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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 real-world situations.

Review Queue

1. What are the congruent angles? List each pair.
2. Write the similarity statement.
3. If AB=8,BD=20\begin{align*}AB = 8, BD = 20\end{align*}, and BC=25\begin{align*}BC = 25\end{align*}, find BE\begin{align*}BE\end{align*}.
1. Solve the following proportions.
1. 68=21x\begin{align*}\frac{6}{8} = \frac{21}{x}\end{align*}
2. x+26=2x115\begin{align*}\frac{x+2}{6} = \frac{2x-1}{15}\end{align*}

Know What? Recall from Chapter 2, that the game of pool relies heavily on angles. In Section 2.5, we discovered that m1=m2\begin{align*}m \angle 1 = m \angle 2\end{align*}.

You decide to hit the cue ball so it follows the yellow path to the right. Are the two triangles similar?

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?

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, the sides are all in the same proportion, 69=812=1015\begin{align*}\frac{6}{9} = \frac{8}{12} = \frac{10}{15}\end{align*}.

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

If ABYZ=BCZX=ACXY\begin{align*}\frac{AB}{YZ} = \frac{BC}{ZX} = \frac{AC}{XY}\end{align*}, then ABCYZX\begin{align*}\triangle ABC \sim \triangle YZX\end{align*}.

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

Solution: Compare two triangles at a time.

ABC\begin{align*}\triangle ABC\end{align*} and DEF\begin{align*}\triangle DEF\end{align*}: 2015=2216=2418\begin{align*}\frac{20}{15} = \frac{22}{16} = \frac{24}{18}\end{align*}

Reduce each fraction to see if they are equal. 2015=43,2216=118\begin{align*}\frac{20}{15} = \frac{4}{3}, \frac{22}{16} = \frac{11}{8}\end{align*}, and 2418=43\begin{align*}\frac{24}{18} = \frac{4}{3}\end{align*}.

43118,ABC\begin{align*}\frac{4}{3} \neq \frac{11}{8}, \triangle ABC\end{align*} and DEF\begin{align*}\triangle DEF\end{align*} are not similar.

DEF\begin{align*}\triangle DEF\end{align*} and GHI\begin{align*}\triangle GHI\end{align*}: 1530=1633=1836\begin{align*}\frac{15}{30} = \frac{16}{33} = \frac{18}{36}\end{align*}

1530=12,1633=1633\begin{align*}\frac{15}{30} = \frac{1}{2}, \frac{16}{33} = \frac{16}{33}\end{align*}, and 1836=12\begin{align*}\frac{18}{36} = \frac{1}{2}\end{align*}. 121633,DEF\begin{align*}\frac{1}{2} \neq \frac{16}{33}, \triangle DEF\end{align*} is not similar to GHI\begin{align*}\triangle GHI\end{align*}.

ABC\begin{align*}\triangle ABC\end{align*} and GHI\begin{align*}\triangle GHI\end{align*}: 2030=2233=2436\begin{align*}\frac{20}{30} = \frac{22}{33} = \frac{24}{36}\end{align*}

2030=23,2233=23\begin{align*}\frac{20}{30} = \frac{2}{3}, \frac{22}{33} = \frac{2}{3}\end{align*}, and 2436=23\begin{align*}\frac{24}{36} = \frac{2}{3}\end{align*}. All three ratios reduce to 23\begin{align*}\frac{2}{3}\end{align*}, ABCGIH\begin{align*}\triangle ABC \sim \triangle GIH\end{align*}.

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

Solution: According to the similarity statement, the corresponding sides are: ABDE=BCEF=ACDF\begin{align*}\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}\end{align*}. Substituting in what we know, we have 96=4x110=18y\begin{align*}\frac{9}{6} = \frac{4x-1}{10} = \frac{18}{y}\end{align*}.

969(10)9096x=4x110=6(4x1)=24x6=24x=4 96=18y9y=18(6)9y=108 y=12\begin{align*}\frac{9}{6} &= \frac{4x-1}{10} && \quad \ \frac{9}{6} = \frac{18}{y}\\ 9(10) &= 6(4x-1) && \quad 9y =18(6)\\ 90 &= 24x-6 && \quad 9y = 108\\ 96 &= 24x && \quad \ y = 12\\ x &= 4\end{align*}

SAS for Similar Triangles

SAS is the last way to show two triangles are similar.

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 45\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 45\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 both are congruent, then the two triangles are similar.

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

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

Solution: BZ\begin{align*}\angle B \cong \angle Z\end{align*} because they are both right angles and 1015=2436\begin{align*}\frac{10}{15} = \frac{24}{36}\end{align*}. So, ABXZ=BCZY\begin{align*}\frac{AB}{XZ} = \frac{BC}{ZY}\end{align*} and ABCXZY\begin{align*}\triangle ABC \sim \triangle XZY\end{align*} by SAS.

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

Solution: A\begin{align*}\angle A\end{align*} is shared by EAB\begin{align*}\triangle EAB\end{align*} and DAC\begin{align*}\triangle DAC\end{align*}, so it is congruent to itself. Let’s see if AEAD=ABAC\begin{align*}\frac{AE}{AD} = \frac{AB}{AC}\end{align*}.

99+3912=1212+5=341217The two triangles are not similar.\begin{align*}\frac{9}{9+3} &= \frac{12}{12+5}\\ \frac{9}{12} &= \frac{3}{4} \neq \frac{12}{17} && \text {The two triangles are} \ not \ \text{similar.}\end{align*}

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*}. \begin{align*}AC = AB + BC\end{align*} and \begin{align*}16 = 12 + BC\end{align*}. \begin{align*}BC\end{align*} should equal 4.

Know What? Revisited Yes, the two triangles are similar because they both have a right angle and, from early in this text learned that \begin{align*}m \angle 1 = m \angle 2\end{align*}.

Review Questions

• Questions 1-5 are vocabulary.
• Questions 6-18 are similar to Examples 1, 3, and 4 and review.
• Questions 19-24 are similar to Examples 3 and 4.
• Questions 25-28 are similar to Example 2.
• Questions 29 and 30 are a review of the last section.

Fill in the blanks.

1. Two triangles are similar if two angles in each triangle are _____________.
2. If all three sides in one triangle are __________________ to the three sides in another, then the two triangles are similar.
3. Two triangles are congruent if the corresponding sides are _____________.
4. Two triangles are similar if the corresponding sides are _____________.
5. If two sides in one triangle are _________________ to two sides in another and the ________________ angles are _________________, then the triangles are ______________.

Use the following diagram for questions 6-8. The diagram is to scale.

3. What is the scale factor for the two triangles?

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

1. \begin{align*}\triangle ABC \sim \triangle\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*}?
4. Find the perimeter of \begin{align*}\triangle ABC\end{align*} and \begin{align*}\triangle DEF\end{align*}. Find the ratio of the perimeters.

Use the diagram to the right for questions 13-18.

1. \begin{align*}\triangle ABC \sim \triangle\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*} true?
6. Is \begin{align*}\frac{AD}{DB} = \frac{AC}{DE}\end{align*} true?

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?

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*}

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