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7.3: Similarity by AA

Difficulty Level: At Grade Created by: CK-12
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Learning Objectives

  • Determine whether triangles are similar.
  • Understand AA for similar triangles.
  • Solve problems involving similar triangles.

Review Queue

    1. Find the measures of \begin{align*}x\end{align*}x and \begin{align*}y\end{align*}y.
    2. The two triangles are similar. Find \begin{align*}w\end{align*}w and \begin{align*}z\end{align*}z.
  1. Use the true proportion \begin{align*}\frac{6}{8}=\frac{x}{28}=\frac{27}{y}\end{align*}68=x28=27y to answer the following questions.
    1. Find \begin{align*}x\end{align*}x and \begin{align*}y\end{align*}y.
    2. Write another true proportion.
    3. Is \begin{align*}\frac{28}{8}=\frac{6+x}{12}\end{align*}288=6+x12 true? If you solve for \begin{align*}x\end{align*}x, is it the same as in part a?

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 was 85 feet long. How tall is the flagpole?

Angles in Similar Triangles

The Third Angle Theorem states if two angles are congruent to two angles in another triangle, the third angles are congruent too. Because a triangle has \begin{align*}180^\circ\end{align*}180, the third angle in any triangle is \begin{align*}180^\circ\end{align*}180 minus the other two angle measures. Let’s investigate what happens when two different triangles have the same angle measures. We will use Investigation 4-4 (Constructing a Triangle using ASA) to help us with this.

Investigation 7-1: Constructing Similar Triangles

Tools Needed: pencil, paper, protractor, ruler

  1. Draw a \begin{align*}45^\circ\end{align*}45 angle. Extend the horizontal side and then draw a \begin{align*}60^\circ\end{align*}60 angle on the other side of this side. Extend the other side of the \begin{align*}45^\circ\end{align*}45 angle and the \begin{align*}60^\circ\end{align*}60 angle so that they intersect to form a triangle. What is the measure of the third angle? Measure the length of each side.
  2. Repeat Step 1 and make the horizontal side between the \begin{align*}45^\circ\end{align*}45 and \begin{align*}60^\circ\end{align*}60 angle at least 1 inch longer than in Step 1. This will make the entire triangle larger. Find the measure of the third angle and measure the length of each side.
  3. Find the ratio of the sides. Put the sides opposite the \begin{align*}45^\circ\end{align*}45 angles over each other, the sides opposite the \begin{align*}60^\circ\end{align*}60 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.

The AA Similarity Postulate is a shortcut for showing that two triangles are similar. If you know that two angles in one triangle are congruent to two angles in another, which is now enough information to show that the two triangles are similar. Then, you can use the similarity to find the lengths of the sides.

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

Solution: Find the measure of the third angle in each triangle. \begin{align*}m \angle G = 48^\circ\end{align*}mG=48 and \begin{align*}m \angle M = 30^\circ\end{align*}mM=30 by the Triangle Sum Theorem. Therefore, all three angles are congruent, so the two 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*}. The angles are not equal, \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. Therefore, by the AA Similarity Postulate, \begin{align*}\triangle ABE \sim \triangle DBC\end{align*}.

Indirect Measurement

An application of similar triangles is to measure lengths indirectly. The length to be measured would be some feature that was not easily accessible to a person, such as: the width of a river or canyon and the height of a tall object. To measure something indirectly, you would need to set up a pair of similar triangles.

Example 4: 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. From the picture to the right, we see that the tree and Ellie are parallel, therefore the two triangles are similar to each other. Write a proportion.

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

Notice that our measurements are not all in the same units. Change both numerators to inches and then we can cross multiply.

\begin{align*}\frac{58in}{xft}=\frac{66in}{125ft} \longrightarrow 58(125) &= 66(x)\\ 7250 &= 66x\\ x & \approx 109.85 \ ft\end{align*}

Know What? Revisited It is safe to assume that George and the flagpole stand vertically, making right angles with the ground. Also, the angle where the sun’s rays hit the ground is the same for both. The two trianglesare similar. Set up a proportion.

\begin{align*}\frac{10}{85} = \frac{6}{x} \longrightarrow 10x &= 510\\ x &= 51 \ ft.\end{align*}

The height of the flagpole is 51 feet.

Review Questions

Use the diagram to complete each statement.

  1. \begin{align*}\triangle SAM \sim \triangle \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  2. \begin{align*}\frac{SA}{?}=\frac{SM}{?}=\frac{?}{RI}\end{align*}
  3. \begin{align*}SM = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
  4. \begin{align*}TR = \underline{\;\;\;\;\;\;\;\;\;}\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!
  5. Writing How many angles need to be congruent to show that two triangles are similar? Why?
  6. Writing How do congruent triangles and similar triangles differ? How are they the same?

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 _______.

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} \bot \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} \bot \overline{NA}\end{align*}. \begin{align*}OC = 50 \ feet, CE = 30 \ feet, NA = 80 \ feet\end{align*}.

  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. Janet wants to measure the height of her apartment building. She places a pocket mirror on the ground 20 ft from the building and steps backwards until she can see the top of the build in the mirror. She is 18 in from the mirror and her eyes are 5 ft 3 in above the ground. The angle formed by her line of sight and the ground is congruent to the angle formed by the reflection of the building and the ground. You may wish to draw a diagram to illustrate this problem. How tall is the building?
  6. Sebastian is curious to know how tall the announcer’s box is on his school’s football field. On a sunny day he measures the shadow of the box to be 45 ft and his own shadow is 9 ft. Sebastian is 5 ft 10 in tall. Find the height of the box.
  7. Juanita wonders how tall the mast of a ship she spots in the harbor is. The deck of the ship is the same height as the pier on which she is standing. The shadow of the mast is on the pier and she measures it to be 18 ft long. Juanita is 5 ft 4 in tall and her shadow is 4 ft long. How tall is the ship’s mast?
  8. Use shadows or a mirror to measure the height of an object in your yard or on the school grounds. Draw a picture to illustrate your method.

Use the diagram below to answer questions 27-31.

  1. Draw the three separate triangles in the diagram.
  2. Explain why \begin{align*}\triangle GDE \cong \triangle DFE \cong \triangle GFD\end{align*}.

Complete the following proportionality statements.

  1. \begin{align*}\frac{GF}{DF}=\frac{?}{FE}\end{align*}
  2. \begin{align*}\frac{GF}{GD}=\frac{?}{GE}\end{align*}
  3. \begin{align*}\frac{GE}{DE}=\frac{DE}{?}\end{align*}

Review Queue Answers

  1. \begin{align*}x=52^\circ, y=80^\circ\end{align*}
  2. \begin{align*}\frac{w}{20} = \frac{15}{25} \qquad \qquad \qquad \quad \ \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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