Skill: 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.