Similar Triangles

Written by

Malcolm McKinsey

January 11, 2023

Fact-checked by

Paul Mazzola

Similar Triangles (Definition, Proving, & Theorems)

Similarity in mathematics does not mean the same thing that similarity in everyday life does. Similar triangles are triangles with the same shape but different side measurements.

Similar triangles definition

Mint chocolate chip ice cream and chocolate chip ice cream are similar, but not the same. This is an everyday use of the word "similar," but it not the way we use it in mathematics.

In geometry, two shapes are similar if they are the same shape but different sizes. You could have a square with sides 21 cm and a square with sides 14 cm; they would be similar. An equilateral triangle with sides 21 cm and a square with sides 14 cm would not be similar because they are different shapes.

Similar triangles are easy to identify because you can apply three theorems specific to triangles. These three theorems, known as Angle-Angle (AA), Side-Angle-Side (SAS), and Side-Side-Side (SSS), are foolproof methods for determining similarity in triangles.

1. Angle-Angle (AA)

2. Side-Angle-Side (SAS)

3. Side-Side-Side (SSS)

Corresponding angles

In geometry, correspondence means that a particular part on one polygon relates exactly to a similarly positioned part on another. Even if two triangles are oriented differently from each other, if you can rotate them to orient in the same way and see that their angles are alike, you can say those angles correspond.

The three theorems for similarity in triangles depend upon corresponding parts. You look at one angle of one triangle and compare it to the same-position angle of the other triangle.

Proportion

Similarity is related to proportion. Triangles are easy to evaluate for proportional changes that keep them similar. Their comparative sides are proportional to one another; their corresponding angles are identical.

You can establish ratios to compare the lengths of the two triangles' sides. If the ratios are congruent, the corresponding sides are similar to each other.

Included angle

The included angle refers to the angle between two pairs of corresponding sides. You cannot compare two sides of two triangles and then leap over to an angle that is not between those two sides.

Proving triangles similar

Here are two congruent triangles. To make your life easy, we made them both equilateral triangles.

△FOX is compared to △HEN. Notice that ∠O on △FOX corresponds to ∠E on △HEN. Both ∠O and ∠E are included angles between sides FO and OX on △FOX, and sides HE and EN on △HEN.

Side FO is congruent to side HE; side OX is congruent to side EN, and ∠O and ∠E are the included, congruent angles.

The two equilateral triangles are the same except for their letters. They are the same size, so they are identical triangles. If they both were equilateral triangles but side EN was twice as long as side HE, they would be similar triangles.

Triangle similarity theorems

Angle-Angle (AA) theorem

Angle-Angle (AA) says that two triangles are similar if they have two pairs of corresponding angles that are congruent. The two triangles could go on to be more than similar; they could be identical. For AA, all you have to do is compare two pairs of corresponding angles.

Trying angle-angle

Here are two scalene triangles △JAM and △OUT. We have already marked two of each triangle's interior angles with the geometer's shorthand for congruence: the little slash marks.

A single slash for interior ∠A and the same single slash for interior ∠U mean they are congruent. Notice ∠M is congruent to ∠T because they each have two little slash marks.

Since ∠A is congruent to ∠U, and ∠M is congruent to ∠T, we now have two pairs of congruent angles, so the AA Theorem says the two triangles are similar.

Tricks of the trade

Watch for trickery from textbooks, online challenges, and mathematics teachers. Sometimes the triangles are not oriented in the same way when you look at them. You may have to rotate one triangle to see if you can find two pairs of corresponding angles.

Another challenge: two angles are measured and identified on one triangle, but two different angles are measured and identified on the other one.

Because each triangle has only three interior angles, one each of the identified angles has to be congruent. By subtracting each triangle's measured, identified angles from 180°, you can learn the measure of the missing angle. Then you can compare any two corresponding angles for congruence.

Side-Angle-Side (SAS) theorem

The second theorem requires an exact order: a side, then the included angle, then the next side. The Side-Angle-Side (SAS) theorem states if two sides of one triangle are proportional to two corresponding sides of another triangle, and their corresponding included angles are congruent, the two triangles are similar.

Trying side-angle-side

Here are two triangles, side by side and oriented in the same way. △RAP and △EMO both have identified sides measuring 37 inches on △RAP and 111 inches on △EMO, and also sides 17 on △RAP and 51 inches on △EMO. Notice that the angle between the identified, measured sides is the same on both triangles: 47°.

Is the ratio $\frac{37}{111}$ the same as the ratio $\frac{17}{51}$? Yes; the two ratios are proportional, since they each simplify to $\frac{1}{3}$. With their included angle the same, these two triangles are similar.

Side-Side-Side (SSS) theorem

The last theorem is Side-Side-Side, or SSS. This theorem states that if two triangles have proportional sides, they are similar. This might seem like a big leap that ignores their angles, but think about it: the only way to construct a triangle with sides proportional to another triangle's sides is to copy the angles.

Trying side-side-side

Here are two triangles, △FLO and △HIT. Notice we have not identified the interior angles. The sides of △FLO measure 15, 20, and 25 cm in length. The sides of △HIT measure 30, 40, and 50 cm in length.

You need to set up ratios of corresponding sides and evaluate them:

They all are the same ratio when simplified. They all are $\frac{1}{2}$. So even without knowing the interior angles, we know these two triangles are similar, because their sides are proportional to each other.

Lesson summary

Now that you have studied this lesson, you are able to define and identify similar figures, and you can describe the requirements for triangles to be similar (they must either have two congruent pairs of corresponding angles, two proportional corresponding sides with the included corresponding angle congruent, or all corresponding sides proportional).

You also can apply the three triangle similarity theorems, known as Angle-Angle (AA), Side-Angle-Side (SAS) or Side-Side-Side (SSS), to determine if two triangles are similar.