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Triangle Angle Relationships

A free Geometry lesson from the “Geometry Foundations” unit, with a worked example and practice problems including step-by-step solutions.

Triangle angle theorems describe how angles inside and around triangles relate. The interior angles of every triangle add to 180 degrees, and an exterior angle equals the sum of the two remote interior angles. These facts matter because they let you solve for missing angles without measuring. When practicing, first decide whether the unknown angle is inside the triangle or outside it, then write an equation using the appropriate theorem. A common mistake is adding all visible angles together without identifying which angles form the triangle.

What you'll learn

Why it matters: Surveyors and animators use triangle angle relationships to reconstruct shapes when only part of a figure can be measured.

Worked example

Problem. A triangle has angles 42 degrees and 68 degrees. Find the third angle.

  1. Triangle angles add to 180 degrees.
  2. 42 + 68 = 110.
  3. 180 - 110 = 70.

Answer: 70 degrees

Practice problems

1. A triangle has angles 50 and 60 degrees. Find the third angle.

Show solution
  1. Warm-up: First identify exactly what the question is asking: A triangle has angles 50 and 60 degrees. Find the third angle.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. 50 + 60 = 110.
  4. 180 - 110 = 70.
  5. Check the result by substituting or estimating: the response should match 70 and make sense in the original problem.

Answer: 70

2. An equilateral triangle has each angle equal to what measure?

Show solution
  1. Warm-up: First identify exactly what the question is asking: An equilateral triangle has each angle equal to what measure?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. 180 divided by 3 is 60.
  4. Check the result by substituting or estimating: the response should match 60 and make sense in the original problem.

Answer: 60

3. A right triangle has one acute angle of 35 degrees. Find the other acute angle.

Show solution
  1. Core Practice: First identify exactly what the question is asking: A right triangle has one acute angle of 35 degrees. Find the other acute angle.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. 90 - 35 = 55.
  4. Check the result by substituting or estimating: the response should match 55 and make sense in the original problem.

Answer: 55

4. An exterior angle is 124 degrees and one remote interior angle is 49 degrees. Find the other.

Show solution
  1. Core Practice: First identify exactly what the question is asking: An exterior angle is 124 degrees and one remote interior angle is 49 degrees. Find the other.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. 124 - 49 = 75.
  4. Check the result by substituting or estimating: the response should match 75 and make sense in the original problem.

Answer: 75

5. Triangle angles are x, 2x, and 3x. Find x.

Show solution
  1. Challenge: First identify exactly what the question is asking: Triangle angles are x, 2x, and 3x. Find x.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. 6x = 180.
  4. Check the result by substituting or estimating: the response should match 30 and make sense in the original problem.

Answer: 30

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