Angle Sum of a Quadrilateral
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Theory
The four angles inside any quadrilateral add to \(360^\circ\). This is the angle sum of a quadrilateral. To find a missing angle, add the angles you know and subtract from \(360^\circ\). This Year 7 maths page shows why, and uses the rule with right angles, reflex angles, special quadrilaterals and pronumerals.
For a quadrilateral with angles \(a^\circ\), \(b^\circ\), \(c^\circ\) and \(d^\circ\): \(a + b + c + d = 360\). Reason: angle sum of a quadrilateral is \(360^\circ\).
Why \(360^\circ\)? A diagonal splits any quadrilateral into two triangles. Each triangle has an angle sum of \(180^\circ\), and together the six triangle angles fill the four corners exactly: \(180 + 180 = 360\). This works for a non-convex quadrilateral too, using the diagonal from the reflex angle.
Finding a missing angle. Write the equation and its reason, add the known angles, then subtract from \(360\): \(a + 95 + 120 + 80 = 360\), so \(a = 360 - 295 = 65\).
Special quadrilaterals give more facts. In a parallelogram opposite angles are equal, and angles next to each other are co-interior, so they add to \(180^\circ\). In a kite the two angles between the unequal sides are equal. In an isosceles trapezium the two angles on each parallel side are equal.
The reflex angle counts. In a non-convex quadrilateral the angle at the dent is more than \(180^\circ\). It is one of the four angles.
The rule (angle sum of a quadrilateral) and the special cases:
| Quadrilateral | Extra angle facts |
|---|---|
| parallelogram | opposite angles equal; angles next to each other add to \(180^\circ\) |
| rectangle, square | every angle \(90^\circ\) |
| kite | the angles between the unequal sides are equal |
| isosceles trapezium | the two angles on each parallel side are equal |
How to find a missing angle in a quadrilateral
- Read every corner: a right-angle mark is \(90^\circ\); a reflex angle is used as it is.
- Write the four angles \(= 360\), with the reason.
- Add the known angles and subtract from \(360\).
- Check that all four angles add to \(360\).
When the angles have pronumerals
- Add the four angles and make them equal to \(360\).
- Collect like terms and solve.
- Substitute to find each angle.
The right-angle mark at \(A\) shows \(90^\circ\).
| \(a + 90 + 67 + 118\) | \(=\) | \(360\) |
| \(a + 275\) | \(=\) | \(360\) |
| \(a\) | \(=\) | \(85\) |
\(a = 85\) (angle sum of a quadrilateral is \(360^\circ\)).
| \(x + 70 + 80 + 100\) | \(=\) | \(360\) |
| \(x + 250\) | \(=\) | \(360\) |
| \(x\) | \(=\) | \(110\) |
\(x = 110^\circ\)
| \(58 + 42 + x + 33\) | \(=\) | \(360\) |
| \(x + 133\) | \(=\) | \(360\) |
| \(x\) | \(=\) | \(227\) |
\(x = 227\). It is more than \(180\), so it is reflex, as the diagram shows.
| \(x + 64\) | \(=\) | \(180\) |
| \(x\) | \(=\) | \(116\) |
\(x = 116\) (co-interior angles, \(AD \parallel BC\)). \(y = 64\) and \(z = 116\) (opposite angles of a parallelogram are equal). Check: \(64 + 116 + 64 + 116 = 360\).
| \(2x + (x + 15) + 3x + 105\) | \(=\) | \(360\) |
| \(6x + 120\) | \(=\) | \(360\) |
| \(6x\) | \(=\) | \(240\) |
| \(x\) | \(=\) | \(40\) |
\(x = 40\). The angles are \(80^\circ\), \(55^\circ\), \(120^\circ\) and \(105^\circ\) (check: they add to \(360\)).
| \(2x + 10 + 40 + 90 + 90\) | \(=\) | \(360\) |
| \(2x + 230\) | \(=\) | \(360\) |
| \(2x\) | \(=\) | \(130\) |
| \(x\) | \(=\) | \(65\) |
\(x = 65\), so \(\angle BCD = 140^\circ\).
(ii) \(ABCD\) is an isosceles trapezium with \(AB \parallel DC\) and \(AD = BC\). \(\angle D = 70^\circ\). Find \(\angle A\) and \(\angle B\).
| \(360 - 100 - 60\) | \(=\) | \(200\) |
| \(200 \div 2\) | \(=\) | \(100\) |
| \(180 - 70\) | \(=\) | \(110\) |
(i) \(\angle Q = \angle S\) (the angles between the unequal sides), so \(\angle Q = 100^\circ\).
(ii) \(\angle A = 110^\circ\) (co-interior angles, \(AB \parallel DC\)) and \(\angle B = \angle A = 110^\circ\) (isosceles trapezium). Check: \(110 + 110 + 70 + 70 = 360\).
Common pitfalls
Frequently asked questions
What do the angles of a quadrilateral add up to?
\(360^\circ\).
Why is the angle sum of a quadrilateral 360 degrees?
A diagonal splits it into two triangles, and \(180 + 180 = 360\).
How do you find the missing angle of a quadrilateral?
Add the three known angles and subtract from \(360\). For \(90^\circ\), \(67^\circ\) and \(118^\circ\): \(360 - 275 = 85\), so the angle is \(85^\circ\).
Does the rule work for non-convex quadrilaterals?
Yes. Include the reflex angle: \(80 + 22 + 236 + 22 = 360\).
What angle facts does a parallelogram have?
Opposite angles are equal, and angles next to each other add to \(180^\circ\).