AA Similarity
If two angles of one triangle are congruent to two angles of another, the triangles are similar. This is the most-used similarity shortcut — you only need two angles.
Introduction
Similar triangles have the same shape but can be different sizes. Their corresponding angles are equal and their corresponding sides are proportional. The AA (Angle-Angle) Postulate is the easiest way to prove similarity — you only need two pairs of congruent angles (the third automatically follows from the Triangle Angle Sum).
Past Knowledge
Dilations (7.1.3–7.1.4). Angle Sum Theorem (5.1.2). Proportions (7.1.1).
Today's Goal
Prove triangles similar using AA, then use similarity to find missing lengths.
Future Success
SSS & SAS Similarity (7.2.2–7.2.3), indirect measurement (7.3.1).
Key Concepts
AA Similarity Postulate
If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
Similar Triangles — What It Means
If , then:
- Corresponding angles are congruent:
- Corresponding sides are proportional:
Theorem & Proof
Two-Column Proof: AA Similarity Theorem
Given: and
Prove:
Strategy: Dilate so one side matches, then show the triangles are congruent via ASA.
| # | Statement | Reason |
|---|---|---|
| 1 | and | Given |
| 2 | Let . Apply dilation centered at with factor to get | A dilation exists for any center and scale factor |
| 3 | Dilation scales lengths by | |
| 4 | and | Dilations preserve angle measures; step 1 |
| 5 | ASA Congruence: , , | |
| 6 | A dilation produces a similar figure | |
| 7 | Steps 5–6: similar to a congruent copy means similar to the original |
∎ Two matching angles force the third to match (angle sum = 180°), and the dilation argument shows all sides must be proportional.
Worked Examples
Proving Similarity by AA
in . in . Are the triangles similar?
✓
✓
Two pairs of congruent angles → AA Similarity.
Yes, by AA
Finding a Missing Side
with . Find .
Corresponding sides are proportional:
Cross-multiply: →
Parallel Lines Create AA
In , with on and on . Prove .
(reflexive/shared angle)
(corresponding angles, )
Two pairs → AA Similarity.
by AA (parallel lines guarantee corresponding angles)
Common Pitfalls
Mismatching Corresponding Vertices
In , vertex corresponds to , not or . The order of letters defines the correspondence.
Confusing Similar and Congruent
Congruent () means same shape AND size. Similar () means same shape, possibly different size. Congruence is similarity with .
Real-Life Applications
Shadow Problems
A person and a tree both cast shadows at the same time. The sun's angle is the same for both, creating two triangles with two congruent angles (the sun angle and 90° at the ground). AA similarity lets you find the tree height from shadow lengths.
Perspective Drawing
Artists use similar triangles to draw objects that appear smaller as they recede. The vanishing point acts like the center of a dilation, and parallel lines create AA-similar triangles at different depths.
Practice Quiz
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