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Corresponding Sides in Geometry - Definition, Examples

TL;DR

Corresponding sides are sides in the same relative position in two figures that are either similar or congruent. In congruent figures corresponding sides are equal; in similar figures they are proportional - their ratio is constant. This article shows how to match them from vertex order, how to use the ratio to find a missing length, and the mistakes to avoid.

What Are Corresponding Sides In Geometry?

Corresponding sides are a pair of matching sides that sit in the same relative position in two different figures, where the two figures must be either congruent (same shape and size) or similar (same shape, possibly different size). Each side in one figure has a partner - its corresponding side - in the other.

What the pairing tells you depends on the relationship:

• Congruent figures - corresponding sides are equal in length, and corresponding angles are equal too.
• Similar figures - corresponding sides are proportional: the ratio of every matched pair is the same number.

The matching itself is not arbitrary. Corresponding sides are the sides opposite equal (corresponding) angles. Match the angles first, and the sides fall into place.

How Do You Identify Corresponding Sides?

There are two reliable ways to pair sides, and they agree with each other.

Method 1 - Match the angles. Corresponding sides lie opposite equal angles. Find which angle in the first figure equals which angle in the second, then the sides across from those equal angles are partners.

Method 2 - Read the vertex order. When a statement names the figures in matched order - "△ABC∼△DEF" or "△ABC≅△DEF" — the order is the correspondence. The letters line up position by position:

• A↔D, B↔E, C↔F (vertices)
• side AB↔DE, side BC↔EF, side AC↔DF (sides)

The symbol ∼ means "is similar to"; the symbol ≅ means "is congruent to." The order of the letters is doing real work - it is not decoration.

The similarity ratio

For similar figures, the constant ratio between corresponding sides is called the similarity ratio (or scale factor). If △ABC∼△DEF, then:

[ \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k ]
where k is the scale factor. To find a missing length, set two of these equal ratios in a proportion and solve.

Where Do Corresponding Sides Show Up?

Corresponding sides are the working part of two big ideas: congruence and similarity. Once figures are known to be similar or congruent, corresponding sides let you transfer measurements from one to the other without re-measuring.

• Scale models and maps - a floor plan drawn at 1:100 has every wall corresponding to a real wall 100 times longer. Read one, compute the other.
• Congruence proofs - the rule CPCTC ("corresponding parts of congruent triangles are congruent") is used constantly once two triangles are proven congruent by a rule such as SAS.
• Indirect measurement - the height of a tall tree or building is found by comparing its shadow to the shadow of a stick of known height, using similar triangles.

Examples Of Corresponding Sides

Example 1

Triangle ABC is congruent to triangle DEF. If AB = 7 cm, what is the length of DE?

Since △ABC≅△DEF, the vertex order gives AB↔DE.

In congruent figures, corresponding sides are equal.

DE = 7 cm.

Example 2

Triangle ABC is similar to triangle PQR, with AB = 4, BC = 6, and PQ = 8. Find QR.

The correct method uses the ratio. Since △ABC∼△PQR: [ \frac{AB}{PQ} = \frac{BC}{QR} ]
Cross-multiply to find [ QR = 12 ].

Example 3

Two similar triangles have corresponding sides in the ratio 3:5. If a side of the smaller triangle is 9 cm, find the corresponding side of the larger.

Let the larger side be x. The ratio of corresponding sides is constant:

[ \frac{3}{5} = \frac{9}{x} ]
Cross-multiply: [ x = 15 ].

Example 4

Given △ABC ∼ △DEF with AB = 6, DE = 9, and DF = 15, find AC.

The scale factor from ABC to DEF is [ \frac{DE}{AB} = \frac{9}{6} = \frac{3}{2} ].

Corresponding sides obey the same ratio: [ AC = \frac{2}{3} \times DF \Rightarrow AC = 10 ].

Example 5

Are two triangles with sides 6, 8, 10 and 9, 12, 15 similar?

All three ratios equal ( \frac{2}{3} ), so the corresponding sides are proportional. The triangles are similar, with similarity ratio 2/3.

Example 6

A stick casts a 2 m shadow. A flagpole casts a 10 m shadow. Find the flagpole's height.

The height corresponds to height, shadow to shadow:

[ \frac{1.5}{2} = \frac{h}{10} ]
Final answer: The flagpole is 7.5 m tall.

Why Corresponding Sides Matter - "Same shape, any size"

The idea is old and practical: Measuring the unreachable - the Greek mathematician Thales of Miletus is said to have found the height of a pyramid using corresponding sides of similar triangles.

What Are the Most Common Mistakes With Corresponding Sides?

Mistake 1: Ignoring the vertex order

Read a statement like "△ABC∼△DEF" and pairing sides by size or by eye instead of by the letter order.

Mistake 2: Adding instead of multiplying to scale

Finding a missing side in similar figures by adding a constant instead of multiplying will lead to an incorrect solution.

Mistake 3: Comparing sides that aren't corresponding

Setting up the proportion with mismatched pairs can lead to wrong answers.

Conclusion

• Corresponding sides are matching sides in the same relative position in similar or congruent figures.
• In congruent figures they are equal; in similar figures they are proportional.
• Match them by equal angles or by the vertex order in the similarity/congruence statement.
• The constant ratio of corresponding sides in similar figures is the similarity ratio (scale factor).
• Scale by multiplying by the ratio, never by adding a fixed amount.