Quadrilaterals • Topic 3 of 3

Non-Parallelogram Quadrilaterals: Trapezium & Kite

What are trapeziums and kites? Not all quadrilaterals have two pairs of parallel sides. Some have only one pair, and some have none at all! These are the non-parallelogram members of the quadrilateral family.

Trapezium: A trapezium (often called a trapezoid) is a quadrilateral with exactly one pair of opposite sides that are parallel. The other two sides are non-parallel. Think of a popcorn bucket profile or a handbag shape.

Isosceles Trapezium: A special type of trapezium where the two non-parallel side legs are exactly equal in length. In this case, the base corner angles are also equal to each other!

Kite: A kite is a quadrilateral that looks exactly like the flying toy you use at the park. It has no parallel sides. Instead, it features two distinct pairs of adjacent sides that are equal in length.

Special Diagonal Properties: The diagonals of a kite cross each other perpendicularly ($90^\circ$). Also, the longer diagonal cuts the shorter diagonal exactly in half like a crossbow structure.

Solved Examples

Worked Example

In a trapezium $ABCD$, side $AB$ is parallel to side $CD$. If $\angle A = 110^\circ$ and $\angle B = 85^\circ$, calculate the values of the remaining base angles $\angle D$ and $\angle C$.

Solution*Step 1: Identify consecutive interior angles between parallel lines. Because $AB \parallel CD$, the angles along the transversal legs ($\angle A$ and $\angle D$) must add up to $180^\circ$. *Step 2: Calculate $\angle D$:* $110^\circ + \angle D = 180^\circ \implies \angle D = 180^\circ - 110^\circ = 70^\circ$. *Step 3: Similarly, the angles along the other leg ($\angle B$ and $\angle C$) must also add up to $180^\circ$:* $\angle B + \angle C = 180^\circ$. *Step 4: Calculate $\angle C$:* $85^\circ + \angle C = 180^\circ \implies \angle C = 180^\circ - 85^\circ = 95^\circ$.

Key Points

A trapezium contains exactly one pair of parallel opposite sides.
In an isosceles trapezium, the non-parallel legs are equal, and the base corner angles match.
A kite has zero parallel sides, but features two pairs of equal adjacent sides.
The diagonals of a kite cross at right angles ($90^\circ$), and the longer diagonal splits the shorter one perfectly in half.
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Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.A trapezium has exactly ___ pair of parallel sides.

Explanation: One pair.

Q2.A kite has two pairs of ___ equal sides.

Explanation: Adjacent.

Q3.The area of a trapezium is:

Explanation: $\tfrac12(a+b)h$.

Q4.The diagonals of a kite are:

Explanation: Perpendicular.

Practice more MCQs → 📝 Assignment · 35 marks