Quadrilateral angle facts
A quadrilateral has four interior angles whose total is 360°. Properties of special quadrilaterals can also give equal, supplementary or right angles
- Quadrilateral sum The four interior angles total 360° \(\displaystyle a+b+c+d=360^\circ\)
- Parallelogram Both opposite side pairs are parallel. Opposite angles are equal and adjacent angles total 180° \(\displaystyle \begin{gathered}A=C\\A+B=180^\circ\end{gathered}\)
- Rhombus A rhombus is a parallelogram with four equal sides. Its angles need not be right angles \(\displaystyle AB=BC=CD=DA\)
- Rectangle and square Both have four right angles. A square also has four equal sides. Every square is a rectangle, a rhombus and a parallelogram \(\displaystyle A=B=C=D=90^\circ\)
- Trapezium In these exercises, a trapezium has exactly one pair of parallel sides. The angles along each side joining the parallel sides total 180°. For PQRS with PQ parallel to SR, angles P and S form one such pair \(\displaystyle P+S=180^\circ\)
- Kite In these exercises, a kite has two equal adjacent side pairs of unequal lengths. The opposite angles between the unequal sides are equal. If AB = AD and CB = CD, then B = D. \(\displaystyle B=D\)
- Why the total is 360 degrees An interior diagonal divides a simple quadrilateral into two triangles. Their angles recombine into the four interior angles. Include the reflex angle for a concave quadrilateral \(\displaystyle 2\times180^\circ=360^\circ\)
- Why opposite angles are equal In a parallelogram, adjacent pairs are cointerior angles between parallel sides. Equating two 180-degree totals and subtracting their common angle gives equal opposite angles
Worked example
A parallelogram ABCD represents a paving section. The interior angle at A is 116°. Find the interior angle at B.
- \(\displaystyle \angle A+\angle B=180^\circ\)Adjacent interior angles of a parallelogram are supplementary
- \(\displaystyle 116^\circ+x=180^\circ\)Substitute the given angle
- \(\displaystyle x=180^\circ-116^\circ\)Subtract 116° from 180° to isolate the unknown angle
- \(\displaystyle \angle B=64^\circ\)The interior angle at B is 64°
Common mistakes
Difficulty levels
Questions about this topic
Why do the interior angles of a quadrilateral add to 360°?+
Choose a diagonal lying inside the quadrilateral. It divides the shape into two triangles, each with a 180° interior total, giving 360° altogether
What is the difference between a rhombus and a square?+
Both have four equal sides. A square also has four right angles, while a rhombus does not have to have right angles
Which angles are equal in a parallelogram?+
Opposite interior angles are equal. Each pair of adjacent interior angles adds to 180°
Does the 360-degree total include concave quadrilaterals?+
Yes, for a simple quadrilateral. Include its reflex interior angle and choose a diagonal inside the shape. Self-crossing figures are outside this rule
Is every rectangle a square?+
No. A rectangle with unequal adjacent sides is a counterexample. Every square is a rectangle, but the reverse implication does not hold
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 7: Geometry and similarity