Triangle Angle Sum Theorem

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The triangle angle sum theorem says the three interior angles of any triangle add up to 180°. If you know two angles, subtract their sum from 180° to find the third. A related fact, the exterior angle theorem, says an exterior angle equals the sum of the two interior angles that are not next to it.

Before this lesson: You should know Parallel Lines and Transversals, Angle Pairs and Two-Column Proofs.

Why Do a Triangle’s Angles Add to 180°?

Cut a triangle out of paper. Tear off its three corners. Now set the corners side by side, tips touching. They always form a straight line, which is 180°. It works for every triangle, big or small, skinny or wide.

Tearing paper shows the idea, but it is not a proof. A proof uses a line through one vertex, parallel to the opposite side. The picture shows how.

Triangle ABC with base AC along the bottom and vertex B at the top. Its angles are numbered 1 at A, 2 at B and 3 at C. A dashed green line through B is parallel to AC. At B, angle 4 is between the parallel line and side BA, and angle 5 is between side BC and the parallel line. Angle 4 matches angle 1 and angle 5 matches angle 3 as alternate interior angles. Angles 4, 2 and 5 together make a straight line, so they add to 180 degrees.
A line through B parallel to AC copies angles 1 and 3 to the top. Together with angle 2 they form a straight line, 180°.

Triangle Angle Sum Theorem

m\angle A + m\angle B + m\angle C = 180^\circ

Here m\angle A means “the measure of angle A.” The theorem works for every triangle: acute, right or obtuse.

An exterior angle is formed when you extend one side of a triangle past a vertex. It forms a linear pair with the interior angle at that vertex. The two interior angles at the other vertices are the remote interior angles.

Exterior Angle Theorem

m\angle(\text{exterior}) = m\angle(\text{remote 1}) + m\angle(\text{remote 2})
Triangle ABC with side AC extended past C to point D. The interior angle at A is 52 degrees and the interior angle at B is 71 degrees, both marked with solid blue arcs as the remote interior angles. The interior angle at C is 57 degrees, marked in gray. The exterior angle BCD is marked with a dashed orange arc and labeled 123 degrees. A note says 52° + 71° = 123°.
The exterior angle, 123°, equals 52° + 71°, the two interior angles that are not next to it.

Two quick facts follow from the triangle angle sum theorem. These are called corollaries, results that follow right away from a theorem.

  • The two acute angles of a right triangle are complementary. They add to 90°.
  • A triangle can have at most one right angle or one obtuse angle.
  • Each angle of an equiangular triangle measures 180^\circ \div 3 = 60^\circ.

How to Find a Missing Angle in a Triangle

  1. Label what you know. Mark each angle as interior or exterior.
  2. For a missing interior angle, add the two known interior angles.
  3. Subtract that sum from 180°.
  4. For an exterior angle, add the two remote interior angles.
  5. If the angles have variables, write an equation, solve for x, and substitute back.
  6. Check: the three interior angles should add to exactly 180°.

Worked Examples

Example 1: Find the third angle

A triangle has angles of 48° and 67°. Find the third angle.

  1. Add the two known angles: 48^\circ + 67^\circ = 115^\circ
  2. Subtract from 180°: 180^\circ - 115^\circ = 65^\circ
  3. Check: 48^\circ + 67^\circ + 65^\circ = 180^\circ

Answer: The third angle is 65°.

Example 2: Word problem with a variable

A triangular garden bed has three corners. The middle-sized angle is twice the smallest angle. The largest angle is 3 times the smallest. Find all three angles.

  1. Let the smallest angle be x. Then the others are 2x and 3x.
  2. Write the equation: x + 2x + 3x = 180
  3. Combine like terms (see combining like terms): 6x = 180
  4. Divide both sides by 6: x = 30
  5. Find the middle angle: 2(30) = 60
  6. Find the largest angle: 3(30) = 90
  7. Check: 30 + 60 + 90 = 180

Answer: The angles are 30°, 60° and 90°. The garden bed is a right triangle.

