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Congruence Through
Transformations

Learn how translations, reflections, and rotations prove that two figures are congruent: same shape, same size, just moved.

8th Grade Math

What Is Congruence?

Two figures are congruent when they have exactly the same shape and same size. Every side length matches, and every angle matches. Think of it like making a perfect copy of a shape and placing it somewhere else on the page.

The Three Rigid Transformations

Translation
Slide every point the same distance in the same direction: no flipping, no turning.
Reflection
Flip the figure over a line of reflection: creates a mirror image.
Rotation
Turn the figure around a fixed center point by a given angle.

Why Transformations Matter

In earlier grades, you checked congruence by measuring every side and angle. Transformations give you a more powerful tool: if you can describe a sequence of rigid motions that maps one figure onto the other, you've proved they're congruent without measuring anything.

A B C translate A' B' C' ≅

The Congruence Rule

This means you might need more than one transformation. For example, you might translate then reflect. The order matters: try different sequences until the figures overlap perfectly.

Original translate reflect Image (intermediate)

Visualizing Each Transformation

Translation Slide (no turn) Reflection Flip over a line Rotation Turn around a point All three preserve side lengths and angle measures Rigid Motions = Congruence

Congruent vs. Similar

Congruent Similar
Shape Same Same
Size Same Can differ
Angles All equal All equal
Sides All equal Proportional
Transformations Rigid motions only Rigid motions + dilation
Symbol ≅ ~

How to Describe a Sequence

1
Identify corresponding parts. Match each vertex of the first figure to a vertex of the second.
2
Check orientation. If the figures face opposite directions, you need a reflection.
3
Check position. If one figure is shifted left, right, up, or down from the other, you need a translation.
4
Check angle of alignment. If one figure is turned compared to the other, you need a rotation. State the center and degree.
5
Write the full sequence. List transformations in order: e.g., "Reflect over the \( y \)-axis, then translate 3 units right."

Example 1: Translation

A B C A' B' C'

Example 2: Reflection

y-axis P Q R P' Q' R'

Example 3: Rotation

O (center) D E F D' E' F' 90°

Combining Multiple Steps

Step 0: Original Step 1: Reflect Step 2: Translate G H I G' H' I' G'' H'' I'' Reflect over a vertical line, then translate right

Coordinate Notation

When figures are on a coordinate plane, we describe transformations using precise notation:

Translation
\( (x, y) \to (x + a,\, y + b) \)
Shifts by \( a \) horizontally, \( b \) vertically
Reflection
Over \( x \)-axis: \( (x, y) \to (x, -y) \)
Over \( y \)-axis: \( (x, y) \to (-x, y) \)
Rotation about O
\( 90° \) CW: \( (x, y) \to (y, -x) \)
\( 180° \): \( (x, y) \to (-x, -y) \)

Example 4: Coordinate Proof

\( \triangle JKL \) has vertices \( J(1, 2) \), \( K(4, 2) \), \( L(4, 6) \). \( \triangle J'K'L' \) has vertices \( J'(-1, 2) \), \( K'(-4, 2) \), \( L'(-4, 6) \). Describe a sequence of transformations.

1
Compare coordinates: each \( x \)-coordinate is negated, but \( y \)-coordinates stay the same.
2
Identify the rule: \( (x, y) \to (-x, y) \) is a reflection over the \( y \)-axis.
3
Conclusion: One reflection maps \( \triangle JKL \) onto \( \triangle J'K'L' \), so \( \triangle JKL \cong \triangle J'K'L' \).

Common Mistakes

✕ Including Dilation
Dilation changes size, so it is not a rigid motion. If you dilate, you get similarity, not congruence.
✕ Wrong Vertex Matching
Writing \( \triangle ABC \cong \triangle EDF \) when the correct correspondence is \( A \to D \), \( B \to E \), \( C \to F \). Order matters.
✕ Forgetting Orientation
Two figures may look positioned correctly but have opposite orientations. You must include a reflection to fix the flip.

Summary of Transformations

Transformation What It Does Preserves Changes
Translation Slides the figure Size, shape, orientation Position only
Reflection Flips over a line Size, shape Orientation (mirror)
Rotation Turns around a point Size, shape Position, direction
Dilation Scales the figure Shape, angles Size: NOT rigid

Question 1 of 5

Which of the following transformations is not a rigid motion?

Question 2 of 5

\( \triangle ABC \) has vertices \( A(2, 3) \), \( B(5, 3) \), \( C(5, 7) \). After a transformation, the image is \( A'(2, -3) \), \( B'(5, -3) \), \( C'(5, -7) \). What single transformation was performed?

Question 3 of 5

Two figures are congruent. Which statement must be true?

Question 4 of 5

A triangle is reflected over the \( y \)-axis, then translated 4 units up. The resulting image is congruent to the original because:

Question 5 of 5

\( \triangle MNO \) has vertices \( M(0, 0) \), \( N(4, 0) \), \( O(0, 3) \). Which sequence maps it to \( \triangle M'N'O' \) with \( M'(0, 0) \), \( N'(0, 4) \), \( O'(-3, 0) \)?