Identify congruent figures after transformations

Congruent figures have the same size and shape because one can be mapped to the other by a sequence of translations, rotations, and/or reflections; these rigid transformations preserve corresponding side lengths, angle measures, and overall distances, although a reflection reverses orientation. Learners identify congruence by matching corresponding vertices and comparing figures in diagrams or on the coordinate plane, without requiring figures to occupy the same position or introducing more advanced transformation rules.

Detailed Explanation: Identify congruent figures after transformations

To identify whether two figures are congruent, check whether one figure can be moved to the other using only:

a translation (slide),
a rotation (turn), and/or
a reflection (flip).

These are rigid transformations, so they do not change side lengths or angle measures. The figures may be in different locations.

Example

Triangle $ABC$ has vertices

A(1,1), \quad B(4,1), \quad C(1,3).

Triangle $DEF$ has vertices

D(6,2), \quad E(9,2), \quad F(6,4).

Determine whether the triangles are congruent.

Step 1: Match the corresponding vertices.

The right angle in $\triangle ABC$ is at $A$ , because $\overline{AB}$ is horizontal and $\overline{AC}$ is vertical. The right angle in $\triangle DEF$ is at $D$ .

So match:

A \leftrightarrow D.

The other vertices match as follows:

B \leftrightarrow E, \qquad C \leftrightarrow F.

Step 2: Look for a rigid transformation.

From $A(1,1)$ to $D(6,2)$ , move $5$ units right and $1$ unit up.

Check the other vertices:

$B(4,1)$ moved $5$ right and $1$ up becomes $E(9,2)$ .
$C(1,3)$ moved $5$ right and $1$ up becomes $F(6,4)$ .

Thus, $\triangle ABC$ can be translated to exactly match $\triangle DEF$ .

Step 3: State the conclusion.

The triangles are congruent:

\triangle ABC \cong \triangle DEF.

Their corresponding side lengths and angles are the same; only their positions have changed. When naming congruent figures, list corresponding vertices in the same order.

Learn by doing: Identify congruent figures after transformations

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Multiple Transformations - Grid Image & Transformations to Similar or Congruent

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