Example 3: Use the exterior angle theorem

In triangle ABC, side AC is extended past C to point D. The remote interior angles measure 52° and 71°. Find the exterior angle \angle BCD and the interior angle at C. This is the second picture above.

  1. Add the remote interior angles: 52^\circ + 71^\circ = 123^\circ. So m\angle BCD = 123^\circ.
  2. The interior angle at C and the exterior angle form a linear pair. Subtract from 180°: 180^\circ - 123^\circ = 57^\circ
  3. Check the triangle: 52^\circ + 71^\circ + 57^\circ = 180^\circ

Answer: The exterior angle is 123° and the interior angle at C is 57°.

Example 4: Exterior angle with algebra

An exterior angle of a triangle measures (5x - 10)^\circ. The remote interior angles measure (2x + 15)^\circ and (x + 25)^\circ. Find x and each angle.

  1. Use the exterior angle theorem: 5x - 10 = (2x + 15) + (x + 25)
  2. Combine like terms on the right: 5x - 10 = 3x + 40
  3. Subtract 3x from both sides (see equations with variables on both sides): 2x - 10 = 40
  4. Add 10 to both sides: 2x = 50
  5. Divide both sides by 2: x = 25
  6. Exterior angle: 5(25) - 10 = 125 - 10 = 115
  7. First remote angle: 2(25) + 15 = 50 + 15 = 65
  8. Second remote angle: 25 + 25 = 50
  9. Check: 65 + 50 = 115

Answer: x = 25. The exterior angle is 115°, and the remote interior angles are 65° and 50°.

Example 5: Prove the exterior angle theorem

Given: In triangle ABC, \angle 1 is at A, \angle 2 is at B and \angle 3 is at C. \angle 4 is the exterior angle at C, so \angle 3 and \angle 4 form a linear pair.
Prove: m\angle 4 = m\angle 1 + m\angle 2

StatementsReasons
1. m\angle 1 + m\angle 2 + m\angle 3 = 180^\circTriangle angle sum theorem
2. \angle 3 and \angle 4 form a linear pairGiven
3. m\angle 3 + m\angle 4 = 180^\circLinear pair postulate (see proving angle relationships)
4. m\angle 1 + m\angle 2 + m\angle 3 = m\angle 3 + m\angle 4Substitution (steps 1 and 3)
5. m\angle 1 + m\angle 2 = m\angle 4Subtraction property of equality

The proof of the triangle angle sum theorem itself follows the first picture. Draw a line through B parallel to \overline{AC}. Then \angle 4 \cong \angle 1 and \angle 5 \cong \angle 3 by the alternate interior angles theorem. Angles 4, 2 and 5 form a straight angle, so m\angle 4 + m\angle 2 + m\angle 5 = 180^\circ. Substitute to get m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ.

Common Mistakes

MistakeExample of the errorFix
Using 360° for a triangle360 - (48 + 67) = 245A triangle’s angles add to 180°. The 360° total is for quadrilaterals.
Adding the wrong angles for an exterior angleAdding 57°, the angle next to it, to 52°Add only the two remote angles: 52 + 71 = 123
Stopping at xGiving x = 25 as the angleSubstitute x back into each expression.
Thinking a triangle can have two obtuse anglesAngles of 100° and 95° in one triangle100 + 95 = 195, which is more than 180°, so it is impossible.
Mixing up exterior and interior anglesSetting the exterior angle equal to the interior angle next to itThey form a linear pair, so they add to 180°.

Using the Theorem with Other Figures

  • Right triangles: once you know one acute angle, subtract it from 90° to get the other.
  • Isosceles triangles: the two base angles are equal. You will use this with the angle sum in Isosceles and Equilateral Triangles, later in this unit.
  • Polygons: a polygon can be split into triangles from one vertex. Each triangle adds 180°. That idea gives the angle sum of any polygon in Unit 7.
  • Parallel lines: the proof uses the alternate interior angles from parallel lines and transversals, so this theorem is true in flat (Euclidean) geometry.

