Quadrilaterals.

Section 2

The Trapezium

The first special quadrilateral. One pair of sides is parallel — and only one. That single parallel pair unlocks co-interior angle reasoning.

Definition

A trapezium is a quadrilateral that has only two parallel sides.

In the figure, AB ∥ DC while DA is not parallel to CB. Therefore ABCD is a trapezium.

Key angle fact

Because AB ∥ DC with AD as a transversal, ∠A and ∠D are co-interior (same-side interior) angles, so m(∠A) + m(∠D) = 180°. The same holds for ∠B and ∠C.
ABCD
Trapezium ABCD: AB ∥ DC, while DA and CB are not parallel.

Two special trapeziums

ABCD
Isosceles trapezium: the two non-parallel sides are equal in length.

1. Isosceles trapezium — the non-parallel sides are equal in length. Its base angles are equal in measure and its diagonals are equal.

ABCD
Right trapezium: one of its angles is a right angle.

2. Right trapezium — one of its angles is a right angle. In fact it then has two right angles, because co-interior angles are supplementary.

Worked examples

Example 3
In quadrilateral ABCD the angles are m(∠A) = 2x°, m(∠B) = 3x°, m(∠C) = 3x°, m(∠D) = 4x°. Find x and decide whether ABCD is a right trapezium.

Solution

  1. 1.2x + 3x + 3x + 4x = 360 → 12x = 360 → x = 30
  2. 2.m(∠A) = 60°, m(∠B) = m(∠C) = 90°, m(∠D) = 120°
  3. 3.m(∠A) + m(∠D) = 60° + 120° = 180°, and they are co-interior on transversal AD
  4. 4.So AB ∥ DC. Since m(∠A) + m(∠B) = 150° ≠ 180°, AD is not parallel to BC.
  5. 5.ABCD is a right trapezium (it has right angles at B and C).
ABCD
Right trapezium: one of its angles is a right angle.
Try it yourself 3
ABCD is an isosceles trapezium with AB ∥ DC. Given DA = (2a + 3) cm, CB = (9 + a) cm, m(∠C) = 127° and m(∠B) = (36 + b)°. Find a and b.
ABCD
Isosceles trapezium: the two non-parallel sides are equal in length.
Try it yourself 4
In trapezium ABCD (AB ∥ DC), m(∠A) = 3x° and m(∠B) = 4x°... but you are told m(∠C) = 112°. Find m(∠D).