AA Similarity

Learn how two triangles are similar when they share two pairs of congruent angles.

Intermediate25 minLesson
Printable worksheet

Definition

The AA (Angle-Angle) Similarity Postulate states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
Why only two angles? Because if two angles are equal, the third angle must also be equal!
If ∠A=∠D and ∠B=∠E, then △ABC∼△DEF\text{If } \angle A = \angle D \text{ and } \angle B = \angle E, \text{ then } \triangle ABC \sim \triangle DEF
Similar triangles have:
  • The same shape (all corresponding angles equal)
  • Proportional sides (the ratios of corresponding sides are equal)

Try it now

What does the AA in AA Similarity stand for?

Worked Examples

In triangle ABC, ∠A=50°\angle A = 50° and ∠B=70°\angle B = 70°. In triangle DEF, ∠D=50°\angle D = 50° and ∠E=70°\angle E = 70°. Are these triangles similar?
1

Compare the first pair of angles

∠A=50°\angle A = 50° and ∠D=50°\angle D = 50° → ∠A≅∠D\angle A \cong \angle D ✓
2

Compare the second pair of angles

∠B=70°\angle B = 70° and ∠E=70°\angle E = 70° → ∠B≅∠E\angle B \cong \angle E ✓
3

Apply the AA Similarity Postulate

Two pairs of congruent angles found → △ABC∼△DEF\triangle ABC \sim \triangle DEF

Common Mistakes

Matching angles in the wrong order (e.g., saying △ABC∼△FDE\triangle ABC \sim \triangle FDE)

Why it's wrong: The order of letters matters! It tells us which angles correspond. ∠A\angle A corresponds to the first letter, ∠B\angle B to the second, etc.

Correct: Always write similarity statements with corresponding vertices in the same order: if ∠A=∠D\angle A = \angle D, ∠B=∠E\angle B = \angle E, then write △ABC∼△DEF\triangle ABC \sim \triangle DEF.

Thinking you need to check all three angles

Why it's wrong: Since the angles in any triangle sum to 180°180°, if two pairs are equal, the third pair must be equal too.

Correct: AA Similarity requires only TWO pairs of congruent angles. The third angle is automatically congruent.

Confusing similarity with congruence

Why it's wrong: Similar triangles have the same shape but not necessarily the same size. Congruent triangles have both the same shape AND size.

Correct: AA proves SIMILARITY (same shape, proportional sides), not congruence.

Interactive Visual

Similar triangles

ABCDEF△ABC ∼ △DEF

AATwo pairs of angles are equal, so the third pair is too.

  • ∠A = ∠D = 50°
  • ∠B = ∠E = 70°

Move the slider to change the scale factor k. The angles stay the same, every side is multiplied by k.

Angle Explorer

12345678
∠1 = 110°∠2 = 70°∠3 = 70°∠4 = 110°∠5 = 110°∠6 = 70°∠7 = 70°∠8 = 110°

Corresponding angles are in the same position at each crossing. With parallel lines they are equal.

Drag the blue point to tilt the transversal. Pick a pair type, or tap two angle numbers to see how they are related.

Interactive Sandbox

Expression Calculator

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History

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Practice Problems

15 problems
Problem 1 of 15
Easy
What does the AA in AA Similarity stand for?

Why It Matters

AA Similarity is one of the most practical tools in geometry because:
  • Architecture: Architects use similar triangles to scale blueprints to actual buildings
  • Photography: Camera lenses create similar triangles to focus images
  • Navigation: Sailors and pilots use similar triangles to calculate distances
  • Art: Artists use similar triangles to draw objects in perspective
The beauty of AA Similarity is that you only need to measure angles, not sides, to prove triangles are similar!

Real World Applications

Measuring Building Height

You can find the height of a building using your shadow and the building's shadow.

Example:

A 2-meter person casts a 3-meter shadow at the same time a building casts a 45-meter shadow. Using AA similarity (same sun angle creates similar triangles), the building's height is (2×45)÷3=30(2 \times 45) \div 3 = 30 meters.

1Try It Yourself

A flagpole casts a 12-meter shadow. At the same time, a 1.5-meter stick casts a 2-meter shadow.

How tall is the flagpole?

Step 1: Write the mathematical expression

Set up the proportion using similar triangles:

Map Scaling

Maps use similar triangles to represent real distances. A triangle drawn on a map is similar to the actual triangle formed by the locations.

Example:

On a map, three cities form a triangle with angles 40°40°, 60°60°, and 80°80°. The actual triangle formed by these cities has the same angles, making the triangles similar by AA.

2Try It Yourself

On a map, two cities are 5 cm apart. The map scale is 1 cm = 20 km.

What is the actual distance between the cities?

Step 1: Write the mathematical expression

Use the scale ratio:

Key Takeaways

  • 1AA Similarity Postulate: Two triangles are similar if two pairs of corresponding angles are congruent
  • 2You only need to check TWO angles because the third is determined by the 180°180° sum
  • 3Similar triangles have equal corresponding angles and proportional corresponding sides
  • 4Write similarity statements with vertices in corresponding order: △ABC∼△DEF\triangle ABC \sim \triangle DEF
  • 5Parallel lines cutting a triangle create smaller similar triangles

Frequently Asked Questions

Because you only need to check two angles! If two angles of one triangle equal two angles of another, the third angles are automatically equal (since all triangles have angles summing to 180°180°).
Because you only need to check two angles! If two angles of one triangle equal two angles of another, the third angles are automatically equal (since all triangles have angles summing to 180°180°).
No. AA only proves similarity (same shape). Two triangles with the same angles can have different sizes. To prove congruence, you need information about side lengths too.
Yes! The order tells us which vertices correspond. If △ABC∼△DEF\triangle ABC \sim \triangle DEF, then ∠A=∠D\angle A = \angle D, ∠B=∠E\angle B = \angle E, and ∠C=∠F\angle C = \angle F.

Glossary

AA Similarity
A postulate stating that if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar
Similar triangles
Triangles with equal corresponding angles and proportional corresponding sides (same shape, different size)
Congruent angles
Angles that have the same measure
Corresponding angles
Angles in the same position in similar or congruent figures
Today's challengeOne problem, three tries

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