Quadrilaterals.

Section 4

When Does a Quadrilateral Become a Parallelogram?

You do not have to prove everything. Any one of these five conditions is enough on its own to conclude that a quadrilateral is a parallelogram.

The five conditions

A quadrilateral is a parallelogram if any one of the following conditions holds.
  1. 01

    Each two opposite sides are parallel.

  2. 02

    Each two opposite sides are equal in length.

  3. 03

    Two opposite sides are both parallel and equal in length.

  4. 04

    The diagonals bisect each other.

  5. 05

    Each two opposite angles are equal in measure.

Proof template

State what is given → derive one condition using angle rules (corresponding, alternate, co-interior, vertically opposite) or length rules → name the condition → conclude “therefore ABCD is a parallelogram”.

Worked examples

Example 5
Find the values of a and b that make quadrilateral ABCD a parallelogram, where DC = 2a cm, AB = 10 cm, m(∠ABD) = (b − 21)° and m(∠BDC) = 30°.

Solution

  1. 1.Use the condition: two opposite sides both parallel and equal.
  2. 2.DC = AB → 2a = 10 → a = 5
  3. 3.DC ∥ AB → m(∠ABD) = m(∠BDC) (alternate interior angles) → b − 21 = 30
  4. 4.a = 5, b = 51
ABCD10 cm2a cm(b−21)°30°
DC and AB must be both equal and parallel.
Example 6
ABCD is a parallelogram and H lies on ray AB with AB = BH. Prove that BHCD is a parallelogram.

Solution

  1. 1.ABCD is a parallelogram (given) → AB = DC
  2. 2.AB = BH (given) → DC = BH
  3. 3.AB ∥ CD and H ∈ ray AB → BH ∥ DC
  4. 4.BH and DC are opposite sides that are both equal and parallel
  5. 5.∴ BHCD is a parallelogram
ABCDH
AB is extended to H so that AB = BH.
Try it yourself 7
ABCD is a quadrilateral with AB ∥ DC, m(∠DAB) = 70°, m(∠ECB) = 110° where E ∈ DC. Prove that ABCD is a parallelogram.
ABCD
Parallelogram ABCD: AB ∥ DC and AD ∥ BC.
Try it yourself 8
In quadrilateral ABCD, m(∠A) = (4x + 10)° and m(∠C) = (3x − 40)° with m(∠B) = 50° and AB ∥ DC. Find x and prove ABCD is a parallelogram.

Choosing the right condition

What the question gives youCondition to use
Two side lengths in terms of x and yOpposite sides equal in length
Angles at A and C onlyOpposite angles equal in measure
Diagonals meeting at a point M with segments givenDiagonals bisect each other
One pair marked parallel plus one lengthOne pair both parallel and equal
Two pairs of parallel marksOpposite sides parallel