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# 7.4: Similarity by SSS and SAS

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Use SSS and SAS to determine whether triangles are similar.
• Apply SSS and SAS to solve problems about similar triangles.

## Review Queue

1. What are the congruent angles? List each pair.
2. Write the similarity statement.
3. If \begin{align*}AB = 8, BD = 20,\end{align*} and \begin{align*}BC = 25\end{align*}, find \begin{align*}BE\end{align*}.
1. Solve the following proportions.
1. \begin{align*}\frac{6}{8}=\frac{21}{x}\end{align*}
2. \begin{align*}\frac{x+2}{6}=\frac{2x-1}{15}\end{align*}
3. \begin{align*}\frac{x-3}{9}=\frac{4}{x+2}\end{align*}

Know What? Recall from Chapter 2, that the game of pool relies heavily on angles. In Section 2.5, we discovered that \begin{align*}m \angle 1 = m \angle 2\end{align*}.

The dimensions of a pool table are 92 inches by 46 inches. You decide to hit the cue ball so it follows the yellow path to the right. The horizontal and vertical distances are in the picture. Are the two triangles similar? Why? How far did the cue ball travel to get to the red ball?

Link for an interactive game of pool: http://www.coolmath-games.com/0-poolgeometry/index.html

## SSS for Similar Triangles

If you do not know any angle measures, can you say two triangles are similar? Let’s investigate this to see. You will need to recall Investigation 4-2, Constructing a Triangle, given Three Sides.

Investigation 7-2: SSS Similarity

Tools Needed: ruler, compass, protractor, paper, pencil

1. Using Investigation 4-2, construct a triangle with sides 6 cm, 8 cm, and 10 cm.
2. Construct a second triangle with sides 9 cm, 12 cm, and 15 cm.
3. Using your protractor, measure the angles in both triangles. What do you notice?
4. Line up the corresponding sides. Write down the ratios of these sides. What happens?

To see an animated construction of this, click: http://www.mathsisfun.com/geometry/construct-ruler-compass-1.html

From #3, you should notice that the angles in the two triangles are equal. Second, when the corresponding sides are lined up, the sides are all in the same proportion, \begin{align*}\frac{6}{9}=\frac{8}{12}=\frac{10}{15}\end{align*}. If you were to repeat this activity, for a 3-4-5 or 12-16-20 triangle, you will notice that they are all similar. That is because, each of these triangles are multiples of 3-4-5. If we generalize what we found in this investigation, we have the SSS Similarity Theorem.

SSS Similarity Theorem: If the corresponding sides of two triangles are proportional, then the two triangles are similar.

Example 1: Determine if any of the triangles below are similar.

Solution: Compare two triangles at a time. In the proportions, place the shortest sides over each other, the longest sides over each other, and the middle sides over each other. Then, determine if the proportions are equal.

\begin{align*}\triangle ABC\end{align*} and \begin{align*}\triangle DEF\end{align*}: \begin{align*}\frac{20}{15}=\frac{22}{16}=\frac{24}{18}\end{align*}

Reduce each fraction to see if they are equal. \begin{align*}\frac{20}{15}=\frac{4}{3}, \frac{22}{16}=\frac{11}{8}\end{align*}, and \begin{align*}\frac{24}{18}=\frac{4}{3}\end{align*}. Because \begin{align*}\frac{4}{3} \neq \frac{11}{8}, \triangle ABC\end{align*} and \begin{align*}\triangle DEF\end{align*} are not similar.

\begin{align*}\triangle DEF\end{align*} and \begin{align*}\triangle GHI\end{align*}: \begin{align*}\frac{15}{30}=\frac{16}{33}=\frac{18}{36}\end{align*}

\begin{align*}\frac{15}{30}=\frac{1}{2}, \frac{16}{33}=\frac{16}{33},\end{align*} and \begin{align*}\frac{18}{36}=\frac{1}{2}\end{align*}. Because \begin{align*}\frac{1}{2} \neq \frac{16}{33}, \triangle DEF\end{align*} is not similar to \begin{align*}\triangle GHI\end{align*}.

\begin{align*}\triangle ABC\end{align*} and \begin{align*}\triangle GHI\end{align*}: \begin{align*}\frac{20}{30}=\frac{22}{33}=\frac{24}{36}\end{align*}

\begin{align*}\frac{20}{30}=\frac{2}{3}, \frac{22}{33}=\frac{2}{3},\end{align*} and \begin{align*}\frac{24}{36}=\frac{2}{3}\end{align*}. Because all three ratios reduce to \begin{align*}\frac{2}{3}, \triangle ABC \sim \triangle GIH\end{align*}.

Example 2: Algebra Connection Find \begin{align*}x\end{align*} and \begin{align*}y\end{align*}, such that \begin{align*}\triangle ABC \sim \triangle DEF\end{align*}.

