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
\[a+b+c+d=360^\circ\]
- 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° \(\displaystyle A=C,\quad A+B=180^\circ\)
- 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. \(\displaystyle B=D\)
Worked example
A parallelogram ABCD represents a paving section. The interior angle at A is 116°. Find the interior angle at B.
\[116^\circ+x=180^\circ\]
- \(\displaystyle \angle A+\angle B=180^\circ\)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
Using the triangle total
✕\(\displaystyle a+b+c+d=180^\circ\)✓\(\displaystyle a+b+c+d=360^\circ\)
A quadrilateral has four interior angles, so their total is 360°, not the triangle total of 180°Giving every parallelogram right angles
✕\(\displaystyle A=B=C=D=90^\circ\)✓\(\displaystyle A=C,\quad B=D\)
A general parallelogram has equal opposite angles, but all four right angles require a rectangle or squareConfusing a rhombus with a square
✕\(\displaystyle A=B=C=D=90^\circ\)✓\(\displaystyle AB=BC=CD=DA\)
Equal sides alone do not force right angles; a square has equal sides and four right anglesDifficulty levels
introThree known angles are used with the 360° quadrilateral sum
standardOne defining property of a special quadrilateral gives the unknown angle
challengeA compound property classification or a two-step parallelogram angle calculation is required
examA property choice is followed by one exact angle calculation
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°
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 7: Geometry and similarity
In this chapter
Angle facts and parallel linesTriangle angle sum, special triangles and the exterior angle theoremQuadrilateral angle sums and special-quadrilateral propertiesPolygon angle sums and regular polygonsCongruent triangle testsProve congruence and use it to prove a further resultEnlargement transformation and scale factor7G Enlargement and scale: Similar figures and scale factorsTests for similar triangles7H Similar triangle tests: Similar figures and scale factorsApplied similarity and scaleSimilar solids and scaling volumeDesign, test and refine geometrical algorithms