Quadrilaterals.

Section 1

The Sum of the Interior Angles

Split any quadrilateral with one diagonal and you get two triangles. Two triangles carry 180° each — so the quadrilateral carries 360°.

The rule and its proof

The sum of the measures of the interior angles of any quadrilateral equals 360°.

Draw the diagonal AC of quadrilateral ABCD. It creates triangle ABC and triangle ACD.

  1. 1.In △ABC: m(∠1) + m(∠2) + m(∠3) = 180°
  2. 2.In △ACD: m(∠4) + m(∠5) + m(∠6) = 180°
  3. 3.Adding: [m(∠1)+m(∠6)] + m(∠2) + [m(∠3)+m(∠4)] + m(∠5) = 360°
  4. 4.So m(∠A) + m(∠B) + m(∠C) + m(∠D) = 360°

Why it works

∠1 and ∠6 together rebuild the full angle A, and ∠3 with ∠4 rebuild the full angle C. Nothing is lost or added by drawing the diagonal.
ABCD162345
Diagonal AC splits quadrilateral ABCD into two triangles: 180° + 180° = 360°.

Worked examples

Example 1
In the opposite figure, find with proof the value of x. Triangle ABC has angles 50° and 70° at A and B, and CFED is a quadrilateral sharing the vertical angle at C.

Solution

  1. 1.In △ABC: m(∠ACB) + 50° + 70° = 180° → m(∠ACB) = 60°
  2. 2.m(∠DCF) = m(∠ACB) = 60° (vertically opposite angles)
  3. 3.CFED is a quadrilateral → x + 130 + 85 + 60 = 360
  4. 4.x = 360 − 275 = 85
ACBFED50°70°130°85°x°
Example 1: triangle ABC meets quadrilateral CFED at the vertical angles at C.
Example 2
In quadrilateral ABCD, BC ∥ AE and BF is a transversal. Given m(∠B) = 72°, m(∠C) = 125° and m(∠D) = 120°, find x = m(∠EAD).

Solution

  1. 1.m(∠FAE) = m(∠B) = 72° (corresponding angles)
  2. 2.m(∠DAB) = 360° − (125° + 120° + 72°) = 43°
  3. 3.m(∠FAE) + m(∠EAD) + m(∠DAB) = 180° (straight line)
  4. 4.72 + x + 43 = 180 → x = 65
ABCDx°125°120°72°
Quadrilateral with three known angles.
Try it yourself 1
A quadrilateral has angles 96°, 88° and 104°. Find the fourth angle x, giving your reason.
ABCDx°96°88°104°
Quadrilateral with three known angles.
Try it yourself 2
The four angles of a quadrilateral are in the ratio 2 : 3 : 3 : 4. Find each angle.

Common mistakes

  • Using 180° instead of 360° — that is the triangle rule, not the quadrilateral rule.
  • Adding an exterior angle into the sum. Convert it first: interior = 180° − exterior.
  • Forgetting to justify each step; every jump needs a reason in a proof question.