Lesson Explainer: Triangle Similarity Criteria and Their Applications

In this explainer, we will learn how to use the properties of similar triangles to solve problems.

We can begin by understanding the meaning of similar.

Definition: Similar Triangles

Two triangles are similar if corresponding angles are congruent and the lengths of their corresponding sides are in the same proportion.

More colloquially, we might say that similar triangles are the same shape, but they can be a different size. As an aside, triangles that are the same shape and the same size are defined as congruent.

Below is an example of two triangles that are similar.

The angle pairs, and , and , and and , are of equal measures. The lengths of the corresponding sides, and , and , and and , are in the same proportion. In this example, we can also say that the scale factor from to is .

We will now investigate the geometry of similar triangles. We can take a triangle, such as the triangle .

Performing a dilation of scale factor would produce the following triangle .

In a dilation, all side lengths are multiplied by the scale factor, and all the angle measures are preserved. If the scale factor is greater than 1, then the figure is enlarged. And if the scale factor is less than 1, then the figure is reduced.

We can say that triangle is similar to triangle , and we can write this as

When we write the similarity relationship between triangles, the ordering of the lettering is important, as it indicates the angles and sides that correspond in the triangles.

If a given pair of triangles have equal corresponding angles, then the lengths of the corresponding sides are in the same proportion. And if the triangles have the lengths of the corresponding sides in the same proportion, then the corresponding angles are equal. In order to prove that two triangles are similar, rather than needing to prove that all corresponding angles are equal and all the lengths of corresponding sides are in proportion, there are a number of similarity criteria we can use.

The first criterion we could use is the angle–angle (AA) criterion.

Definition: Angle–Angle (AA) Similarity Criterion

If two pairs of corresponding angles in a pair of triangles are congruent, then the triangles are similar.

We can begin with the following pair of triangles.

Here, we are given that and . We can show that with just two given pairs of angles equal, the third pair of angles, and , must also be equal.

We can recall that the internal angles in a triangle sum to . Therefore, to work out the measure of angle , we could calculate

In , we could calculate the measure of as

Since we know that and , we can say that

Therefore,

Thus, when two pairs of corresponding angles in a triangle are congruent, the third pair of corresponding angles are also congruent and the triangles are similar.

The second similarity criterion involves the sides of the triangles.

Definition: Side-Side-Side (SSS) Similarity Criterion

If all three pairs of corresponding side lengths of two triangles are proportional, then the two triangles are similar.

We can apply this criterion in the following way. We can take the triangles and .

If we can demonstrate that the sides in these triangles have a proportional relationship such that then

For example, we can consider the triangles below, and .

We can write that since

The proportions of corresponding sides are all equivalent to . Hence,

Note that it would also be valid to write

We simply need to ensure that we keep all the sides of each triangle either as the numerators or as the denominators.

As there is an equivalent proportional relationship between all corresponding side lengths of the triangle, the triangles are similar.

We will now discuss the final similarity criterion.

Definition: Side-Angle-Side (SAS) Similarity Criterion

If the lengths of two sides in one triangle are proportional to the lengths of two sides in another triangle and the included angles in both are congruent, then the two triangles are similar.

We can illustrate this criterion with the following figure.

Here, we have two pairs of corresponding side lengths in the same proportion, since

Both of these proportions are equivalent to . And, in both triangles, the included angles between these sides are congruent. Thus,

To fulfill the criterion for the SAS rule, we just need two pairs of sides in proportion, but the pair of angles have to be the angles between these sides in each triangle. Note that this rule is different from the SAS congruency criterion, where we must demonstrate that corresponding sides are equal for triangles to be congruent.

We will now see some examples of how we can apply these similarity criteria to prove that a pair of triangles are similar.

Example 1: Proving Whether Two Triangles are Similar

The figure shows a triangle , where line segment is parallel to .

Which angle is equivalent to ? Why?
1. , because the angles are corresponding
2. , because the angles are alternate
3. , because the angles are alternate
4. , because the angles are corresponding
5. , because the angles are corresponding
Which angle is equivalent to ? Why?
1. , because the angles are corresponding
2. , because the angles are corresponding
3. , because the angles are alternate
4. , because the angles are alternate
5. , because the angles are corresponding
Hence, are triangles and similar? If yes, how?
1. Yes, they are similar by the SSS criterion.
2. Yes, they are similar by the SAS criterion.
3. Yes, they are similar by the AA criterion.
4. No, they are not similar.

Answer

Part 1

In the diagram, we observe that there is a pair of parallel line segments, and . Line is a transversal of these; hence, the angle that is equal to is

Part 2

To find the angle equivalent to , we use the properties of the angles in parallel lines, along with the transversal . Thus, the angle that is equivalent to is

Part 3

We have now demonstrated that there are two pairs of corresponding angles equal in triangles and :

The AA similarity criterion states that if two pairs of corresponding angles in a pair of triangles are congruent, then the triangles are similar. Therefore, we can give the answer that triangles and are similar by the AA criterion.

