Two angles are enough
Two angles decide the third. That is enough to fix a shape.
- A triangle with angles 50°, 60° and one more.
- Mark the two known angles.
- The third angle must be 70°.
- A different-sized triangle, same two angles.
Two angles are enough
The three angles of a triangle add up to 180°. So two angles decide the third. Here 180° − 50° − 60° = 70°.
Take two triangles with two matching angles. The third angle of each is 180° minus the other two. So all three angles match. Then their sides are in one ratio. This is the angle-angle test, or AA. It needs no side lengths.
The three angles of a triangle add up to 180°. Suppose two angles of one triangle equal two angles of another. Then the triangles are similar.
Example: take triangles PQR and ABC. ∠P = ∠A = 50° and ∠Q = ∠B = 60°. So ∠R = 70°, and ∠C = 70°. By AA, triangle PQR is similar to triangle ABC.
Watch out!
Do not use AA for four-sided figures. A square and a 3 by 5 rectangle have equal angles. The square has side 3. They are not similar. Their side ratios, 3 ÷ 3 and 5 ÷ 3, differ.
Slide one angle until the triangles are similar.
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