Skill: Determine corresponding sides and angles in similar figures

To find corresponding parts of similar figures:

1. Match angles with the same measure.
2. Match each vertex to the vertex with its equal angle.
3. Match sides by looking at their endpoints: a side corresponds to the side connecting the two matching vertices.
4. Remember that figures may be rotated, reflected, enlarged, or reduced, so do not match parts only by their position on the page.

Example

Triangles $\triangle ABC$ and $\triangle DEF$ are similar. Their angle measures are:

$$\angle A=50^\circ,\quad \angle B=70^\circ,\quad \angle C=60^\circ$$

and

$$\angle D=70^\circ,\quad \angle E=60^\circ,\quad \angle F=50^\circ.$$

Determine the corresponding angles and sides.

Step 1: Match the angles.

Match equal angle measures:

• $\angle A=50^\circ$ corresponds to $\angle F=50^\circ$.
• $\angle B=70^\circ$ corresponds to $\angle D=70^\circ$.
• $\angle C=60^\circ$ corresponds to $\angle E=60^\circ$.

So the vertex correspondence is

$$A\leftrightarrow F,\qquad B\leftrightarrow D,\qquad C\leftrightarrow E.$$

Step 2: Match the sides using their endpoints.

Since $A$ matches $F$ and $B$ matches $D$, side $\overline{AB}$ corresponds to $\overline{FD}$:

$$\overline{AB}\leftrightarrow\overline{FD}.$$

Using the other pairs of matching vertices:

$$\overline{BC}\leftrightarrow\overline{DE}$$

and

$$\overline{AC}\leftrightarrow\overline{FE}.$$

Therefore, the corresponding parts are

$$\angle A\leftrightarrow\angle F,\quad \angle B\leftrightarrow\angle D,\quad \angle C\leftrightarrow\angle E$$

and

$$\overline{AB}\leftrightarrow\overline{FD},\quad \overline{BC}\leftrightarrow\overline{DE},\quad \overline{AC}\leftrightarrow\overline{FE}.$$

The side correspondence can also be checked by matching opposite angles. For example, $\overline{AB}$ is opposite $\angle C$, so it corresponds to the side opposite $\angle E$, which is $\overline{FD}$.