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AA Similarity

AA Similarity

Angle-angle similarity. When two triangles have corresponding angles that are congruent as shown below, the triangles are similar.

 

Two similar triangles: △ABC (smaller, left) and △DEF (larger, right) with congruent angle marks at B and E.

 

See also

Worked Example

Problem: Triangle ABC has angles A = 50° and B = 60°. Triangle DEF has angles D = 50° and F = 70°. Are the two triangles similar by AA Similarity? If so, find the ratio of their sides given that AB = 8 and DE = 12.
Step 1: Find the missing angle in each triangle. Since the angles in any triangle sum to 180°:
∠C=180°−50°−60°=70°\angle C = 180° - 50° - 60° = 70°
Step 2: Find the missing angle in triangle DEF:
∠E=180°−50°−70°=60°\angle E = 180° - 50° - 70° = 60°
Step 3: Match the corresponding angles between the two triangles. Angle A = 50° matches angle D = 50°, and angle B = 60° matches angle E = 60°. Two pairs of corresponding angles are congruent.
∠A=∠D=50°,∠B=∠E=60°\angle A = \angle D = 50°, \quad \angle B = \angle E = 60°
Step 4: By AA Similarity, triangle ABC is similar to triangle DEF. Since the sides opposite equal angles are proportional, the scale factor (ratio of corresponding sides) is:
DEAB=128=32\frac{DE}{AB} = \frac{12}{8} = \frac{3}{2}
Answer: Yes, △ABC ~ △DEF by AA Similarity. The scale factor from △ABC to △DEF is 3/2, so every side of △DEF is 1.5 times the corresponding side of △ABC.

Another Example

Problem: A 2-meter tall post casts a 3-meter shadow at the same time a tree casts a 15-meter shadow. Use AA Similarity to find the height of the tree.
Step 1: Both the post and the tree stand vertically, so each makes a 90° angle with the ground. That gives us one pair of congruent angles.
Step 2: The sun hits both objects at the same angle, so the angle of elevation from the tip of each shadow to the top of each object is the same. That gives us a second pair of congruent angles.
Step 3: By AA Similarity, the two right triangles (object, shadow, sun ray) are similar. Set up a proportion using corresponding sides:
height of treeheight of post=tree shadowpost shadow\frac{\text{height of tree}}{\text{height of post}} = \frac{\text{tree shadow}}{\text{post shadow}}
Step 4: Substitute the known values and solve:
h2=153  ⟹  h=2×5=10 meters\frac{h}{2} = \frac{15}{3} \implies h = 2 \times 5 = 10 \text{ meters}
Answer: The tree is 10 meters tall.

Frequently Asked Questions

Why do you only need two angles instead of three to prove similarity?
The three interior angles of any triangle always sum to 180°. If two angles of one triangle match two angles of another, the third angles are automatically equal as well. So checking two pairs is enough — the third pair is guaranteed.
What is the difference between AA Similarity and AAA Similarity?
They are effectively the same thing. AAA (Angle-Angle-Angle) states that if all three pairs of angles are congruent, the triangles are similar. AA is the more efficient version because proving two pairs automatically proves the third. Most textbooks use AA since it requires less work.

AA Similarity vs. SAS Similarity

AA Similarity requires two pairs of congruent angles and no side information at all. SAS Similarity requires two pairs of proportional sides with the included angle between them congruent. Use AA when you know angle measures; use SAS when you know side lengths and one angle.

Why It Matters

AA Similarity is one of the most frequently used tools in geometry because angle measures are often easier to find than side lengths. It is the basis for indirect measurement — calculating heights of buildings, trees, or mountains using shadows or sightlines. It also underpins core results in trigonometry, since the ratios of sides in a right triangle depend only on the acute angles, a fact that follows directly from AA Similarity.

Common Mistakes

Mistake: Matching angles that are not actually corresponding — for example, pairing the largest angle in one triangle with the smallest angle in another.
Correction: Always check the actual degree measures (or use geometric reasoning such as vertical angles or parallel-line theorems) to confirm which specific angles in one triangle correspond to which angles in the other.
Mistake: Thinking AA Similarity means the triangles are congruent (same size).
Correction: Similar triangles have the same shape but can be different sizes. AA Similarity tells you the sides are proportional, not necessarily equal. Congruence requires additional information about side lengths.

Related Terms

  • Similar — Definition of similar figures and proportionality
  • Similarity Tests for Triangles — All methods to prove triangle similarity
  • Angle — The fundamental measurement AA Similarity uses
  • Triangle — The polygon AA Similarity applies to
  • Corresponding — Matching angles and sides between triangles
  • Congruent — Equal in measure — required for the angle pairs
  • Proportion — The equal ratios that result from similarity