Properties of a Trapezium

Learn about the quadrilateral with exactly one pair of parallel sides

What is a Trapezium?

Definition:

A trapezium is a quadrilateral with exactly ONE pair of parallel sides

(Different from parallelogram which has TWO pairs of parallel sides)

Look at trapezium PQRS below. Notice only the top and bottom sides are parallel:

Trapezium PQRS with PQ // SR (one pair of parallel sides)

Parallel Sides:

$PQ \parallel SR$

(Only one pair!)

Non-Parallel Sides:

PS and QR are NOT parallel

(They would meet if extended)

The Key Angle Property

Sum of Angles Between Parallel Sides = 180°

The two angles on the same slanted side of the trapezium (between the parallel sides) add up to 180°.

∠w + ∠z = 180° (left side) and ∠x + ∠y = 180° (right side)

Left Side:

$\angle w + \angle z = 180°$

Right Side:

$\angle x + \angle y = 180°$

Why Does This Work?

PQ and SR are parallel, and each slanted side (PS or QR) runs from one to the other. The two angles at the ends of a slanted side lie between the parallel sides, and two such angles always add up to 180°.

Example: 64° + 116° = 180° ✓

Trapezium vs Parallelogram

It's important to know the difference between a trapezium and a parallelogram:

Trapezium

ONE pair of parallel sides

Only 1 pair of parallel sides
Opposite angles are NOT equal
The two angles on each slanted side sum to 180°

Parallelogram

TWO pairs of parallel sides

2 pairs of parallel sides
Opposite angles ARE equal
ALL adjacent angles sum to 180°

Special Case: Isosceles Trapezium

An isosceles trapezium has equal non-parallel sides. The two angles on each parallel side are equal.

Isosceles trapezium: ∠A = ∠B and ∠C = ∠D

In an isosceles trapezium: $\angle A = \angle B$ and $\angle C = \angle D$

Worked Examples

Example 1: Finding an angle on the same slanted side

PQRS is a trapezium. PQ // RS. ∠SPQ = 116°. Find ∠PSR.

Find ∠PSR

Solution:

∠SPQ and ∠PSR are on the same slanted side, PS (on the left side, between the parallel sides PQ and SR)

The two angles on the same slanted side add up to 180°

$\angle PSR = 180° - 116°$

Answer: $\angle PSR = 64°$

Example 2: Finding two unknown angles

WXYZ is a trapezium. WX // ZY. ∠WXY = 125° and ∠WZY = 78°. Find ∠a and ∠c.

Find ∠a and ∠c

Solution:

Finding ∠a: ∠a and ∠WZY are on the same slanted side (left side)

$\angle a = 180° - 78° = 102°$

Finding ∠c: ∠c and ∠WXY are on the same slanted side (right side)

$\angle c = 180° - 125° = 55°$

Answer: $\angle a = 102°$ , $\angle c = 55°$

Example 3: Finding the angles and checking

ABCD is a trapezium. AB // DC. ∠CDA = 74° and ∠ABC = 100°. Find ∠y and ∠DAB. Then check that the four angles add up to 360°.

Find ∠y and ∠DAB

Solution:

Step 1: Find ∠y. ∠y (that is, ∠BCD) and ∠ABC are on the same slanted side, BC (right side):

$\angle y = 180° - 100° = 80°$

Step 2: Find ∠DAB. ∠DAB and ∠CDA are on the same slanted side, AD (left side):

$\angle DAB = 180° - 74° = 106°$

Step 3: Check that the four angles add up to 360°:

$74° + 80° + 100° + 106° = 360°$ ✓

Answer: $\angle y = 80°$ , $\angle DAB = 106°$

Practice Problems

Practice 1: Find the unknown angle

EFGH is a trapezium. EF // HG. ∠FEH = 72°. Find ∠EHG.

Find ∠EHG

ShowHide solution

∠FEH and ∠EHG are on the same slanted side, EH (left side), so they add up to 180°

$\angle EHG = 180° - 72°$

$\angle EHG = 108°$

Practice 2: Multiple unknown angles

JKLM is a trapezium. JK // ML. ∠KJM = 95° and ∠KLM = 120°. Find ∠JKL and ∠JML.

Find ∠JKL and ∠JML

ShowHide solution

∠JML (on the same slanted side as ∠KJM):

$\angle JML = 180° - 95° = 85°$

∠JKL (on the same slanted side as ∠KLM):

$\angle JKL = 180° - 120° = 60°$

Answer: ∠JKL = 60°, ∠JML = 85°

Practice 3: Isosceles trapezium

RSTU is an isosceles trapezium. RS // UT. ∠SRU = 65°. Find ∠TUR, ∠RST, and ∠STU.

Isosceles trapezium RSTU

ShowHide solution

∠RST (on the same parallel side as ∠SRU, so equal to it in an isosceles trapezium):

$\angle RST = \angle SRU = 65°$

∠TUR (on the same slanted side as ∠SRU):

$\angle TUR = 180° - 65° = 115°$

∠STU (on the same parallel side as ∠TUR, so equal to it in an isosceles trapezium):

$\angle STU = \angle TUR = 115°$

Answer: ∠TUR = 115°, ∠RST = 65°, ∠STU = 115°

Practice 4: Verify your understanding

Mei says that in trapezium ABCD, with AB // DC, ∠A = 80° and ∠D = 100°. Can she be right?

Can ∠A = 80° and ∠D = 100°?

ShowHide solution

Check: ∠A and ∠D are on the same slanted side, AD (left side), so they must add up to 180°

$\angle A + \angle D = 80° + 100° = 180°$ ✓

Yes, she can be right.

Key Takeaways

A trapezium has exactly ONE pair of parallel sides
The two angles on the same slanted side (between the parallel sides) sum to 180°
This is because each slanted side runs from one parallel side to the other
An isosceles trapezium has equal non-parallel sides, and the two angles on each parallel side are equal
All four angles still add up to 360°