Solution: According to the similarity statement, the corresponding sides are: \begin{align*}\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}\end{align*}. Substituting in what we know, we have:

\begin{align*}\frac{9}{6} = \frac{4x-1}{10}=\frac{18}{y}\end{align*}

\begin{align*}\frac{9}{6} &= \frac{4x-1}{10} && \frac{9}{6}=\frac{18}{y}\\ 9(10) &= 6(4x-1) && 9y=18(6)\\ 90 &= 24x-6 && 9y=108\\ 96 &= 24x && \ y=12\\ x &= 4\end{align*}

## SAS for Similar Triangles

SAS is the last way to show two triangles are similar. If we know that two sides are proportional AND the included angles are congruent, then the two triangles are similar. For the following investigation, you will need to use Investigation 4-3, Constructing a Triangle with SAS.

Investigation 7-3: SAS Similarity

Tools Needed: paper, pencil, ruler, protractor, compass

1. Using Investigation 4-3, construct a triangle with sides 6 cm and 4 cm and the included angle is \begin{align*}45^\circ\end{align*}.
2. Repeat Step 1 and construct another triangle with sides 12 cm and 8 cm and the included angle is \begin{align*}45^\circ\end{align*}.
3. Measure the other two angles in both triangles. What do you notice?
4. Measure the third side in each triangle. Make a ratio. Is this ratio the same as the ratios of the sides you were given?

SAS Similarity Theorem: If two sides in one triangle are proportional to two sides in another triangle and the included angle in the first triangle is congruent to the included angle in the second, then the two triangles are similar.

In other words,

If \begin{align*}\frac{AB}{XY}=\frac{AC}{XZ}\end{align*} and \begin{align*}\angle A \cong \angle X\end{align*}, then \begin{align*}\triangle ABC \sim \triangle XYZ\end{align*}.

Example 3: Are the two triangles similar? How do you know?

Solution: \begin{align*}\angle B \cong \angle Z\end{align*} because they are both right angles. Second, \begin{align*}\frac{10}{15}=\frac{24}{36}\end{align*} because they both reduce to \begin{align*}\frac{2}{3}\end{align*}. Therefore, \begin{align*}\frac{AB}{XZ}=\frac{BC}{ZY}\end{align*} and \begin{align*}\triangle ABC \sim \triangle XZY\end{align*}.

Notice with this example that we can find the third sides of each triangle using the Pythagorean Theorem. If we were to find the third sides, \begin{align*}AC = 39\end{align*} and \begin{align*}XY = 26\end{align*}. The ratio of these sides is \begin{align*}\frac{26}{39}=\frac{2}{3}\end{align*}.

Example 4: Are there any similar triangles? How do you know?

Solution: \begin{align*}\angle A\end{align*} is shared by \begin{align*}\triangle EAB\end{align*} and \begin{align*}\triangle DAC\end{align*}, so it is congruent to itself. If \begin{align*}\frac{AE}{AD}=\frac{AB}{AC}\end{align*} then, by SAS Similarity, the two triangles would be similar.

\begin{align*}\frac{9}{9+3} &= \frac{12}{12+5}\\ \frac{9}{12} &= \frac{3}{4} \neq \frac{12}{17}\end{align*} Because the proportion is not equal, the two triangles are not similar.

Example 5: From Example 4, what should \begin{align*}BC\end{align*} equal for \begin{align*}\triangle EAB \sim \triangle DAC\end{align*}?

Solution: The proportion we ended up with was \begin{align*}\frac{9}{12}=\frac{3}{4} \neq \frac{12}{17}\end{align*}. \begin{align*}AC\end{align*} needs to equal 16, so that \begin{align*}\frac{12}{16}=\frac{3}{4}\end{align*}. Therefore, \begin{align*}AC = AB + BC\end{align*} and \begin{align*}16 = 12 + BC\end{align*}. \begin{align*}BC\end{align*} should equal 4 in order for \begin{align*}\triangle EAB \sim \triangle DAC\end{align*}.

## Similar Triangles Summary

Let’s summarize what we’ve found that guarantees two triangles are similar.Two triangles are similar if and only if:

Name Description Picture
AA Two angles in one triangle are congruent to two angles in another triangle.
SSS for Similar Triangles All three sides in one triangle are proportional to three sides in another triangle.
SAS for Similar Triangles Two sides in one triangle are proportional with two sides in another triangle AND the included angles are congruent.

Know What? Revisited First, we need to find the vertical length of the larger triangle. The two triangles are similar by AA, two right angles and \begin{align*}\angle 1 \cong \angle 2\end{align*}. Set up a proportion.

\begin{align*}\frac{17.5}{23}=\frac{29}{v}\end{align*}

Doing cross-multiplication, \begin{align*}v = 38.1\end{align*}. Second, to find the distance that the cue ball travels, use the Pythagorean Theorem. \begin{align*}17.5^2+23^2=d_1^2\end{align*} and \begin{align*}38.1^2+29^2=d_2^2\end{align*}, the lengths 28.9 and 47.9, and the total length is 76.8 inches.

