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Determine corresponding sides and angles in similar figures

Similar figures have the same angle measures and proportional corresponding side lengths, even when one figure is enlarged, reduced, rotated, or reflected. The correspondence is determined by matching vertices, angles, and sides in the same relative order; a side corresponds only to the side opposite the matching angle or occupying the equivalent position. This understanding supports scale-factor reasoning, proportional calculations, and similarity proofs.

Detailed Explanation: 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 â–³ABC\triangle ABC and â–³DEF\triangle DEF are similar. Their angle measures are:

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

and

∠D=70∘,∠E=60∘,∠F=50∘.\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:

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

So the vertex correspondence is

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

Step 2: Match the sides using their endpoints.

Since AA matches FF and BB matches DD, side AB‾\overline{AB} corresponds to FD‾\overline{FD}:

AB‾↔FD‾.\overline{AB}\leftrightarrow\overline{FD}.

Using the other pairs of matching vertices:

BC‾↔DE‾\overline{BC}\leftrightarrow\overline{DE}

and

AC‾↔FE‾.\overline{AC}\leftrightarrow\overline{FE}.

Therefore, the corresponding parts are

∠A↔∠F,∠B↔∠D,∠C↔∠E\angle A\leftrightarrow\angle F,\quad \angle B\leftrightarrow\angle D,\quad \angle C\leftrightarrow\angle E

and

AB‾↔FD‾,BC‾↔DE‾,AC‾↔FE‾.\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, AB‾\overline{AB} is opposite ∠C\angle C, so it corresponds to the side opposite ∠E\angle E, which is FD‾\overline{FD}.

Learn by doing: Determine corresponding sides and angles in similar figures

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