Section 3

The Parallelogram

Two pairs of parallel sides instead of one. That extra pair forces the sides, the angles and even the diagonals into a strict pattern you can use in every proof.

Definition

A parallelogram is a quadrilateral in which each two opposite sides are parallel: AB ∥ DC and AD ∥ BC.

Parallelogram ABCD: AB ∥ DC and AD ∥ BC.

The diagonals meet at M and bisect each other.

The four properties

Property 1

Each two opposite sides are equal in length

AB = DC and BC = AD.

Property 2

The diagonals bisect each other

If the diagonals meet at M, then AM = CM and BM = DM.

Property 3

Each two opposite angles are equal in measure

m(∠A) = m(∠C) and m(∠B) = m(∠D).

Property 4

Each two consecutive angles are supplementary

m(∠A) + m(∠B) = 180°, m(∠B) + m(∠C) = 180°, and so on around the shape.

Reading the notation

m(∠A) means “the measure of angle A”. A pair of consecutive angles sits at the two ends of the same side; a pair of opposite angles sits across the diagonal from each other.

Worked examples

Example 4

ABCD is a parallelogram with m(∠A) = (3x − 50)°, m(∠C) = 2x° and m(∠B) = y°. Find x and y.

Solution

1.Opposite angles are equal: 3x − 50 = 2x → x = 50
2.m(∠A) = 2 × 50° = 100°
3.Consecutive angles: m(∠A) + m(∠B) = 180° → 100 + y = 180
4.x = 50, y = 80

Opposite angles are equal; consecutive angles are supplementary.

Example 4b

ABCD is a parallelogram with AB = (4x − 7) cm, DC = (2x) cm, AD = (y − 5) cm and BC = 5 cm. Find x and y.

Solution

1.AB = DC → 4x − 7 = 2x → 2x = 7 → x = 3.5
2.AD = BC → y − 5 = 5 → y = 10
3.x = 3.5, y = 10

Opposite sides of a parallelogram are equal in length.

Try it yourself 5

ABCD is a parallelogram whose diagonals meet at M, with MD = 6 cm, MC = 8 cm and DC = 9 cm. Calculate the perimeter of triangle CMD.

Parallelogram ABCD: AB ∥ DC and AD ∥ BC.

Try it yourself 6

Architecture: a balcony railing forms parallelogram ABCD with m(∠B) = (3x + 37)° and m(∠D) = (9x + 1)°. What is the measure of ∠C?