The Isosceles Trapezium | Moosa Academy

The Isosceles Trapezium — Moosa Academy

A trapezium has just one pair of parallel sides, so it is not a parallelogram at all. But if its two slanted sides are equal, it gains a line of symmetry — and with it, equal base angles and equal diagonals. Slide the shorter base out until the two bases match, and the shape turns into a rectangle.

Concept The shape and its parts

AB

and

DC

are the two bases — the parallel pair.

AD

and

BC

are the legs. In an isosceles trapezium the legs are equal, and the dashed line is its axis of symmetry.

Theorem The four properties

Equal legs:

AD = BC

Equal base angles:

\angle A = \angle B

and

\angle C = \angle D

Equal diagonals:

AC = BD

One line of symmetry, through the midpoints of the two bases

Because $AB \parallel DC$ , each leg is a transversal across the parallel bases, so a base angle and the angle above it are co-interior: $\angle A + \angle D = 180^\circ$ .

Concept Why the diagonals are equal

Compare triangles $ABC$ and $BAD$ :

AB

is shared by both

BC = AD

— the legs are equal

\angle ABC = \angle BAD

— the base angles are equal

The triangles are congruent by SAS, so their remaining sides match:

AC = BD

Note what this does not say: unlike a parallelogram, the diagonals of a trapezium do not bisect each other. They are the same length, but they cross away from their midpoints.

Example Finding all four angles

In an isosceles trapezium $ABCD$ with $AB \parallel DC$ , the base angle $\angle A = 62^\circ$ . Find the other three angles.

\angle B = \angle A = 62^\circ

— base angles are equal

\angle D = 180^\circ - 62^\circ = 118^\circ

— co-interior with

\angle A

\angle C = \angle D = 118^\circ

— the other pair of base angles

Check: $62 + 62 + 118 + 118 = 360^\circ$ , as any quadrilateral must give.

Note When it becomes a rectangle

Stretch the shorter base until the two bases are equal. The legs stop slanting, and every base angle becomes 90°:

Property	Isosceles trapezium	Rectangle
Bases	Unequal	Equal
Angles	Equal in pairs	All 90°
Diagonals	Equal	Equal and bisecting
Lines of symmetry	1	2

Equal diagonals is the property the two shapes share. What the rectangle adds is that its diagonals also bisect each other, because it is a parallelogram and the trapezium is not.

Summary

A trapezium has exactly one pair of parallel sides, called the bases.
It is isosceles when the two legs are equal.
Its base angles are then equal in pairs, and adjacent angles add to 180°.
Its diagonals are equal, but they do not bisect each other.
It has exactly one line of symmetry, through the midpoints of the bases.
When the two bases become equal, the shape is a rectangle.