# 7.3: Similarity by AA

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Determine whether triangles are similar using the AA Postulate.
• Solve problems involving similar triangles.

## Review Queue

1. Find the measures of \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.
2. The two triangles are similar. Find \begin{align*}w\end{align*} and \begin{align*}z\end{align*}.
1. Use the true proportion \begin{align*}\frac{6}{8} = \frac{x}{28} = \frac{27}{y}\end{align*} to answer the following questions.
1. Find \begin{align*}x\end{align*}.
2. Find \begin{align*}y\end{align*}.

Know What? George wants to measure the height of a flagpole. He is 6 feet tall and his shadow is 10 feet long. At the same time, the shadow of the flagpole is 85 feet long. How tall is the flagpole?

## Angles in Similar Triangles

The Third Angle Theorem says if two angles are congruent to two angles in another triangle the third angles are congruent too. Let’s see what happens when two different triangles have the same angle measures.

Investigation 7-1: Constructing Similar Triangles

Tools Needed: pencil, paper, protractor, ruler

1. Draw a \begin{align*}45^{\circ}\end{align*} angle. Make the horizontal side 3 inches and draw a \begin{align*}60^{\circ}\end{align*} angle on the other endpoint.

2. Extend the other sides of the \begin{align*}45^{\circ}\end{align*} and \begin{align*}60^{\circ}\end{align*} angles so that they intersect to form a triangle.

Find the measure of the third angle and measure the length of each side.

3. Repeat Steps 1 and 2, but make the horizontal side between the \begin{align*}45^{\circ}\end{align*} and \begin{align*}60^{\circ}\end{align*} angle 4 inches.

Find the measure of the third angle and measure the length of each side.

4. Find the ratio of the sides. Put the sides opposite the \begin{align*}45^{\circ}\end{align*} angles over each other, the sides opposite the \begin{align*}60^{\circ}\end{align*} angles over each other, and the sides opposite the third angles over each other. What happens?

AA Similarity Postulate: If two angles in one triangle are congruent to two angles in another triangle, the two triangles are similar.

If \begin{align*}\angle A \cong \angle Y\end{align*} and \begin{align*}\angle B \cong \angle Z\end{align*}, then \begin{align*}\triangle ABC \sim \triangle YZX\end{align*}.

Example 1: Determine if the following two triangles are similar. If so, write the similarity statement.

Solution: \begin{align*}m \angle G = 48^{\circ}\end{align*} and \begin{align*}m \angle M = 30^\circ\end{align*} So, \begin{align*}\angle F \cong \angle M, \angle E \cong \angle L\end{align*} and \begin{align*}\angle G \cong \angle N\end{align*} and the triangles are similar. \begin{align*}\triangle FEG \sim \triangle MLN\end{align*}.

Example 2: Determine if the following two triangles are similar. If so, write the similarity statement.

Solution: \begin{align*}m \angle C = 39^{\circ}\end{align*} and \begin{align*}m \angle F = 59^{\circ}\end{align*}. \begin{align*}m \angle C \neq m \angle F\end{align*}, So \begin{align*}\triangle ABC\end{align*} and \begin{align*}\triangle DEF\end{align*} are not similar.

Example 3: Are the following triangles similar? If so, write the similarity statement.

Solution: Because \begin{align*}\overline {AE}\| \overline{CD}, \angle A \cong \angle D\end{align*} and \begin{align*}\angle C \cong \angle E\end{align*} by the Alternate Interior Angles Theorem. By the AA Similarity Postulate, \begin{align*}\triangle ABE \sim \triangle DBC\end{align*}.

Example 4: \begin{align*}\triangle LEG \sim \triangle MAR\end{align*} by AA. Find \begin{align*}GE\end{align*} and \begin{align*}MR\end{align*}.

Solution: Set up a proportion to find the missing sides.

\begin{align*}\frac{24}{32} &= \frac{MR}{20} && \qquad \ \frac{24}{32} = \frac{21}{GE}\\ 480 &= 32MR && \quad 24GE = 672\\ 15 &= MR && \qquad GE = 28\end{align*}

When two triangles are similar, the corresponding sides are proportional. But, what are the corresponding sides? Using the triangles from Example 4, we see how the sides line up in the diagram to the right.

## Indirect Measurement

An application of similar triangles is to measure lengths indirectly. You can use this method to measure the width of a river or canyon or the height of a tall object.

Example 5: A tree outside Ellie’s building casts a 125 foot shadow. At the same time of day, Ellie casts a 5.5 foot shadow. If Ellie is 4 feet 10 inches tall, how tall is the tree?

