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7. Similarity
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Geometry

7.03 Proving triangles similar

Lesson
Practice

Proving triangles similar

Exploration

Drag the triangles and move the sliders to explore how \triangle ABC and \triangle DEF change.

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  1. What do you notice about the corresponding parts of the triangles?

  2. Is it possible to create triangles such that the ratios of their corresponding parts are not proportional?

In the previous lesson, we verified that similar figures will have corresponding sides that are proportional and corresponding angles that are congruent. We can build on this to formalize similarity theorems for triangles. Two triangles can be shown to be similar using several different theorems involving their angles and sides.

Angle-Angle similarity (AA \sim) theorem

If two angles in a triangle are congruent to two corresponding angles in another triangle, then the triangles are similar

A larger triangle A B C, and a smaller triangle X Y Z. Angles A and Z are congruent, as well as angles B and Y.
Side-Angle-Side similarity (SAS \sim) theorem

If an angle in one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar

Two triangles. The larger triangle has 2 sides of length x and y. The angle formed by the two sides is marked. The smaller triangle has 2 sides of length a x and a y. The angle formed by the two sides is marked. The marked angles on the top and bottom triangles are congruent.
Side-Side-Side similarity (SSS \sim) theorem

If the lengths of the corresponding sides of a two triangles are proportional, then the triangles are similar

Two triangles. The larger triangle has sides of length x, y, and z. The smaller triangle has sides of length a x, a y, and a z.

Examples

Example 1

Explain why similarity theorems work using transformations.

a

Why can triangles be proven similar with only two corresponding congruent angles, and not all 3?

Worked Solution
Create a strategy

We need to consider why the AA similarity theorem works using transformations. We know that the triangle sum theorem states the sum of the angles in a triangle is 180 \degree.

Draw a simple diagram as an example and use it to show why AA similarity works.

Apply the idea

Given: \angle L \cong \angle X and \angle M \cong \angle Y, we can prove: \triangle{LMN} \sim \triangle{XYZ}

Triangles L M N and X Y Z. Angles L and X are congruent as well as angles M and Y.
To prove: \triangle{LMN} \sim \triangle{XYZ}
StatementsReasons
1.\begin{aligned} \angle L \cong \angle X \\ \angle M \cong \angle Y \end{aligned} Given
2.\begin{aligned} m\angle L =m \angle X \\ m \angle M = m \angle Y \end{aligned}Definition of congruent angles
3.\begin{aligned} m \angle L + m \angle M + m \angle N= 180 \\ m \angle X + m \angle Y + m \angle Z = 180 \end{aligned}Triangle sum theorem
4. m \angle L + m \angle M + m \angle N=m \angle X + m \angle Y + m \angle ZTransitive property of equality
5. m \angle X + m \angle Y + m \angle N=m \angle X + m \angle Y + m \angle ZCongruent angles
6.m \angle N = m \angle ZSubtraction property of equality
7.\triangle{LMN} \sim \triangle{XYZ}Since the triangles have three corresponding congruent angles, a similarity transformation exists that maps \triangle{LMN} to \triangle{XYZ}

Since we know transformations lead to similar figures, we can conclude that triangles can be proven similar with only two corresponding congruent angles.

b

Why can triangles be proven similar using only their side lengths?

Worked Solution
Create a strategy

We need to consider why the SSS similarity theorem works using transformations.

Apply the idea

Recall that two figures are similar when their corresponding angle measures are congruent and their corresponding side lengths are proportional.

If all corresponding side lengths from the pre-image to the image are proportional, then the figures are similar, based on the properties of a dilation transformation and we know that dilations preserve angle measure. For triangles, we only need to find a common scale factor between all three corresponding sides to show two triangles are similar.

Example 2

State whether Rectangle P\sim Rectangle Q. If so, describe the similarity transformation.

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Worked Solution
Create a strategy

To determine if the rectangles are similar, we will examine whether their corresponding sides are proportional and their corresponding angles are congruent. Rectangles have congruent opposite sides and all angles are right angles, so we'll focus on the proportionality of their sides.

Apply the idea

First, calculate the lengths of the sides for both rectangles.

For Rectangle P, the length from x=2 to x=6 is 4 units. The width from y=1 to y=3 is 2 units. For Rectangle Q, the length from x=7 to x=13 is 6 units. The width from y=4 to y=6 is 2 units.

Next, we will compare the ratios of the corresponding sides. The length ratio is \dfrac{4}{6} \left(\text{or }\dfrac{2}{3}\right), and the width ratio is \dfrac{2}{2} \left(\text{or }\dfrac{1}{1}\right).

