Identify congruent figures

Congruent figures have the same shape and size: corresponding side lengths and angle measures are equal, even when one figure is translated, rotated, or reflected and therefore has a different position or orientation. Congruence can be identified from visual, geometric, or coordinate representations by matching corresponding points and determining whether a sequence of rigid motions maps one figure exactly onto the other; dilation and formal proof with advanced theorems are not included.

Detailed Explanation: Identify congruent figures

To identify whether two figures are congruent, check whether they have the same shape and size. A figure can be moved by a translation, rotation, or reflection without changing its size or shape.

Example

Triangle $ABC$ has vertices

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

Triangle $A'B'C'$ has vertices

A'(6,4),\quad B'(9,4),\quad C'(6,6).

Determine whether the triangles are congruent.

Step 1: Match corresponding points

The labels show the matching points:

$A$ corresponds to $A'$
$B$ corresponds to $B'$
$C$ corresponds to $C'$

Step 2: Check for the same movement

Compare each point with its corresponding point:

A(1,1)\to A'(6,4)

This moves $5$ units right and $3$ units up.

Check the other points:

B(4,1)\to B'(9,4)

This also moves $5$ units right and $3$ units up.

C(1,3)\to C'(6,6)

This also moves $5$ units right and $3$ units up.

Every point moves by the same amount, so triangle $ABC$ is translated to form triangle $A'B'C'$ .

Step 3: State the conclusion

A translation does not change side lengths or angle measures. Therefore,

\triangle ABC \cong \triangle A'B'C'.

The triangles are congruent because one can be moved exactly onto the other without stretching or resizing it.

Learn by doing: Identify congruent figures

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2D Shape Congruent or Similar - Image and Property to Image

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