Solution: Draw a picture. We see that the tree and Ellie are parallel, so the two triangles are similar.

\begin{align*}\frac{4 \ ft, 10 \ in}{x \ ft} = \frac{5.5 \ ft}{125 \ ft}\end{align*}

The measurements need to be in the same units. Change everything into inches and then we can cross multiply.

\begin{align*}\frac{58 \ in}{x \ ft} &= \frac{66 \ in}{1500 \ ft}\\ 87000 &= 66x\\ x & \approx 1318. \overline{18} \ in \ \text{or} \ 109.85 \ ft\end{align*}

Know What? Revisited It is safe to assume that George and the flagpole stand vertically, making them parallel. This is very similar to Example 4. Set up a proportion.

\begin{align*}\frac{10}{85} = \frac{6}{x} \longrightarrow \ 10x &= 510\\ x &= 51 \ ft. \quad \text{The height of the flagpole is 51 feet.}\end{align*}

## Review Questions

• Questions 1-13 are similar to Examples 1-4 and review.
• Question14 compares the definitions of congruence and similarity.
• Questions 15-23 are similar to Examples 1-3.
• Questions 24-30 are similar to Example 5 and the Know What?

Use the diagram to complete each statement.

1. \begin{align*}\triangle SAM \sim \triangle\end{align*} ______
2. \begin{align*}\frac{SA}{?} = \frac{SM}{?} = \frac{?}{RI}\end{align*}
3. \begin{align*}SM\end{align*} = ______
4. \begin{align*}TR\end{align*} = ______
5. \begin{align*}\frac{9}{?} = \frac{?}{8}\end{align*}

Answer questions 6-9 about trapezoid \begin{align*}ABCD\end{align*}.

1. Name two similar triangles. How do you know they are similar?
2. Write a true proportion.
3. Name two other triangles that might not be similar.
4. If \begin{align*}AB = 10, AE = 7,\end{align*} and \begin{align*}DC = 22\end{align*}, find \begin{align*}AC\end{align*}. Be careful!

Use the triangles to the left for questions 5-9.

\begin{align*}AB = 20, DE = 15\end{align*}, and \begin{align*}BC = k\end{align*}.

1. Are the two triangles similar? How do you know?
2. Write an expression for \begin{align*}FE\end{align*} in terms of \begin{align*}k\end{align*}.
3. If \begin{align*}FE = 12,\end{align*}, what is \begin{align*}k\end{align*}?
4. Fill in the blanks: If an acute angle of a _______ triangle is congruent to an acute angle in another ________ triangle, then the two triangles are _______.
5. Writing How do congruent triangles and similar triangles differ? How are they the same?

Are the following triangles similar? If so, write a similarity statement.

In order to estimate the width of a river, the following technique can be used. Use the diagram on the left.

Place three markers, \begin{align*}O, C,\end{align*} and \begin{align*}E\end{align*} on the upper bank of the river. \begin{align*}E\end{align*} is on the edge of the river and \begin{align*}\overline {OC} \perp \overline{CE}\end{align*}. Go across the river and place a marker, \begin{align*}N\end{align*} so that it is collinear with \begin{align*}C\end{align*} and \begin{align*}E\end{align*}. Then, walk along the lower bank of the river and place marker \begin{align*}A\end{align*}, so that \begin{align*}\overline{CN} \perp \overline{NA}\end{align*}. \begin{align*}OC = 50\end{align*} feet, \begin{align*}CE = 30\end{align*} feet, \begin{align*}NA = 80\end{align*} feet.

1. Is \begin{align*}\overline {OC} \| \overline {NA}?\end{align*} How do you know?
2. Is \begin{align*}\triangle OCE \sim \triangle ANE?\end{align*} How do you know?
3. What is the width of the river? Find \begin{align*}EN\end{align*}.
4. Can we find \begin{align*}EA\end{align*}? If so, find it. If not, explain.
5. The technique above was used to measure the distance across the Grand Canyon. Using the same set up and marker letters, \begin{align*}OC = 72 \ ft , CE = 65 \ ft\end{align*}, and \begin{align*}NA = 14,400 \ ft\end{align*}. Find \begin{align*}EN\end{align*} (the distance across the Grand Canyon).
6. Cameron is 5 ft tall and casts a 12 ft shadow. At the same time of day, a nearby building casts a 78 ft shadow. How tall is the building?
7. The Empire State Building is 1250 ft. tall. At 3:00, Pablo stands next to the building and has an 8 ft. shadow. If he is 6 ft tall, how long is the Empire State Building’s shadow at 3:00?

1. \begin{align*}x=52^\circ, y=80^\circ\end{align*}
2. \begin{align*}{\;} \ \frac{w}{20} = \frac{15}{25} \qquad \qquad \qquad \ \ \frac{15}{25} = \frac{18}{z}\!\\ 25w = 15(20) \qquad \quad 25(18) = 15z\!\\ 25w = 300 \qquad \qquad \quad \ \ 450=15z\!\\ {\;} \ \ w = 12 \qquad \qquad \qquad \ \ 30=z\end{align*}
1. \begin{align*}168 = 8x \qquad 6y = 216\!\\ {\;} \ \ x = 21 \qquad \ \ y = 36\end{align*}
2. Answers will vary. One possibility: \begin{align*}\frac{28}{8} = \frac{21}{6}\end{align*}
3. \begin{align*}28(12) = 8(6+x)\!\\ {\;} \ \ \ 336 = 48+8x\!\\ {\;} \ \ \ 288 = 8x\!\\ {\;} \quad \ 36 = x \quad \text{Because}\ x \neq 21, \ \text{like in part a, this is not a true proportion.}\end{align*}

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