Two angles are enough

The idea

If two angles of one triangle are respectively equal to two angles of another, the third angles are also equal by the angle sum property, so the triangles are similar (AA similarity criterion).

Only two angles known

Triangle ABC has ∠A = 40° and ∠B = 75°. Triangle DEF has ∠D = 40° and ∠E = 75°.

You know nothing else about DEF: not its size and not the angle ∠F.

Is Δ DEF similar to Δ ABC?

• Yes. The third angles must be equal too.
• No. All three angles must be checked.
• Only if ∠F is measured and found equal to ∠C.

Build a triangle from two angles

Triangle ABC is fixed. Build DEF with the dials: you set ∠D and ∠E and the size. You do not set ∠F. Watch what ∠F does.

Two angles are enough

If two angles of one triangle are equal to two angles of another triangle, then the triangles are similar (AA criterion).

The third angles are equal too, by the angle sum property: ∠F = 180° − ∠D − ∠E = 180° − ∠A − ∠B = ∠C. So all three angles are equal and the AAA criterion applies.

Example: in Δ ABC and Δ DEF, ∠A = ∠D and ∠B = ∠E, so Δ ABC ~ Δ DEF (AA).

Notes

If two angles of one triangle are equal to two angles of another triangle, the triangles are similar (AA criterion). The third angles are equal by the angle sum property.

Check yourself

Which pair of triangles is similar?

A lamp-post AB is 3.6 m tall. A girl CD is 0.9 m tall and stands 4.8 m from the foot B of the post. The tip of her shadow is E, so her shadow is DE. How long is DE, in m?

Answer: 1.6

Let DE = x m. Both triangles have a right angle and share ∠E, so Δ ABE ~ Δ CDE (AA). Then (4.8 + x)/x = 3.6/0.9 = 4. So 4.8 + x = 4x, 3x = 4.8 and x = 1.6 m.

In Δ ABC and Δ DEF, ∠A = ∠D = 90° and ∠B = ∠E. Which statement is correct?

Why is it enough to check only two pairs of angles?

• Triangle 1 has angles 40° and 80°. Triangle 2 has angles 40° and 70°.. The third angles are 60° and 70°. The angles are 40°, 80°, 60° and 40°, 70°, 70°. They do not match.
• Triangle 1 has angles 50° and 60°. Triangle 2 has angles 60° and 70°. — correct. Yes! The third angles are 70° and 50°. Both triangles have the angles 50°, 60° and 70°.
• Triangle 1 has angles 30° and 60°. Triangle 2 has angles 30° and 80°.. The third angles are 90° and 70°. The angles are 30°, 60°, 90° and 30°, 80°, 70°. They do not match.
• Δ ABC ~ Δ DEF by the AA criterion. — correct. Yes! Two angles are equal, so the third angles are equal too (∠C = ∠F).
• They need not be similar. The third angles may differ.. The third angle is fixed by the angle sum: ∠C = 90° − ∠B and ∠F = 90° − ∠E. They are equal.
• They are similar only if AB = DE.. AA does not need equal sides. The triangles may be of different sizes.
• The third angle is always 90°.. No. The third angle can be any size that makes the three add up to 180°.
• The third angles are equal by the angle sum property. — correct. Yes! Each third angle is 180° minus the other two, so the third angles are equal too.
• The third angle does not matter.. It does matter. It is equal too, and that is why two pairs are enough.