Practice Problems

  1. A triangle has angles of 35° and 90°. Find the third angle.
  2. A triangle has angles of 112° and 29°. Find the third angle.
  3. One acute angle of a right triangle measures 41°. Find the other acute angle.
  4. Find the measure of each angle of an equiangular triangle.
  5. The angles of a triangle measure (x + 10)^\circ, (2x)^\circ and (3x - 10)^\circ. Find each angle.
  6. The remote interior angles of an exterior angle measure 38° and 64°. Find the exterior angle.
  7. An exterior angle measures 140°. One remote interior angle measures 85°. Find the other remote interior angle.
  8. An exterior angle measures (4x + 6)^\circ. Its remote interior angles measure (x + 20)^\circ and (2x + 11)^\circ. Find x and the exterior angle.
  9. Complete the proof. Given: Triangle ABC with a right angle at C. Prove: \angle A and \angle B are complementary. Statements: (1) \angle C is a right angle. (2) m\angle C = 90^\circ. (3) m\angle A + m\angle B + m\angle C = 180^\circ. (4) m\angle A + m\angle B + 90^\circ = 180^\circ. (5) m\angle A + m\angle B = 90^\circ. (6) \angle A and \angle B are complementary. Give a reason for each statement.
  10. A ladder leans against a vertical wall. It makes a 66° angle with the level ground. What angle does it make with the wall?
Show answers
  1. Add: 35 + 90 = 125
    Subtract: 180 - 125 = 55
    The third angle is 55°.
  2. Add: 112 + 29 = 141
    Subtract: 180 - 141 = 39
    The third angle is 39°.
  3. The acute angles add to 90°
    Subtract: 90 - 41 = 49
    The other angle is 49°.
  4. All three angles are equal
    Divide: 180 \div 3 = 60
    Each angle is 60°.
  5. Write the equation: (x + 10) + 2x + (3x - 10) = 180
    Combine like terms: 6x = 180
    Divide by 6: x = 30
    Angles: 30 + 10 = 40, 2(30) = 60, 3(30) - 10 = 80
    Check: 40 + 60 + 80 = 180
  6. Add the remote angles: 38 + 64 = 102
    The exterior angle is 102°.
  7. Exterior angle = sum of the remote angles, so subtract: 140 - 85 = 55
    The other remote angle is 55°.
  8. Write the equation: 4x + 6 = (x + 20) + (2x + 11)
    Combine like terms: 4x + 6 = 3x + 31
    Subtract 3x: x + 6 = 31
    Subtract 6: x = 25
    Exterior angle: 4(25) + 6 = 106
    Check: (25 + 20) + (2(25) + 11) = 45 + 61 = 106
  9. (1) Given
    (2) Definition of a right angle
    (3) Triangle angle sum theorem
    (4) Substitution property of equality
    (5) Subtraction property of equality
    (6) Definition of complementary angles
  10. The wall and the ground meet at a right angle, so the triangle has a 90° angle
    The acute angles add to 90°
    Subtract: 90 - 66 = 24
    The ladder makes a 24° angle with the wall.

FAQ

What is the triangle angle sum theorem?

It says the interior angles of every triangle add up to 180°. It is also called the triangle sum theorem.

What is the exterior angle theorem?

An exterior angle of a triangle equals the sum of the two remote interior angles. The remote interior angles are the two angles not next to the exterior angle.

Can a triangle have two right angles?

No. Two right angles already add to 180°, leaving 0° for the third angle. A triangle can have at most one right angle or one obtuse angle.

How do you find the third angle of a triangle?

Add the two angles you know, then subtract the sum from 180°.

Summary

  • The interior angles of a triangle add to 180°.
  • Missing angle = 180° minus the sum of the other two.
  • An exterior angle equals the sum of the two remote interior angles.
  • An exterior angle and the interior angle next to it add to 180°.
  • The acute angles of a right triangle add to 90°.
  • With variables, write an equation, solve, and substitute back.

Previous lesson: Symmetry

Next lesson: Congruent Figures (coming soon)

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