Angle-Angle (AA) Similarity

Chapter 44

Concept Explanation:

The AA Similarity Postulate states that if two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. This means their corresponding sides are proportional, and their corresponding angles are congruent.

If triangles $\triangle ABC$ and $\triangle DEF$ have $\angle A \cong \angle D$ and $\angle B \cong \angle E$ , then the triangles are similar, written as $\triangle ABC \sim \triangle DEF$ .

*** QuickLaTeX cannot compile formula:

\begin{tikzpicture}
\draw<a href="0,0">thick</a> -- (3,0) -- (1.5,2.5) -- cycle;
\draw<a href="5,0">thick</a> -- (7.5,0) -- (6.25,1.25) -- cycle;
\draw<a href="0,0">fill=blue</a> circle [radius=0.05];
\draw<a href="5,0">fill=blue</a> circle [radius=0.05];
\draw<a href="3,0">fill=blue</a> circle [radius=0.05];
\draw<a href="7.5,0">fill=blue</a> circle [radius=0.05];
\node at (-0.3,-0.2) {A};
\node at (3.3,-0.2) {B};
\node at (1.6,2.7) {C};
\node at (4.7,-0.2) {D};
\node at (7.8,-0.2) {E};
\node at (6.1,1.45) {F};
\end{tikzpicture}


*** Error message:
Error: Cannot create svg file

In this diagram, $\triangle ABC \sim \triangle DEF$ .

Common Mistakes:

Not Checking All Angles: Make sure both pairs of angles are congruent before concluding that triangles are similar.
Confusing Congruence with Similarity: Similar triangles have proportional sides, not necessarily congruent sides.

Helpful Tips:

Focus on Angles: To prove triangles are similar, concentrate on identifying two pairs of congruent angles.
Use Proportions for Sides: Once similarity is established, set up ratios between corresponding sides.

Hard Questions:

Q1: Given that $\triangle ABC \sim \triangle DEF$ with $\angle A = \angle D$ , $\angle B = \angle E$ , and $AB = 8$ , $AC = 10$ , $DE = 4$ , find $DF$ .

Step-by-Step Solution:

Since $\triangle ABC \sim \triangle DEF$ , corresponding sides are proportional:

$\frac{AB}{DE} = \frac{AC}{DF}$

Substitute the known values:

$\frac{8}{4} = \frac{10}{DF}$

Simplify:

$2 = \frac{10}{DF} \quad \Rightarrow \quad DF = 5$

Answer:

$DF = 5$

Q2: In $\triangle ABC$ , if $\angle A = 50^\circ$ and $\angle B = 80^\circ$ , find $\angle C$ and determine if $\triangle ABC \sim \triangle DEF$ , where $\angle D = 50^\circ$ and $\angle E = 80^\circ$ .