Angles in a Quadrilateral

Use the fact that the angles in a quadrilateral add up to 360°, with the extra angle rules for a parallelogram, rhombus and kite, to find missing angles and solve for x.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024 · 26 real past-paper questions below

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Four Angles Make 360°

Draw one diagonal across any quadrilateral. It makes two triangles, and each has $18 0^{\circ}$ .

Formula

a + b + c + d = 36 0^{\circ}

The 3-mark question: a quadrilateral has angles $5 8^{\circ}$ , $9 6^{\circ}$ and two equal angles $x$ .

Worked example

1
$58 + 96 + 2 x = 360$ — Angles in a quadrilateral add up to 360°. The two equal angles together make 2x.
2
$154 + 2 x = 360$ — Add the two angles you know: 58 + 96 = 154.
3
$2 x = 206$ — Take 154 from both sides, so the x's are alone: 360 − 154 = 206.
4
$x = 10 3^{\circ}$ — Two equal angles share the 206°, so divide by 2.
5
$58 + 96 + 103 + 103 = 360$ — Check: all four angles add up to 360°, so the answer is right.

Common mistake

Taking 154 from 360 and stopping at 206. Two angles share it, so halve it.

Key idea

Write the first line, $58 + 96 + 2 x = 360$ , in full. That line alone earns a mark for working, even if a later step slips.

Parallelogram, Rhombus and Kite

The 3-mark question names a special quadrilateral. It gives you extra facts, so one angle can be enough.

Shape	Angle facts
Parallelogram, rhombus	opposite angles are equal
Parallelogram, rhombus	neighbours add to 180°
Kite	one pair of opposite angles equal
Trapezium	the two angles between the parallel sides, on the same side, add to 180°

Worked example

1
$∠ B = 11 8^{\circ}$ — The question gives one angle of rhombus $A B C D$ .
2
$∠ A + ∠ B = 18 0^{\circ}$ — $A D$ is parallel to $B C$ , so A and B are co-interior angles. Co-interior angles add up to 180°.
3
$∠ A = 180 - 118 = 6 2^{\circ}$ — Take 118 from 180 to leave angle A.
4
$∠ C = ∠ A = 6 2^{\circ}$ — C is opposite A, and opposite angles of a rhombus are equal.
5
$∠ D = ∠ B = 11 8^{\circ}$ — D is opposite B, so it matches B.
6
$62 + 118 + 62 + 118 = 360$ — Check: the four angles add up to 360°.

Key idea

Kite: the equal angles are the two between a long side and a short side. With $7 0^{\circ}$ and $12 4^{\circ}$ at the ends, each is $\frac{360 - 70 - 124}{2} = 8 3^{\circ}$ .

Angles Written With x

The 3-mark question: the angles are expressions. Add them all, set the total to $36 0^{\circ}$ , and solve.

The angles are $x$ , $2 x + 10$ , $3 x - 30$ and $8 0^{\circ}$ .

Worked example

1
$x + (2 x + 10) + (3 x - 30) + 80 = 360$ — Angles in a quadrilateral add up to 360°. Brackets keep each angle together.
2
$6 x + 10 - 30 + 80 = 360$ — Collect the x terms: x + 2x + 3x = 6x.
3
$6 x + 60 = 360$ — Now the numbers: +10 − 30 + 80 = +60. Keep the minus on the 30.
4
$6 x = 300$ — Take 60 from both sides, so the x term is alone.
5
$x = 50$ — Divide both sides by 6.
6
$50, 110, 120, 80$ — Put x = 50 back into each angle: 2 × 50 + 10 = 110 and 3 × 50 − 30 = 120.
7
$50 + 110 + 120 + 80 = 360$ — Check: they add up to 360°, so x is right.

Common mistake

Getting the signs wrong when you collect: $+ 10 - 30 + 80 = + 60$ .

Angles in a Ratio

Some questions give the four angles as a ratio. The parts share the 360° between them.

The angles of a quadrilateral are in the ratio $2 : 3 : 4 : 6$ . Find the largest angle.

