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Triangle Congruence
from Rigid Motions

Understand how translations, rotations, and reflections prove that two triangles are congruent: corresponding sides and angles must all match.

Geometry

What Does Congruent Mean?

Two figures are congruent when one can be mapped exactly onto the other using only rigid motions: translations, rotations, and reflections. No stretching or shrinking allowed.

The Three Rigid Motions

Translation
Slide every point the same distance in the same direction. Shape and size stay unchanged.
Rotation
Turn the figure around a fixed center point by a specified angle. Distances are preserved.
Reflection
Flip the figure across a line of reflection. Orientation reverses, but size and shape stay the same.

Why Rigid Motions Matter

Before rigid motions, congruence was defined by listing matching parts. Now we have a single definition that works for any shape: two figures are congruent if and only if a sequence of rigid motions maps one onto the other.

A B C rigid motion D E F
\(\cong\)

The Congruence Theorem

This biconditional has two directions:

1
If congruent, then all parts match. A rigid motion preserves every distance and every angle, so corresponding parts must be equal.
2
If all parts match, then congruent. When all six pairs (3 sides + 3 angles) are equal, a rigid motion can map one triangle onto the other.

Corresponding Parts in Detail

A B C
\(c\)
\(a\)
\(b\)
\(\cong\)
 D E F
\(c\)
\(a\)
\(b\)

Congruence vs. Similarity

Congruence Similarity
Transformations Rigid motions only Rigid motions + dilations
Side lengths Equal Proportional
Angle measures Equal Equal
Size Same size and shape Same shape, may differ in size
Symbol \(\cong\) \(\sim\)

How to Prove Congruence

1
Identify the correspondence. Match vertices of the two triangles: which vertex maps to which?
2
Describe a rigid motion sequence. Find translations, rotations, and/or reflections that carry one triangle onto the other.
3
Verify all six pairs. Confirm that all 3 pairs of corresponding sides and all 3 pairs of corresponding angles are congruent.
4
State the conclusion. Write the congruence statement with vertices in corresponding order: \(\triangle ABC \cong \triangle DEF\).

Example 1: Translation

P Q R Translate 280 units right S T U

Example 2: Rotation

center
\(180^{\circ}\)
 A B C D E F

Example 3: Reflection

line of reflection J K L M N O

Combining Multiple Motions

Sometimes a single rigid motion is not enough. A sequence of rigid motions (for example, a translation followed by a reflection) can map one triangle onto another. As long as every step is a rigid motion, the composite is also rigid.

original translate
rotate \(180^{\circ}\)
 image

CPCTC: The Direct Consequence

Once you establish \(\triangle ABC \cong \triangle DEF\), you can immediately conclude:

Sides
\(\overline{AB} \cong \overline{DE}\)
\(\overline{BC} \cong \overline{EF}\)
\(\overline{CA} \cong \overline{FD}\)
Angles
\(\angle A \cong \angle D\)
\(\angle B \cong \angle E\)
\(\angle C \cong \angle F\)

When Congruence Fails

P Q R
\(x\)
 X Y Z
\(2x\)
\(\not\cong\)
 not congruent

Common Mistakes to Avoid

✕ Wrong Vertex Order
Writing \(\triangle ABC \cong \triangle EDF\) instead of \(\triangle ABC \cong \triangle DEF\). The vertex order must reflect the correspondence.
✕ Confusing \(\cong\) and \(\sim\)
Concluding congruence when only angles match but sides are proportional. Equal angles alone mean similarity, not congruence.
✕ Including Dilations
Treating a dilation as a rigid motion. Dilations change size, so they break congruence. Only translations, rotations, and reflections count.

Summary: Rigid Motions at a Glance

Property Translation Rotation Reflection
Preserves distances ✓ ✓ ✓
Preserves angles ✓ ✓ ✓
Preserves orientation ✓ ✓ ✕
Fixed points None Center Points on the line
Can prove congruence ✓ ✓ ✓

Question 1 of 5

Which of the following is not a rigid motion?

Question 2 of 5

If \(\triangle ABC \cong \triangle XYZ\), which statement must be true?

Question 3 of 5

Two triangles have all three pairs of corresponding angles equal, but one triangle is twice the size of the other. These triangles are:

Question 4 of 5

\(\triangle PQR\) is reflected over a line to produce \(\triangle P'Q'R'\). Which property does the reflection not preserve?

Question 5 of 5

A student writes: "I translated \(\triangle ABC\) by 5 units right, then dilated it by a factor of 2, so \(\triangle ABC \cong \triangle A'B'C'\)." What is the error?