We will now look at another example.

Example 2: Proving Whether Two Triangles are Similar

The figure shows two triangles and , where line segment is parallel to .

Which angle is equivalent to ? Why?
1. , because the angles are corresponding
2. , because the angles are alternate
3. , because the angles are corresponding
4. , because the angles are alternate
5. , because the angles are vertically opposite
Which angle is equivalent to ? Why?
1. , because the angles are alternate
2. , because the angles are corresponding
3. , because the angles are corresponding
4. , because the angles are alternate
5. , because the angles are vertically opposite
Hence, are triangles and similar? If yes, how?
1. Yes, they are similar by the SSS criterion.
2. Yes, they are similar by the SAS criterion.
3. Yes, they are similar by the AA criterion.
4. No, they are not similar.

Answer

Part 1

In the diagram, the line segments and are marked as parallel. If we consider , then, using the properties of the angles in parallel lines cut by a transversal, we can identify that the angle that is its equivalent is

Part 2

Using the same properties, with the transversal , the angle that is equivalent to is

Part 3

We have shown that

We can recall the AA similarity criterion, which states that if two pairs of corresponding angles in a pair of triangles are equal, then the triangles are similar. Hence, we can give the answer that triangles and are similar by the AA criterion.

We can generalize the methods used in the previous two questions in the following corollary of triangle similarity.

Definition: Corollary of Triangle Similarity

If the lengths of two sides in one triangle are proportional to the lengths of two sides in another triangle and the included angles in both are congruent, then the two triangles are similar.

In each of the figures above, we can state that if and intersects and at and , respectively, then .

In the following example, we will calculate the scale factor between a pair of similar triangles.

Example 3: Finding the Similarity Scale Factor

In the figure, and . Since triangles and are similar, what is the scale factor?

Answer

We can begin by marking the given lengths onto the figure.

We are given that is similar to . We can recall that similar triangles have corresponding pairs of angles that are congruent and corresponding side lengths that are in proportion.

We can use the ordering of the letters to help us identify that the lengths of and are corresponding. Thus, we can write the proportion of their lengths from to as

We can give the answer as a fraction or a decimal; thus, the scale factor from triangle to triangle is 1.6.

Note that a scale factor in the reverse direction, from to , would be written as the proportion

We will now look at an example where we first need to prove that two triangles are similar and then use these properties to help us identify missing lengths.

Example 4: Calculating Unknowns Using Similarity and Finding the Perimeter of a Triangle

The figure shows triangle .

Work out the value of .
Work out the value of .
Work out the perimeter of .

Answer

Within the larger triangle , we observe that there is a smaller triangle, which we can label with points and to define .

In order to find the missing lengths, we first determine if and are similar. Similar triangles have corresponding angles that are congruent and corresponding side lengths that are in proportion.

We note that in the figure, and are marked as parallel. This means that we can identify two pairs of corresponding angles using the transversals and .

We can write that and that

The AA similarity criterion states that if two pairs of corresponding angles in a pair of triangles are congruent, then the triangles are similar. Alternatively, we could also have demonstrated that since angle is common to both triangles, then . Any two pairs out of these three pairs of angles would be sufficient to prove that

The proportion of the corresponding sides can be written as

We can now use this proportionality relationship to help us find the missing lengths, and .

Part 1

The length, , forms part of , and the corresponding side in triangle is the side .

When working with similar triangles, we usually have, or can calculate, the lengths of two corresponding sides. This allows us to find the proportion, or scale factor, between the given triangles. Here, we are given the lengths of and . Hence, we can write that

The length of can be written in terms of as . We substitute in the values for the lengths to give

Multiplying both sides by 6 and simplifying give

Hence, we have found the value of as 4.5.

Part 2

The unknown length, , is part of line segment . The corresponding side in is . We can use the proportionality relationship:

We can represent length as . Substituting in the values for the lengths and simplifying, we have

Therefore, the value of is 3.75.

Part 3

The perimeter is the distance around the outside of a shape. Thus, to work out the perimeter of , we have the following:

Substituting the lengths from the figure, we have

We calculated and ; therefore, we can simplify to give

We can then give the answer that the perimeter of is 26.25.

In the following question, we will use the similarity of triangles to allow us to form and solve algebraic equations in order to find an unknown length.

Example 5: Forming and Solving an Equation Using Similarity to Find an Unknown

Triangles and are similar. Find to the nearest integer.

Answer

Within the larger triangle, , in the figure, we observe that there is a smaller triangle, . We cannot immediately calculate the lengths of and ; however, we can use the given information that these two triangles are similar.