Worked example

1
$2 + 3 + 4 + 6 = 15$ — Add the ratio numbers. The full 360° is cut into 15 equal parts.
2
$360 \div 15 = 2 4^{\circ}$ — Share 360° between the 15 parts to get one part.
3
$6 \times 24 = 14 4^{\circ}$ — The largest angle has 6 parts, so multiply one part by 6.
4
$48, 72, 96, 144$ — The other angles are 2 × 24, 3 × 24 and 4 × 24. Check: they add up to 360°.

Key idea

Sometimes one angle is compared with the other three, like $w : (x + y + z) = 7 : 11$ . That is still 360° in 7 + 11 = 18 parts. One part is 360 ÷ 18 = 20°, so $w = 7 \times 20 = 14 0^{\circ}$ .

Common mistake

Dividing 360 by 4 because there are four angles. Divide by the total of the ratio numbers.

Reflex Angles and Arrowheads

Some quadrilaterals have one corner that points inwards, like an arrowhead. The angle inside that dent is reflex: more than 180°. The four angles still add up to 360°.

An arrowhead has angles of $3 3^{\circ}$ and $4 6^{\circ}$ , and $y$ at its tip. At the dent, the angle outside the shape is $14 5^{\circ}$ . Find $y$ .

Worked example

1
$360 - 145 = 21 5^{\circ}$ — Angles at a point add up to 360°. The reflex angle inside the dent is the rest of the full turn.
2
$y + 33 + 46 + 215 = 360$ — Angles in a quadrilateral add up to 360°, and the reflex angle counts as one of the four.
3
$y + 294 = 360$ — Add the three angles you know: 33 + 46 + 215 = 294.
4
$y = 6 6^{\circ}$ — Take 294 from both sides.

Common mistake

Using the 145° as the angle inside. Inside the dent the angle is reflex, so it is 360 − 145 = 215°.

A reflex angle on a parallelogram. ABCD is a parallelogram and the reflex angle at D is $23 6^{\circ}$ . Find angle A.

Worked example

1
$∠ A D C = 360 - 236 = 12 4^{\circ}$ — Angles at a point add up to 360°. The angle inside the parallelogram is what the reflex angle leaves.
2
$∠ D A B = 180 - 124 = 5 6^{\circ}$ — $A B$ is parallel to $D C$ , so A and D are co-interior angles. They add up to 180°.

A Quadrilateral in a Bigger Diagram

The 5-mark question joins a quadrilateral to a straight line and an equilateral triangle. Give a reason for each stage.

In quadrilateral $A B C D$ , $B = 9 8^{\circ}$ and $∠ A D C = 11 0^{\circ}$ . $D C F$ is a straight line with $∠ B C F = 12 4^{\circ}$ . $A D E$ is an equilateral triangle outside it. Find angle $E A B$ .

Worked example

1
$∠ B C D = 180 - 124 = 5 6^{\circ}$ — $D C F$ is a straight line, and angles BCD and BCF sit side by side on it. Angles on a straight line add up to 180°.
2
$∠ D A B = 360 - 98 - 56 - 110$ — Now three angles of ABCD are known. Angles in a quadrilateral add up to 360°.
3
$∠ D A B = 9 6^{\circ}$ — 98 + 56 + 110 = 264, and 360 − 264 = 96. This is the part of angle EAB inside the quadrilateral.
4
$∠ E A D = 6 0^{\circ}$ — $A D E$ is equilateral, so each of its angles is 60°. This is the part outside.
5
$∠ E A B = 96 + 60 = 15 6^{\circ}$ — Angle EAB is made of the two parts side by side, so add them.

Key idea

“Give a reason for each stage” earns its own marks. Write the whole fact in words, such as “angles on a straight line add up to 180°”. Just “straight line” or “360” does not score.

Key idea

"Show that it is not a straight line": work out the angle and show it is not $18 0^{\circ}$ .

Which Method? Read the Question

Count the corners, then look for a special name.

You see	You do
Three angles known	360 − the three
Two equal unknowns	360 − the others, then halve
Parallelogram or rhombus	opposite equal, neighbours 180°
Kite	one pair of equal angles
Angles written with x	add, set to 360, solve
Angles in a ratio	add the ratio numbers, share 360
One corner points inwards	the angle inside is reflex; still 360
Parallel sides, angles with x	Angles in Parallel Lines
Five or more sides	Interior and Exterior Angles of Polygons

Key idea

Check your answer: the four angles must add to exactly 360°.

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