What if you were given a pair of triangles and the angle measures for two of their angles? How could you use this information to determine if the two triangles are similar? After completing this Concept, you'll be able to use AA Similarity to decide if two triangles are similar.
Watch This
CK-12 Foundation: Chapter7AASimilarityA
Watch this video beginning at the 2:09 mark.
James Sousa: Similar Triangles
James Sousa: Similar Triangles by AA
Guidance
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*}, the third angle in any triangle is \begin{align*}180^\circ\end{align*} minus the other two angle measures. Let’s investigate what happens when two different triangles have the same angle measures.
Investigation: Constructing Similar Triangles
Tools Needed: pencil, paper, protractor, ruler
- Draw a \begin{align*}45^\circ\end{align*} angle. Extend the horizontal side and then draw a \begin{align*}60^\circ\end{align*} angle on the other side of this side. Extend the other side of the \begin{align*}45^\circ\end{align*} angle and the \begin{align*}60^\circ\end{align*} angle so that they intersect to form a triangle. What is the measure of the third angle? Measure the length of each side.
- Repeat Step 1 and make the horizontal side between the \begin{align*}45^\circ\end{align*} and \begin{align*}60^\circ\end{align*} 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.
- 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.
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 A
Determine if the following two triangles are similar. If so, write the similarity statement.
Find the measure of the third angle in each triangle. \begin{align*}m \angle G = 48^\circ\end{align*} and \begin{align*}m \angle M = 30^\circ\end{align*} 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 B
Determine if the following two triangles are similar. If so, write the similarity statement.
\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 C
Are the following triangles similar? If so, write the similarity statement.
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*}.
Watch this video for help with the Examples above.
CK-12 Foundation: Chapter7AASimilarityB
Vocabulary
Two triangles are similar if all their corresponding angles are congruent (exactly the same) and their corresponding sides are proportional (in the same ratio).
Guided Practice
Are the following triangles similar? If so, write a similarity statement.
1.
2.
3.
Answers:
1. Yes, \begin{align*}\triangle DGE \sim \triangle FGD \sim \triangle FDE\end{align*}.
2. Yes, \begin{align*}\triangle HLI \sim \triangle HMJ\end{align*}.
3. No, though \begin{align*}\angle MNQ \cong \angle ONP\end{align*} because they are vertical angles, we need to have two pairs of congruent angles in order to be able to say that the triangles are similar.
Practice
Use the diagram to complete each statement.
- \begin{align*}\triangle SAM \sim \triangle \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}\frac{SA}{?}=\frac{SM}{?}=\frac{?}{RI}\end{align*}
- \begin{align*}SM = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}TR = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}\frac{9}{?}=\frac{?}{8}\end{align*}
Answer questions 6-9 about trapezoid \begin{align*}ABCD\end{align*}.
- Name two similar triangles. How do you know they are similar?
- Write a true proportion.
- Name two other triangles that might not be similar.
- If \begin{align*}AB = 10, AE = 7,\end{align*} and \begin{align*}DC = 22\end{align*}, find \begin{align*}AC\end{align*}. Be careful!
- Writing How many angles need to be congruent to show that two triangles are similar? Why?
- Writing How do congruent triangles and similar triangles differ? How are they the same?
Use the triangles below for questions 12-15.
\begin{align*}AB = 20, DE = 15,\end{align*} and \begin{align*}BC = k\end{align*}.
- Are the two triangles similar? How do you know?
- Write an expression for \begin{align*}FE\end{align*} in terms of \begin{align*}k\end{align*}.
- If \begin{align*}FE = 12\end{align*}, what is \begin{align*}k\end{align*}?
- 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 _______.
Use the diagram below to answer questions 16-20.
- Draw the three separate triangles in the diagram.
- Explain why \begin{align*}\triangle GDE \sim \triangle DFE \sim \triangle GFD\end{align*}.
Complete the following proportionality statements.
- \begin{align*}\frac{GF}{DF}=\frac{?}{FE}\end{align*}
- \begin{align*}\frac{GF}{GD}=\frac{?}{GE}\end{align*}
- \begin{align*}\frac{GE}{DE}=\frac{DE}{?}\end{align*}