Quadrilateral Geometry
Each special quadrilateral has its own angle and diagonal properties: a parallelogram's opposite angles are equal, a rhombus's diagonals meet at right angles, and a kite has one pair of equal angles. This NSW Year 9 Advanced maths subtopic revises them from Stage 4 (NESA outcome MA4‑GEO‑C‑01).
You will find unknown angles in kites, rhombuses, rectangles, squares and parallelograms, using the angle sum of \(360^\circ\) and giving reasons. You will form equations for angles in a parallelogram and a kite, find an angle made by an equilateral triangle inside a square, prove that opposite angles of a parallelogram are equal, and use an angle bisector to find a length. It is part of Year 9 Advanced Properties of Geometric Figures.
Theory
Every quadrilateral has an angle sum of \(360^\circ\), and the special quadrilaterals add properties of their own: parallel sides, equal sides and diagonals that bisect or meet at right angles. This Year 9 Advanced maths page revises the earlier properties of parallelograms, rectangles, rhombuses, squares, kites and trapeziums, and uses them in multi-step angle problems with a reason for each step.
A quadrilateral is a closed shape with four straight sides. A diagonal splits it into two triangles, so its angles add to \(2 \times 180^\circ = 360^\circ\).
A parallelogram has both pairs of opposite sides parallel. A rectangle is a parallelogram with right angles, a rhombus is a parallelogram with four equal sides, and a square is both. A kite has two pairs of equal adjacent sides. A trapezium has at least one pair of parallel sides.
Most angle questions combine three kinds of fact: parallel lines (alternate angles are equal, co-interior angles add to \(180^\circ\)), triangles (angle sum \(180^\circ\), equal angles opposite equal sides) and the diagonal properties of the shape.
A diagonal of a rectangle or square is cut in half by the other diagonal. So the four half-diagonals are equal, and the diagonals make isosceles triangles. In a rhombus the diagonals also bisect the angles and cross at \(90^\circ\).
In geometry answers, every step needs a short reason in brackets, for example (co-interior angles, \(AB \parallel DC\)).
Angle facts used in quadrilateral problems:
Properties of the special quadrilaterals:
| Shape | Sides | Angles | Diagonals |
|---|---|---|---|
| Parallelogram | opposite sides parallel and equal | opposite angles equal | bisect each other |
| Rectangle | opposite sides parallel and equal | all \(90^\circ\) | equal, bisect each other |
| Rhombus | all four equal, opposite sides parallel | opposite angles equal | bisect each other at \(90^\circ\), bisect the angles |
| Square | all four equal, opposite sides parallel | all \(90^\circ\) | equal, bisect each other at \(90^\circ\), make \(45^\circ\) with the sides |
| Kite | two pairs of equal adjacent sides | one pair of equal angles, between the unequal sides | meet at \(90^\circ\); the axis of symmetry bisects the other diagonal |
| Trapezium | one pair of parallel sides | co-interior angles between the parallel sides add to \(180^\circ\) | no special property |
How to find an unknown angle in a quadrilateral
- Name the shape and mark everything you know on the diagram: parallel sides, equal sides, right angles.
- List the properties of that shape that involve angles or diagonals (see the Formulas table).
- Find a triangle or a pair of parallel lines that contains the unknown and at least two known facts.
- Write each step with its reason. When the angles are expressions, set up one equation and solve it.
- Check that the four angles add to \(360^\circ\), or that each triangle adds to \(180^\circ\).
Writing reasons
- Parallel lines: (alternate angles, \(AB \parallel DC\)), (co-interior angles, \(AD \parallel BC\)).
- Shapes: (opposite angles of a parallelogram), (diagonals of a rhombus bisect its angles), (diagonals of a rectangle are equal and bisect each other).
- Triangles: (angle sum of a triangle), (angles opposite equal sides), (exterior angle of a triangle).
- To prove a property, start from facts you are given, never from the result you are trying to show. For example, opposite angles of a parallelogram are equal because each one is \(180^\circ\) minus the same neighbouring angle (co-interior angles twice).
(i) \(\angle WXZ\)
(ii) \(\angle MWX\).