## Review Questions

Use the following diagram for questions 1-3. The diagram is to scale.

3. What is the scale factor for the two triangles?
4. Writing How come there is no ASA Similarity Theorem?

Fill in the blanks in the statements below. Use the diagram to the left.

1. \begin{align*}\triangle ABC \sim \triangle \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
2. \begin{align*}\frac{AB}{?}=\frac{BC}{?}=\frac{AC}{?}\end{align*}
3. If \begin{align*}\triangle ABC\end{align*} had an altitude, \begin{align*}AG = 10\end{align*}, what would be the length of altitude \begin{align*}\overline{DH}\end{align*}?

Use the diagram to the right for questions 8-12.

1. \begin{align*}\triangle ABC \sim \triangle \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
2. Why are the two triangles similar?
3. Find \begin{align*}ED\end{align*}.
4. \begin{align*}\frac{BD}{?}=\frac{?}{BC}=\frac{DE}{?}\end{align*}
5. Is \begin{align*}\frac{AD}{DB}=\frac{CE}{EB}\end{align*} a valid proportion? How do you know?

Determine if the following triangles are similar. If so, write the similarity theorem and statement.

Algebra Connection Find the value of the missing variable(s) that makes the two triangles similar.

1. At a certain time of day, a building casts a 25 ft shadow. At the same time of day, a 6 ft tall stop sign casts a 15 ft shadow. How tall is the building?
2. A child who is 42 inches tall is standing next to the stop sign in #21. How long is her shadow?
3. An architect wants to build 3 similar right triangles such that the ratio of the middle triangle to the small triangle is the same as the ratio of the largest triangle to the middle triangle. The smallest one has side lengths 5, 12 and 13. The largest triangle has side lengths 45, 108 and 117. What are the lengths of the sides of the middle triangle?
4. Jaime wants to find the height of a radio tower in his neighborhood. He places a mirror on the ground 30 ft from the tower and walks backwards 3 ft until he can see the top of the tower in the mirror. Jaime is 5 ft 6 in tall. How tall is the radio tower?

For questions 25-27, use \begin{align*}\triangle ABC\end{align*} with \begin{align*}A(-3, 0), B(-1.5, 3)\end{align*} and \begin{align*}C(0, 0)\end{align*} and \begin{align*}\triangle DEF\end{align*} with \begin{align*}D(0, 2), E(1, 4)\end{align*} and \begin{align*}F(2, 2)\end{align*}.

1. Find \begin{align*}AB, BC, AC, DE, EF\end{align*} and \begin{align*}DF\end{align*}.
2. Use these values to find the following proportions: \begin{align*}\frac{AB}{DE}, \frac{BC}{EF}\end{align*} and \begin{align*}\frac{AC}{DF}\end{align*}.

For questions 28-31, use \begin{align*}\triangle CAR\end{align*} with \begin{align*}C(-3, 3), A(-3, -1)\end{align*} and \begin{align*}R(0, -1)\end{align*} and \begin{align*}\triangle LOT\end{align*} with \begin{align*}L(5, -2), O(5, 6)\end{align*} and \begin{align*}T(-1, 6)\end{align*}.

1. Find the slopes of \begin{align*}\overline{CA}, \overline{AR}, \overline{LO}\end{align*} and \begin{align*}\overline{OT}\end{align*}.
2. What are the measures of \begin{align*}\angle A\end{align*} and \begin{align*}\angle O\end{align*}? Explain.
3. Find \begin{align*}LO, OT, CA\end{align*} and \begin{align*}AR\end{align*}. Use these values to write the ratios \begin{align*}LO:CA\end{align*} and \begin{align*}OT:AR\end{align*}.

1. \begin{align*}\angle A \cong \angle D, \angle E \cong \angle C\end{align*}
2. \begin{align*}\triangle ABE \sim \triangle DBC\end{align*}
3. \begin{align*}BE = 10\end{align*}
1. \begin{align*}\frac{6}{8} = \frac{21}{x}, x=28\end{align*}
2. \begin{align*}{\;} \ 15(x+2) = 6(2x-1)\!\\ {\;} \ 15x+30 = 12x-6\!\\ {\;} \quad \quad \ \ \ 3x = -36\!\\ {\;} \quad \quad \ \ \ \ x = -12\end{align*}
1. \begin{align*}(x-3)(x+2)=36\!\\ {\;} \quad \ x^2-x-6=36\!\\ {\;} \ \ \ x^2-x-42=0\!\\ (x-7)(x+6)=0\!\\ {\;} \qquad \qquad \quad x=7,-6\end{align*}

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