Similar triangles are characterized by having corresponding sides with the same proportions, but not necessarily with the same measurements. On the other hand, congruent triangles have corresponding sides with the same measurements. This means that congruent triangles share the same shape and size, while similar triangles only share the same shape.

Here, we will learn more about similar and congruent triangles.

## Definition of similar triangles and congruent triangles

**Congruence** of two figures means that the figures are exactly the same. For example, when we have two line segments with the same length, the segments are congruent. Similarly, for two geometric figures to be congruent, they must have the same shape and size.

The triangles ABC and DEF are congruent. We can see that these triangles are exactly the same. The triangles have the same shape and size. Therefore, all the lengths of its sides are equal, as are all the measures of its angles.

For two figures to be **similar**, they must have the same shape, but not necessarily the same size. Therefore, similar figures are not congruent.

The triangles PQR and XYZ are similar. We can see that they are exactly the same shape, but they have different sizes.

## Congruent triangle criteria

For two triangles to be congruent, they must meet one of the following criteria:

**(Side, Side, Side):** The three pairs of corresponding sides are the same.

**(Side, Angle, Side):** Two pairs of corresponding sides and the angles between those sides are equal.

**(Angle, Side, Angle):** Two pairs of corresponding angles and the sides between those angles are equal.

**(Angle, angle, side):** Two pairs of corresponding angles and one pair of corresponding sides, which are not between the angles, are equal

**(Hypotenuse, side):** In right triangles, the hypotenuses are equal and another pair of corresponding sides are also equal.

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## Similar triangle criteria

For two triangles to be similar, they must meet one of the following criteria:

**(Angle, Angle):** Two pairs of corresponding angles are equal.

**(Side, Side, Side):** The three corresponding pairs of sides have the same proportions.

**(Side, Angle, Side):** Two pairs of corresponding sides are proportional and the angles between them are equal.

## Solved problems of similar and congruent triangles

### EXAMPLE 1

Determine if the following triangles are similar, congruent, or neither.

**Solution:** We know the measures of two corresponding sides of the triangles. We see that these measurements are the same. Also, we know the measure of a pair of corresponding angles. However, the angle that we know is opposite to one of the sides that we also know.

We do not know the measure of the third side, nor its opposite angle, so we cannot determine if the triangles are similar or congruent.

### EXAMPLE 2

Are the following triangles similar, congruent, or neither?

**Solution:** We know the measures of two pairs of corresponding angles. The measurements are the same. This means that the triangles are similar. We cannot determine if the triangles are congruent since we do not know the measures of their sides.

### EXAMPLE 3

Determine if the triangles are similar, congruent, or neither.

**Solution:** We know the measures of a corresponding pair of angles and the lengths of a corresponding pair of sides. The sides are opposite the known angle. Therefore, we cannot determine whether these triangles are similar or congruent.

## See also

Interested in learning more about similar figures, congruency, and other geometry topics? Take a look at these pages:

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