Congruence and Similarity — Definitions & Examples

Written by

Malcolm McKinsey

January 11, 2023

Fact-checked by

Paul Mazzola

What are congruence and similarity?

All congruent figures are similar, but not all similar figures are congruent. Congruence means two objects (whether two dimensional or three dimensional) are identical in size and shape. Everything about them - their angles, lengths of sides, overall dimensions - are identical. Similar figures have the same shape and proportions but are not necessarily the same size.

Congruence

Two objects can be the same size and shape but not be oriented the same way. They are still congruent, like sea stars turned different ways.

Or consider chess pieces with one knight is on a high shelf and the other on a low shelf. Simply because they are in different planes in three dimensions does not rule out their congruence. They are still congruent.

In geometry, congruent figures have three properties:

1. Same size

2. Same shape

3. Corresponding parts are congruent

That last part is why, in geometry proofs, we sometimes see CPCFC, which means, "Corresponding parts of congruent figures are congruent."

Similarity

Similarity means the same shape and proportions, but not necessarily the same size. Angles of similar figures will be equal, but lengths of sides usually are not equal.

Pairs of shapes that are congruent are automatically similar, but this relationship does not work in reverse. All congruent figures are similar, but not all similar figures are congruent.

Both rectangles have the same proportions. All circles are similar. Neither pair of rectangles or circles is congruent, though. Only the squares, being congruent, are also similar to each other.

Using congruence and similarity

Knowing the properties of congruence and similarity allows you to use them in proofs. You can establish ratios between corresponding parts of two similar figures.

Draw two two rectangles, one measuring 5cm x 3 cm and the other measuring 2.5 cm x 1.5 cm.

Here the ratios of width to length are the same:

The ratios of corresponding parts are also the same:

Using these ratios and the congruence of angles, you know the shapes are similar. They are the same shape and are proportional to each other. You can use similarity in more challenging proofs.

Draw two overlapping triangles: △PYH and △TON. Here the ratios of width to length are the same:

• Given: Right $\triangle PYH$ and $\triangle TON$, $ST\cong SH$

• Prove: $\frac{PY}{NO}=\frac{PH}{NT}$

Statement Reason:

• $\angle PYT$ and $\angle TON$ are right angles (Given)

• $ST\cong SH$

• $\angle PYT\cong \angle TON$; Right angles are all $\cong$

• $\angle STH\cong \angle SHT$; If two sides of a $\triangle$ are $\cong$, $\angle$s opposite those sides are also $\cong$.

• $\triangle PYH\sim \triangle TON$ Angle Angle: If two $\angle$s of one $\triangle$ are $\cong$ to the two ∠s of another $\triangle$, the two $\triangle$s are similar.

• $\frac{PY}{NO}=\frac{PH}{NT}$ Corresponding sides of similar $\triangle$s are in proportion.