Lesson 4 · The Triangle

Section 01

Sum of Measures of the Interior Angles

Every triangle — big, small, thin or wide — has interior angles that add up to exactly 180°. Here is why, and how to use it.

Rule

The sum of the measures of the interior angles of any triangle is 180°.
132BCAm∠1 + m∠2 + m∠3 = 180°
Angles 1, 2 and 3 are the interior angles of triangle ABC
mnABC41523
Proof: line m through A is parallel to line n through BC

The Proof

Straight lines m and n are parallel. Line k cuts them at A and B, and line p cuts them at A and C.

  1. ∵ ∠4, ∠1 and ∠5 together form a straight angle at A
  2. ∴ m(∠4) + m(∠1) + m(∠5) = 180°
  3. ∵ m(∠4) = m(∠2) (alternate interior angles)
  4. ∵ m(∠5) = m(∠3) (alternate interior angles)
  5. ∴ m(∠2) + m(∠1) + m(∠3) = 180° ∎

Example 1 — Angles written as expressions

The three interior angles of a triangle measure (3x − 1)°, (4x − 2)° and (2x + 21)°. Find the value of x.

  1. ∵ The sum of the interior angles = 180°
  2. ∴ (3x − 1) + (4x − 2) + (2x + 21) = 180
  3. ∴ 9x + 18 = 180
  4. ∴ 9x = 162
  5. ∴ x = 162 ÷ 9 = 18

Check: 53° + 70° + 57° = 180° ✔

Example 2 — With parallel lines

In triangle ABC, DE ∥ BC, m(∠A) = 75° and m(∠C) = 60°. Find, with proof, m(∠ADE).

  1. ∵ DE ∥ BC and AC is a transversal
  2. ∴ m(∠AED) = m(∠ACB) = 60° (corresponding angles)
  3. In △ADE: sum of interior angles = 180°
  4. ∴ m(∠ADE) = 180° − (75° + 60°) = 45°
ABCDE75°60°?
DE is parallel to BC, so corresponding angles are equal

Try it yourself

  1. A triangle has angles of measure y°, (y + 20)° and (2y − 8)°. Find y. (Answer: y = 42)
  2. In triangle ABC, m(∠A) = 70° and m(∠C) = 50°. Find m(∠B). (Answer: 60°)
  3. If all three angles are equal, what is each one? (Answer: 60°)