In similar triangles, the proportion of sides is the same. Hence, we can write the proportion of corresponding sides as

We can then substitute in the given lengths from the figure, being careful to note that . This gives us

Simplifying, we have

Therefore, we can give the answer that the value of is 5.

We will now see an example of how we can use the similarity of triangles to find measurements in a real-life situation. It is always helpful to first represent the given information in a sketch.

Example 6: Using the Similarity of Two Triangles to Find Indirect Measurements

A 1.97-metre-tall man stands 3.49 m away from a streetlight and casts a shadow that is 2.73 m long. How high is the lamp? Round your answer to the nearest tenth.

Answer

It might be useful to begin with a sketch of the information we have been given.

We can then model the situation using triangles, with a line joining the top of the light to the top of the shadow.

It is helpful to label the points. Here, we can define them as , , , , and . We can assume that the man and the streetlight meet the horizontal ground at and that the streetlight and the man are vertical and parallel.

We need to work out the height of the lamp, the length of . If we had the length of , we could apply the Pythagorean theorem. However, perhaps the best approach is to see if the triangle created with the light and the shadow, , is similar to the smaller triangle created by the man and the shadow, .

We recall that similar triangles have corresponding angles that are congruent and corresponding side lengths that are in proportion. One of the ways we can prove that two triangles are similar is by demonstrating that there are two pairs of corresponding angles that are congruent, that is, the AA criterion.

We can write that

As we have found two corresponding angles that are congruent, then we have proven that .

In order to work out the length of , we can use the corresponding side in , side . We know that the proportion between these sides will be the same as the proportion between the corresponding sides, and , whose lengths we are given.

Therefore,

We then substitute the length values, noting that . This gives us

The height of the lamp was defined as , and we round this value to the nearest tenth to give the answer for the height of the lamp, which is 4.5 m.

We can also note a particular instance of triangle similarity that involves an altitude of a right triangle.

Consider the following figure.

The largest triangle, right triangle , is divided into two smaller triangles. Note that forms an altitude of this triangle, since it is the perpendicular drawn from the vertex of the triangle to the opposite side.

Note that as the sum of the angle measures on a straight line is , we can also state that .

Let us consider the angles in these triangles, beginning with . We can sketch the 3 triangles separately, with the right angle in the same position.

Since is a common angle between and and they both have a angle, then by the AA similarity criterion,

Next, we observe that the angle at is common to both triangles and .

Since these triangles also both have an angle of , then by the AA similarity criterion,

When any two triangles are each similar to a third triangle, then all three triangles are similar. Hence,

Hence, we can define a second corollary of triangle similarity below.

Definition: Corollary of Triangle Similarity

In any right triangle, the altitude to the hypotenuse separates the triangle into two triangles that are similar to each other and similar to the original triangle.

We will now see how we can apply this corollary in the following example.

Example 7: Using the Similarity of Triangles Formed by the Altitude of a Right Triangle

The given figure shows a right triangle , where is perpendicular to .

Using similarity, express in terms of and .
Using similarity, express in terms of and .
Express the sum of and in terms of .

Answer

In the figure, we observe that is an altitude of the right triangle . We can recall that in any right triangle, the altitude to the hypotenuse separates the triangle into two triangles that are similar to each other and similar to the original triangle.

Therefore, we can write that

Part 1

We can sketch the 3 triangles separately and in the same orientation with corresponding angles aligned.

As we need a relationship between the sides of lengths , , and , we use and to form a similarity relationship.

We recognize that when triangles are similar, corresponding sides are in the same proportion. So, we have that

We then simplify this equation to give

Thus, we have expressed in terms of and .

Part 2

We use and to form the next similarity relationship, for the sides of lengths , , and .

Hence, we have

And so, we have expressed in terms of and .

Part 3

We can write the sum of and by substituting the values and that we calculated in parts 1 and 2. This gives us

Using the figure, we can see that ; hence,

In fact, we may recognize this to be valid using the Pythagorean theorem. This states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Using similarity, we have demonstrated the validity of this theorem. We can express the sum of and in terms of as

We now summarize the key points.

Key Points

Two triangles are similar if corresponding angles are congruent and corresponding side lengths are in proportion.
We can prove that two triangles are similar using one of the following similarity criteria:
- Angle–Angle (AA): If two pairs of corresponding angles in two triangles are congruent, then the triangles are similar.
- Side-Side-Side (SSS): If all three pairs of corresponding side lengths of two triangles are proportional, then the two triangles are similar.
- Side-Angle-Side (SAS): If two side lengths in one triangle are proportional to two side lengths in another triangle and the included angles in both are congruent, then the two triangles are similar.
In any right triangle, the altitude to the hypotenuse separates the triangle into two triangles that are similar to each other and similar to the original triangle.
We can model real-life situations involving similar triangles by drawing a sketch and using the proportionality of sides and the equivalency of angles to find unknown measurements.