The rectangles have the same width but different lengths. Since the ratios are not the same for both sets of corresponding sides, their side lengths are not proportional.

Rectangle P not similar Rectangle Q.

Reflect and check

Remember, dilating a figure will change all sides of a figure by the same factor, not just one dimension.

Any combination of dilations, rotations, translations, and reflections will create similar figures, so we can use transformations in the coordinate plane to justify similarity. If the sides of these rectangles were proportional, we would have described the transformations that justified their similarity.

Example 3

Determine whether the given pairs of triangles are similar. If so, state the theorem which proves their similarity. If not, explain how you know.

a
Two triangles. Left triangle has one side of length 4. Opposite the side of length 4 is an angle measureing 65 degrees. Adjacent to the 65 degreed angle is an angle measuring 85 degrees. Right triangle has one side of length 1.5. The two angles adjacent to the side of lenght 1.5 have measures of 85 degrees and 65 degrees.
Worked Solution
Create a strategy

We can use the given information about the angles in the two triangles to determine whether we can show that they are similar.

We are given two angle measures and a side length in each triangle. The side lengths given are not corresponding sides.

Apply the idea

Since there are two pairs of congruent angles, we can state that the triangles are similar using the AA similarity theorem.

b
Triangles A B C and P R Q. A C has a length of 12, A B has a length of 6, P R has a length of 4, and P Q has a length of 2. Angle A has a measure of 47 degrees, angle C has a measure of 29 degrees, angle P has a measure of 47 degrees, and angle Q has a measure of 104 degrees.
Worked Solution
Create a strategy

There are two angles in the triangle that are 47 \degree which, if the triangles are similar, indicates that these are corresponding angles. We will use this information to determine two pairs of corresponding sides, then use these three pairs of corresponding parts to determine similarity.

Apply the idea

First we must determine if the corresponding side lengths are proportional. Since \angle CAB and \angle RPQ have the same measure, we will assume these are corresponding angles. From this, we can say \overline{AC} corresponds to \overline{PR} and \overline{AB} corresponds to \overline{PQ}.

To check if the sides are proportional, we need to set up our corresponding ratios.

\dfrac{AB}{PQ}=\dfrac{6}{2}=3

\dfrac{AC}{PR}=\dfrac{12}{4}=3

This shows the corresponding side lengths are proportional with a scale factor of 3, and the angle measure between the side lengths is congruent. Therefore, the triangles are similar using the SAS similarity theorem.

c
Triangles A B C and D E F. A C has a length of 16, A B has a length of 25, B C has a length of 12. D E has a length of 24, E F has a length of 18, and D F has a length of 35.
Worked Solution
Create a strategy

We will need to use the side-side-side similarity theorem to show that the pair of triangles are similar, since the only parts of the triangles given are the side lengths.

If a common scale factor exists that will map one triangle to the other, we can state that the corresponding side lengths are proportional and that the triangles are similar.

Apply the idea

We can show that the triangles are similar by showing that their corresponding side lengths are proportional. First, we'll test \overline{BC} and \overline{EF} because they have the shortest side lengths in the triangles:\dfrac{BC}{EF} = \dfrac{12}{18} = \dfrac{2}{3}

We will test the next set of side lengths, \overline{AC} and \overline{DE} because they are the next shortest:\dfrac{AC}{DE} = \dfrac{16}{24}= \dfrac{2}{3}

Finally, if the last pair of sides have a proportional scale factor to the first two sets, we can confirm similarity. We have: \dfrac{AB}{DF}=\dfrac{25}{35}=\dfrac{5}{7}

Since there is not a common factor between the three sides of the triangles, the triangles are not similar.

d
Triangles P Q R and D E F. P Q has a length of 3.4, Q R has a length of 3, R P has a length of 2. D E has a length of 7.8, E F has a length of 8.84, F D has a length of 5.2.
Worked Solution
Create a strategy

We will need to show that the pair of triangles satisfies the side-side-side similarity theorem since there is no information given about the angle measures of the triangles.

Apply the idea

We can test each side that could be corresponding to find a common factor. Since the two shortest sides are 2 from \triangle{PQR} and 5.2 from \triangle{DEF}, \dfrac{DF}{PR}=\dfrac{5.2}{2}=2.6

Next, we will test the next-shortest set of corresponding sides \overline{QR} and \overline{DE}: \dfrac{DE}{QR}=\dfrac{7.8}{3}=2.6

Finally, we'll test the final pair of corresponding sides, \overline{PQ} and \overline{EF}: \dfrac{EF}{PQ}=\dfrac{8.84}{3.4}=2.6

Since the corresponding side lengths of the triangles have a common scale factor and are thus proportional, we can conclude that the triangles are similar by the SSS similarity theorem.