(i) First find the whole angle at \(X\), then halve it.
| \(\angle WXY\) | \(=\) | \(180^\circ - 76^\circ\) |
| \(\) | \(=\) | \(104^\circ\) |
(co-interior angles, \(WZ \parallel XY\))
| \(\angle WXZ\) | \(=\) | \(\dfrac{104^\circ}{2}\) |
| \(\) | \(=\) | \(\mathbf{52^\circ}\) |
(the diagonals of a rhombus bisect its angles)
(ii) The diagonal \(WY\) bisects \(\angle ZWX\).
| \(\angle MWX\) | \(=\) | \(\dfrac{76^\circ}{2}\) |
| \(\) | \(=\) | \(\mathbf{38^\circ}\) |
Check in \(\triangle WMX\), where the diagonals meet at \(90^\circ\): \(38^\circ + 52^\circ + 90^\circ = 180^\circ\).
A kite has one pair of equal angles: the two between the unequal sides, at \(Q\) and \(S\). Call each one \(x\) and use the angle sum of a quadrilateral.
| \(58 + 84 + x + x\) | \(=\) | \(360\) |
| \(142 + 2x\) | \(=\) | \(360\) |
| \(2x\) | \(=\) | \(218\) |
| \(x\) | \(=\) | \(109\) |
\(\therefore\ \angle PQR = \mathbf{109^\circ}\) (angle sum of a quadrilateral, equal angles of a kite).
The \(58^\circ\) and \(84^\circ\) angles sit on the axis of symmetry \(PR\). They are not equal to each other.
\(\angle DAB\) and \(\angle ABC\) are neighbouring angles, so they are co-interior angles between the parallel sides \(AD\) and \(BC\). They add to \(180^\circ\).
| \((2x + 15) + (3x - 5)\) | \(=\) | \(180\) |
| \(5x + 10\) | \(=\) | \(180\) |
| \(5x\) | \(=\) | \(170\) |
| \(x\) | \(=\) | \(\mathbf{34}\) |
| \(\angle DAB\) | \(=\) | \(2(34) + 15 = 83^\circ\) |
| \(\angle ABC\) | \(=\) | \(3(34) - 5 = 97^\circ\) |
Opposite angles of a parallelogram are equal, so \(\angle BCD = \mathbf{83^\circ}\) and \(\angle CDA = \mathbf{97^\circ}\).
Check: \(83 + 97 + 83 + 97 = 360\).
Look for equal sides: \(AD = DC\) (square) and \(DC = DE\) (equilateral triangle), so \(AD = DE\) and \(\triangle ADE\) is isosceles.
| \(\angle ADE\) | \(=\) | \(90^\circ + 60^\circ = 150^\circ\) |
| \(\angle DEA\) | \(=\) | \(\dfrac{180^\circ - 150^\circ}{2}\) |
| \(\) | \(=\) | \(15^\circ\) |
(angles opposite equal sides, angle sum of a triangle)
In the same way \(\triangle BCE\) is isosceles, so \(\angle CEB = 15^\circ\). The angle of the equilateral triangle at \(E\) is \(60^\circ\).
| \(\angle AEB\) | \(=\) | \(60^\circ - 15^\circ - 15^\circ\) |
| \(\) | \(=\) | \(\mathbf{30^\circ}\) |
Common pitfalls
Frequently asked questions
What are the properties of a rhombus?
A rhombus has four equal sides and both pairs of opposite sides parallel. Its opposite angles are equal, and its diagonals bisect each other at right angles and bisect the angles of the rhombus.
Do the diagonals of a rectangle bisect its angles?
No. The diagonals of a rectangle are equal and bisect each other, but they only bisect the angles when the rectangle is a square. They do make isosceles triangles, because the four half-diagonals are equal.
What is the difference between a kite and a rhombus?
A kite has two pairs of equal sides that are next to each other. A rhombus has all four sides equal. Every rhombus is a kite, but most kites are not rhombuses because their sides are not all equal and they have no parallel sides.
How do you find a missing angle in a quadrilateral?
Add the known angles and subtract from \(360^\circ\). If the angles are expressions, add them, set the total equal to \(360\) and solve. For special quadrilaterals use their properties too, such as equal opposite angles in a parallelogram.
Is a square a rhombus?
Yes. A square has four equal sides, so it is a rhombus. It is also a rectangle, a parallelogram and a kite, so it has every property of those shapes.
Why are opposite angles of a parallelogram equal?
Each angle and its neighbour are co-interior angles between parallel sides, so they add to \(180^\circ\). Two opposite angles are both \(180^\circ\) minus the same neighbouring angle, so they are equal.