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 the AA Similarity Postulate to decide if two triangles are similar.
Watch This
CK-12 Foundation: AA Similarity
For additional help, first watch this video beginning at the 2:09 mark.
James Sousa: Similar Triangles
Then watch this video.
James Sousa: Similar Triangles by AA
Guidance
By definition, two triangles are similar if all their corresponding angles are congruent and their corresponding sides are proportional. It is not necessary to check all angles and sides in order to tell if two triangles are similar. In fact, if you only know that two pairs of corresponding angles are congruent that is enough information to know that the triangles are similar. This is called the AA Similarity Postulate.
AA Similarity Postulate: If two angles in one triangle are congruent to two angles in another triangle, then 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 A
Determine if the following two triangles are similar. If so, write the similarity statement.
Compare the angles to see if we can use the AA Similarity Postulate. Using the Triangle Sum Theorem, @$\begin{align*}m \angle G = 48^{\circ}\end{align*}@$ 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 B
Determine if the following two triangles are similar. If so, write the similarity statement.
Compare the angles to see if we can use the AA Similarity Postulate. Using the Triangle Sum Theorem, @$\begin{align*}m \angle C = 39^{\circ}\end{align*}@$ 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 C
@$\begin{align*}\triangle LEG \sim \triangle MAR\end{align*}@$ by AA. Find @$\begin{align*}GE\end{align*}@$ and @$\begin{align*}MR\end{align*}@$.
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 this example, we see how the sides line up in the diagram to the right.
CK-12 Foundation: AA Similarity
-->
Guided Practice
1.Are the following triangles similar? If so, write the similarity statement.
2. Are the triangles similar? If so, write a similarity statement.
3. Are the triangles similar? If so, write a similarity statement.
Answers:
1. 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, @$\begin{align*}\triangle ABE \sim \triangle DBC\end{align*}@$.
2. Yes, there are three similar triangles that each have a right angle. @$\begin{align*}DGE \sim FGD \sim FDE\end{align*}@$.
3. By the reflexive property, @$\begin{align*}\angle H \cong \angle H\end{align*}@$. Because the horizontal lines are parallel, @$\begin{align*}\angle L \cong \angle K\end{align*}@$ (corresponding angles). So yes, there is a pair of similar triangles. @$\begin{align*} HLI \sim HKJ\end{align*}@$.
Explore More
Use the diagram to complete each statement.
- @$\begin{align*}\triangle SAM \sim \triangle\end{align*}@$ ______
- @$\begin{align*}\frac{SA}{?} = \frac{SM}{?} = \frac{?}{RI}\end{align*}@$
- @$\begin{align*}SM\end{align*}@$ = ______
- @$\begin{align*}TR\end{align*}@$ = ______
- @$\begin{align*}\frac{9}{?} = \frac{?}{8}\end{align*}@$
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!
Use the triangles to the left for questions 10-14.
@$\begin{align*}AB = 20, DE = 15\end{align*}@$, 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 _______.
- 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.