Reflect and check

We cannot rely on triangles to have corresponding parts in alphabetical order, so looking at the side lengths in ascending or descending order and pairing them up helps us look for a common factor.

For this pair of triangles, \triangle PQR \sim \triangle FED.

Example 4

Determine whether the pair of triangles are similar. If so, write a similarity statement and justify with a similarity theorem. If not, explain why not.

Triangle M L N is drawn. Point K lies on J M and Point L lies on J N. Segment K L is drawn. The following are the lengths of the segments: J K, 2; J L, 3; K M, 4; and L N, 6.
Worked Solution
Create a strategy

First, we should consider whether the given information is enough to establish similarity between the two triangles. In this case, we are given some side lengths, but also note that there is an angle common to both triangles.

\angle J is shared between the two triangles. We also see that JK is part of the total length of JM, and JL is part of the total length of JN. So, we will test \triangle JKL \sim \triangle JMN. We also know some side lengths on two pairs of sides, so it makes sense to test the SAS similarity theorem.

Apply the idea

Based on the given lengths, we can see that \dfrac{JK}{JM}=\dfrac{JL}{JN}: \dfrac{JK}{JM}= \dfrac{2}{(2+4)} = \dfrac{2}{6} = \dfrac{1}{3} \text{ and } \dfrac{JL}{JN} = \dfrac{3}{(3+6)}=\dfrac{3}{9} = \dfrac{1}{3}The included angle, \angle J belongs to both \triangle JMN and \triangle JKL.

Therefore, \triangle JMN \sim \triangle JKL by the SAS similarity theorem.

Reflect and check

A triangle like this that is split proportionally has special properties, which we will learn about in the next lesson.

Example 5

Find x. Show your work and justify your steps.

Triangle G C D with midpoint F on G D. A point E is outside side G D. Triangle D E F is drawn. C G has a length of 4, C D has a length of 5. Angle C G D has a measure of 54 degrees. E F has a length of 2.5, D E has a length of 2. Angle E D F has a measure of 5 x minus 6 degrees.
Worked Solution
Create a strategy

In order to find x, we're going to need to know the measure of \angle FDE. The only angle measure we're given is \angle CGD. It would be helpful if those two angles happen to be congruent. So, let's see if we can find similar triangles and match those corresponding angles.

Apply the idea

Notice that \dfrac{CG}{DE}=\dfrac{4}{2}=2, and \dfrac{CD}{EF} = \dfrac{5}{2.5} = 2. That gives us two corresponding sides with the same proportions, so now we need to test the third sides. From the diagram, we know DF = FG, so DG = 2DF and \dfrac{DG}{DF}=2.

Since \dfrac{CG}{DE} = \dfrac{CD}{EF} = \dfrac{DG}{DF}=2, we can use SSS similarity theorem to say \triangle GCD \sim \triangle DEF.

Since the triangles are similar, we know that their corresponding angle measures are congruent. \angle{CGD} \cong \angle {EDF}, so we can find x by solving the equation 5x-6=54

\displaystyle 5x-6\displaystyle =\displaystyle 54m \angle{CGD} = m \angle {EDF} by definition of congruent angles
\displaystyle 5x\displaystyle =\displaystyle 60Add 6 to both sides
\displaystyle x\displaystyle =\displaystyle 12Divide by 5 on both sides
Idea summary

Use the theorems about the angles and side lengths of triangles to prove their similarity:

  • Angle-angle similarity, or AA \sim: The two triangles have two pairs of congruent angles
  • Side-side-side similarity, or SSS \sim: The two triangles have three pairs of sides whose lengths are in the same proportion
  • Side-angle-side similarity, or SAS \sim: The two triangles have two pairs of sides whose lengths are in the same proportion, and the angles between these sides are also congruent

Outcomes

G.TR.3

The student will, given information in the form of a figure or statement, prove and justify two triangles are similar using direct and indirect proofs, and solve problems, including those in context, involving measured attributes of similar triangles.

G.TR.3a

Use definitions, postulates, and theorems (including Side-Angle-Side (SAS); Side-Side-Side (SSS); and Angle-Angle (AA)) to prove and justify that triangles are similar.

G.TR.3b

Use algebraic methods to prove that triangles are similar.

G.TR.3d

Describe a sequence of transformations that can be used to verify similarity of triangles located in the same plane.

G.TR.3e

Solve problems, including those in context involving attributes of similar triangles

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