7.4: AA Similarity
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
CK12 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*}
Investigation: Constructing Similar Triangles
Tools Needed: pencil, paper, protractor, ruler
 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.  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.  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 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*}
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*}
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*}
Watch this video for help with the Examples above.
CK12 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*}
Practice
Use the diagram to complete each statement.

\begin{align*}\triangle SAM \sim \triangle \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
△SAM∼△−−−−− 
\begin{align*}\frac{SA}{?}=\frac{SM}{?}=\frac{?}{RI}\end{align*}
SA?=SM?=?RI 
\begin{align*}SM = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
SM=−−−−− 
\begin{align*}TR = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
TR=−−−−− 
\begin{align*}\frac{9}{?}=\frac{?}{8}\end{align*}
9?=?8
Answer questions 69 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*}
AB=10,AE=7, and \begin{align*}DC = 22\end{align*}DC=22 , find \begin{align*}AC\end{align*}AC . 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 1215.
\begin{align*}AB = 20, DE = 15,\end{align*}
 Are the two triangles similar? How do you know?
 Write an expression for \begin{align*}FE\end{align*}
FE in terms of \begin{align*}k\end{align*}k .  If \begin{align*}FE = 12\end{align*}
FE=12 , what is \begin{align*}k\end{align*}k ?  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 1620.
 Draw the three separate triangles in the diagram.
 Explain why \begin{align*}\triangle GDE \sim \triangle DFE \sim \triangle GFD\end{align*}
△GDE∼△DFE∼△GFD .
Complete the following proportionality statements.

\begin{align*}\frac{GF}{DF}=\frac{?}{FE}\end{align*}
GFDF=?FE 
\begin{align*}\frac{GF}{GD}=\frac{?}{GE}\end{align*}
GFGD=?GE 
\begin{align*}\frac{GE}{DE}=\frac{DE}{?}\end{align*}
GEDE=DE?
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AA Similarity Postulate
If two angles in one triangle are congruent to two angles in another triangle, then the two triangles are similar.Dilation
To reduce or enlarge a figure according to a scale factor is a dilation.Triangle Sum Theorem
The Triangle Sum Theorem states that the three interior angles of any triangle add up to 180 degrees.Rigid Transformation
A rigid transformation is a transformation that preserves distance and angles, it does not change the size or shape of the figure.Image Attributions
Here you'll learn how to determine whether or not two triangles are similar using AA Similarity.