Angles in a Triangle

Use the fact that the angles in a triangle add up to 180°, with equal base angles in an isosceles triangle and 60° in an equilateral one, to find angles and solve for x.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024 · 60 real past-paper questions below

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180° in Every Triangle

Tear the three corners off any paper triangle and put them side by side. They make a straight line, so they add to $18 0^{\circ}$ .

The 2-mark question: find $x$ and give a reason. Two angles are $6 7^{\circ}$ and $4 8^{\circ}$ .

Worked example

1
$67 + 48 = 115$ — Add the two angles you know. Together they use 115° of the 180°.
2
$x = 180 - 115$ — Angles in a triangle add up to 180°, so the third angle is what is left.
3
$x = 6 5^{\circ}$ — The third angle. Check: 67 + 48 + 65 = 180.

Key idea

A right-angled triangle already uses $9 0^{\circ}$ . The other two add to $9 0^{\circ}$ : with $3 4^{\circ}$ the last is $5 6^{\circ}$ .

Common mistake

Writing "it is a triangle" as the reason. Write the whole rule: angles in a triangle add up to 180°.

Key idea

Can a triangle have three right angles? Yes, on a globe! Walk from the North Pole down to the equator, turn 90° and walk a quarter of the way round the equator, then turn 90° and walk back to the pole. The angles add up to $27 0^{\circ}$ . The 180° rule is for flat triangles.

Isosceles and Equilateral

The 2-mark question: an isosceles triangle has two equal sides, marked with small ticks. The two angles at the bottom of those sides, the base angles, are equal. Here the top angle is $4 0^{\circ}$ .

Worked example

1
$180 - 40 = 140$ — Angles in a triangle add up to 180°. Take away the top angle, and the two base angles share what is left.
2
$140 \div 2 = 7 0^{\circ}$ — Base angles of an isosceles triangle are equal, so split 140 into two equal halves: each base angle.
3
$40 + 70 + 70 = 180$ — Check: the three angles add to 180°, so the answer fits.

Key idea

Given a base angle of $5 2^{\circ}$ instead? The other base angle is also $5 2^{\circ}$ , so the top is $180 - 104 = 7 6^{\circ}$ .

Rule

An equilateral triangle has three equal sides, so every angle is $180 \div 3 = 6 0^{\circ}$ .

Common mistake

Halving the wrong angle. Look at the ticks: the equal angles are opposite the equal sides.

Angles Written With x

The 4-mark question: the angles are written with x. Use a triangle fact to write an equation, then solve it.

Triangle $A B C$ has $C A = C B$ . Angle $A$ is $(3 x - 5)^{\circ}$ , angle $B$ is $(x + 41)^{\circ}$ and angle $C$ is $y$ . Find $y$ .

Worked example

1
$3 x - 5 = x + 41$ — CA = CB, so the angles at A and B are base angles. Base angles of an isosceles triangle are equal.
2
$2 x - 5 = 41$ — Take x from both sides, so x is on one side only.
3
$2 x = 46$ — Add 5 to both sides to get the x term alone.
4
$x = 23$ — Divide both sides by 2. This is x, not yet the answer.
5
$∠ A = 3 (23) - 5 = 6 4^{\circ}$ — Put x = 23 back into angle A.
6
$∠ B = 23 + 41 = 6 4^{\circ}$ — Put x = 23 into angle B too. Both give 64°, so x is right.
7
$y = 180 - 64 - 64 = 5 2^{\circ}$ — Angles in a triangle add up to 180°, so take both base angles from 180.

Common mistake

Giving $x = 23$ as the answer. The question asks for $y$ , an angle.

Key idea

Show every line. Most of the 4 marks are for the working: the equation, each solving step, and the angle you find. A correct answer with no working can lose them.

A Side Extended Outside

Exams often extend one side of the triangle. The angle outside, between the extended side and the triangle, is the exterior angle.

Formula

e = a + b

$e$ is the exterior angle. $a$ and $b$ are the two inside angles furthest from it, the ones it does not touch. The reason to write: the exterior angle of a triangle equals the sum of the two opposite interior angles.

Triangle $P QR$ has $P Q = P R$ and angle $P = 3 6^{\circ}$ . Side $QR$ is extended to $S$ . Find angle $P R S$ .

Worked example

1
$180 - 36 = 144$ — Angles in a triangle add up to 180°, so the two base angles Q and R share 144°.
2
$144 \div 2 = 7 2^{\circ}$ — PQ = PR, so the base angles are equal. Each one, Q and R, is 72°.
3
$∠ P R S = 180 - 72 = 10 8^{\circ}$ — QRS is a straight line. Angles on a straight line add up to 180°.
4
$36 + 72 = 108$ — Check with the exterior angle rule: the two angles opposite S, at P and Q, also add to 108°.

Common mistake

Adding the angle next to the exterior angle. Use the two angles it does not touch.

Work Back From the Outside Angle

Sometimes the outside angle is given and you must find an angle inside. Start where the outside angle sits.

Triangle $K L M$ has $K M = L M$ . $K L N$ is a straight line and angle $M L N = 12 8^{\circ}$ . Find angle $K M L$ .

Worked example

1
$∠ M L K = 180 - 128 = 5 2^{\circ}$ — KLN is a straight line. Angles on a straight line add up to 180°.
2
$∠ M K L = 5 2^{\circ}$ — KM = LM, so the angles at K and L are base angles. Base angles of an isosceles triangle are equal.
3
$52 + 52 = 104$ — Add the two base angles.
4
$∠ K M L = 180 - 104 = 7 6^{\circ}$ — Angles in a triangle add up to 180°, so the top angle is what is left.
5
$52 + 76 = 128$ — Check with the exterior angle rule: the two angles opposite N add to the outside angle.

Key idea

This is a 3-mark question, and 2 of the marks are for working, like writing 180 − 128. Write each angle you find on the diagram as you go.

Common mistake

Halving the outside angle. Halve only when two equal angles share a total inside the triangle.

Angles in a Ratio

Some questions give the angles as a ratio. The parts share the 180° between them.

One angle of a triangle is $4 8^{\circ}$ . The other two are in the ratio $5 : 6$ . Find them.

Worked example

1
$180 - 48 = 132$ — Angles in a triangle add up to 180°, so the other two share 132°.
2
$5 + 6 = 11$ — Add the ratio numbers. The 132° is cut into 11 equal parts.
3
$132 \div 11 = 12$ — Share 132 between the 11 parts to get one part.
4
$5 \times 12 = 6 0^{\circ}$ — The smaller angle has 5 parts.
5
$6 \times 12 = 7 2^{\circ}$ — The larger angle has 6 parts. Check: 48 + 60 + 72 = 180.

Key idea

Isosceles? The base angles are equal, so write each one in the ratio. Top : base = 1 : 4 becomes $1 : 4 : 4$ . That is 9 parts of $180 \div 9 = 2 0^{\circ}$ , so the top is $2 0^{\circ}$ and each base angle is $8 0^{\circ}$ .

Common mistake

Sharing 180 instead of 132. Take away the angle you know first.

Which Method? Read the Question

Look for ticks on the sides and for any extra lines.

You see	You do
Two angles known	180 − both
Ticks on two sides	two equal base angles
Three equal sides	every angle is 60°
Angles written with x	add to 180, or set equal angles equal
A side extended outside	add the two opposite inside angles
Angles in a ratio	share what is left of 180 into parts
Just naming the triangle	Classifying Triangles

Key idea

Check: your three angles must add to exactly 180°.

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