Every triangle hides the same secret: its three interior angles always add up to 180 degrees. That one fact, plus a few angle rules, finds any missing angle. You can check your work with the free tools on Calc Solver.
The 180-Degree Rule
The interior angles of any triangle, whatever its shape, sum to 180 degrees. If you know two angles, subtract their sum from 180 to find the third. For angles of 50 degrees and 60 degrees, the third angle is 180 minus 110, which is 70 degrees.
Step 1: Write Down What You Know
List the given angles and mark the unknown as x. Check for special clues: a right-angle mark means 90 degrees, and equal side marks mean the base angles are equal. Writing everything down first prevents using the wrong angle later.
Step 2: Apply the Angle Sum
Add the known angles and subtract from 180. For a triangle with angles 35 degrees and 90 degrees, the missing angle is 180 minus 125, which is 55 degrees. If the triangle is isosceles with a vertex angle of 40 degrees, the two base angles are equal, so each is (180 minus 40) divided by 2, which is 70 degrees.
Step 3: Use Exterior Angles When Needed
An exterior angle of a triangle equals the sum of the two opposite interior angles. If an exterior angle is 120 degrees and one opposite interior angle is 50 degrees, the other is 70 degrees. Exterior angles are useful when the problem gives an angle outside the triangle instead of inside it.
Common Mistakes to Avoid
First, adding to 90 instead of 180, which is the rule for right angles, not triangles. Second, forgetting a right-angle mark and treating it as unknown. Third, assuming a triangle is isosceles from its drawing alone: only side marks or given information prove it.
Practice Problems to Try
Find the missing angle when the other two are 65 and 75 degrees, when one is a right angle and the other is 28 degrees, and when an isosceles triangle has a vertex angle of 100 degrees.