The three angles of any triangle add to 180°.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
So two angles fix the third
If two triangles share two angles, their third angles must match as well.
The angle-angle criterion
Two matching angles are enough to make two triangles similar. No side lengths are needed.
Similar is not congruent
The triangles have the same shape but may be any size, so their sides are proportional rather than equal.
Two angles are enough
If two angles of one triangle equal two angles of another, the triangles are similar. You do not need the third angle, because the three must total 180° — knowing two fixes the third automatically.
Why this works only for triangles
A square and a non-square rectangle have all four angles equal but are not similar. Triangles are special: their angles determine their shape completely. That rigidity is why triangles appear everywhere in structural engineering.
Using it to find lengths
Once AA establishes similarity, corresponding sides are proportional, so a missing length comes from a proportion. This is the standard route: prove similarity with angles, then compute with ratios.
Measuring what you cannot reach
The height of a tree can be found from its shadow and your own, because the sun's rays make similar triangles. Surveying, navigation and astronomy all rest on this idea, and it is one of the oldest applications of geometry there is.
Step 2: Try It Yourself
Tap and try it out.
50° is an acute angle.
The two angles together measure 110°. Angle measures add.
Step 3: Watch an Example
One step at a time.
Watch Leo Check Two Triangles
One triangle has angles 50° and 60°. Another has 60° and 70°.
- Step 1
The first triangle’s third angle is 180 − 110 = 70°.
Step 4: Your Turn
Practice makes it stick.
The Third Angle
Problem 1 of 2
A triangle has angles 45° and 65°. What is the third, in degrees?
The Criterion
Problem 2 of 2
How many matching angles are needed for similarity?
Two Is Enough
1 of 4
Angles 30° and 80°. What is the third?
2 of 4
Angles 90° and 45°. What is the third?
3 of 4
Do similar triangles have equal side lengths? 1 for yes, 0 for no.
4 of 4
A triangle with angles 60, 60 and what?
Step 5: Quick Check
Show what you know.
Question 1 of 1
Angles 25° and 95°. What is the third?
What You Learned
- The angles of a triangle always add to 180°.
- Two matching angles force the third to match, so the triangles are similar.
- Similar triangles have proportional sides, not equal ones.