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Corresponding Sides — Definition, Formula & Examples

Corresponding sides are sides that occupy the same relative position in two figures that are similar or congruent. When you match up the vertices of both figures in order, corresponding sides connect the same pairs of matched vertices.

Given two polygons with a specified vertex correspondence, corresponding sides are the pairs of segments that join the same two matched vertices in each polygon. In similar figures, corresponding sides are proportional; in congruent figures, corresponding sides are equal in length.

How It Works

To identify corresponding sides, start by matching the vertices of the two figures in the order given. For example, if △ABC≅△DEF\triangle ABC \cong \triangle DEF, vertex AA matches DD, BB matches EE, and CC matches FF. Side AB‾\overline{AB} corresponds to DE‾\overline{DE} because both connect the first and second matched vertices. Side BC‾\overline{BC} corresponds to EF‾\overline{EF}, and AC‾\overline{AC} corresponds to DF‾\overline{DF}. The order in which the triangles are named tells you exactly which sides match up.

Worked Example

Problem: Triangle PQR is similar to triangle XYZ. If PQ = 6, QR = 9, PR = 12, and XY = 4, find the lengths of YZ and XZ.
Match vertices: The naming order tells us P↔X, Q↔Y, R↔Z. So PQ↔XY, QR↔YZ, and PR↔XZ.
Find the scale factor: Use the known corresponding pair PQ and XY.
XYPQ=46=23\frac{XY}{PQ} = \frac{4}{6} = \frac{2}{3}
Apply the ratio to find YZ and XZ: Multiply each side of triangle PQR by the scale factor.
YZ=9×23=6,XZ=12×23=8YZ = 9 \times \frac{2}{3} = 6, \quad XZ = 12 \times \frac{2}{3} = 8
Answer: YZ = 6 and XZ = 8.

Why It Matters

Setting up correct side correspondences is the first step in every similarity and congruence proof you will write in geometry. If you mismatch the sides, your proportions or equations will be wrong, leading to incorrect missing lengths or a failed proof.

Common Mistakes

Mistake: Matching sides by length alone instead of by vertex order.
Correction: Always use the stated vertex correspondence (the order the figures are named) to pair sides. Two sides may happen to be equal in length without actually corresponding.

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