translations, reflections, and rotations maps one exactly onto the other.

What does CPCTC stand for?

Corresponding Parts of Congruent Triangles are Congruent — once triangles are congruent, matching sides and angles are equal.

How do you know which parts correspond?

Read the congruence statement in order: \(\triangle ABC\cong\triangle DEF\) pairs \(A\) with \(D\), \(B\) with \(E\), and \(C\) with \(F\).

Is a dilated figure congruent to the original?

No. A dilation changes size, so the figures are similar but not congruent.

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