Angles in Triangles and Quadrilaterals

Almost every question on angles in triangles and quadrilaterals comes back to two facts: the three angles of a triangle add up to 180°, and the four angles of a quadrilateral add up to 360°. On this page you will see how to combine those with isosceles triangles, exterior angles, parallel lines and the special quadrilaterals, which is exactly how the multi-step diagram questions on Edexcel IGCSE Maths papers are put together. Every worked example gives a reason for each line of working, because the longer questions ask you to justify what you did. When you are ready, scroll down to the free auto-marked practice questions and work through the four rooms: triangles, quadrilaterals, multi-step and algebraic problems, then a mixed set.

Prior Knowledge This page assumes you can already use the angle facts made by parallel lines (alternate, corresponding and co-interior angles), and that you can solve a simple linear equation, which is what the algebraic angle questions come down to.
⇩ Jump to Practice Questions ⇩

How to find angles in triangles and quadrilaterals

Every question on this topic is the same job: collect the angle facts the diagram gives you, then add them up to 180° for a triangle or 360° for a quadrilateral. The marks in an Edexcel IGCSE Maths paper come from writing a reason beside each step, so get into the habit early.

  1. Copy the diagram and mark it. Write every angle you are given onto your sketch, and add each new angle as you work it out.
  2. Read the markings. Two ticks on two sides mean two equal angles. Arrows mean the sides are parallel, so alternate, corresponding and co-interior angles are available. A small square means 90°.
  3. Look for a straight line or a full turn. Angles on a straight line add up to 180°, and angles round a point add up to 360°. This is how an exterior angle turns into an interior one.
  4. Use the total. Triangle: the three angles make 180°. Quadrilateral: the four angles make 360°.
  5. Write the reason. "angle sum of a triangle", "base angles of an isosceles triangle", "alternate angles", "angles on a straight line". Reasons earn marks.
  6. Answer the question that was asked. If you formed an equation and found \(x\), check whether the question wanted \(x\) or wanted an actual angle.

The four facts you use every time

Triangle angle sum
\(a + b + c = 180^{\circ}\)
The three angles inside any triangle add up to 180°, whatever shape it is.
Exterior angle
\(e = a + b\)
Extend one side and the angle outside equals the two interior angles it does not touch, added together.
Isosceles triangle
\(b = \dfrac{180^{\circ} - a}{2}\)
Two equal sides give two equal angles, and they sit opposite those sides. An equilateral triangle is the case where all three are 60°.
Quadrilateral angle sum
\(a + b + c + d = 360^{\circ}\)
One diagonal splits any quadrilateral into two triangles, so the total is two lots of 180°.

Why a triangle makes 180° and a quadrilateral makes 360°

These two totals are not something to take on trust. Both come from the parallel-line angle facts you already know, and seeing where they come from makes the harder questions much easier to unpick.

a c b a c

Draw a line through the top vertex parallel to the base. The two outer angles are alternate angles, so they equal \(a\) and \(c\). All three now sit on a straight line, so \(a + b + c = 180^{\circ}\).

180° 180°

One diagonal cuts any quadrilateral into two triangles. Two lots of 180° gives the 360° total, and nothing about the shape matters.

The exterior angle result, in one line
\(e = a + b\), because \(e = 180^{\circ} - c\) and \(a + b = 180^{\circ} - c\).

The special quadrilaterals and their symmetry

A named quadrilateral hands you extra angle facts for free, so learning to recognise them from the side and angle markings is worth real marks. The tick marks show equal sides and the arrows show parallel sides.

Square 4 lines of symmetry, rotational order 4
Rectangle 2 lines of symmetry, rotational order 2
Rhombus 2 lines of symmetry, rotational order 2
Parallelogram no lines of symmetry, rotational order 2
Kite 1 line of symmetry, rotational order 1
Trapezium no symmetry in general, one pair of parallel sides
Isosceles trapezium 1 line of symmetry, rotational order 1
Reading the markings is the skill. Equal ticks give you equal sides, which give you equal angles. Arrows give you parallel sides, which give you alternate and co-interior angles.

Worked examples

💡 Example 1: an isosceles triangle

Triangle \(PQR\) has \(PQ = PR\), and the angle at \(P\) is 44°. Work out the angle at \(Q\).

44° b P Q R
\[ \begin{array}{rcl} b + b + 44 &=& 180 \\ 2b &=& 136 \\ b &=& 68 \end{array} \]
What's happening?
  • The ticks say \(PQ = PR\), so the angles at \(Q\) and \(R\) are equal. Call each of them \(b\).
  • The three angles make 180°, so \(2b + 44 = 180\).
  • The angle at \(Q\) is 68°.

💡 Example 2: an exterior angle

\(LNT\) is a straight line. Angle \(NLM\) is 57° and angle \(LMN\) is 68°. Work out angle \(MNT\).

57° 68° e L M N T
\[ \begin{array}{rcl} e &=& 57 + 68 \\ &=& 125 \end{array} \]
What's happening?
  • The exterior angle equals the two interior angles it does not touch, added together.
  • The long way round works too: the angle at \(N\) inside is \(180 - 57 - 68 = 55\), and \(180 - 55 = 125\).
  • Angle \(MNT\) is 125°.

💡 Example 3: a quadrilateral with a straight line

\(ADV\) is a straight line. Work out the size of the angle marked \(x\).

84° 118° x 105° A B C D
\[ \begin{array}{rcl} d &=& 180 - 105 \\ &=& 75 \\ x &=& 360 - 84 - 118 - 75 \\ &=& 83 \end{array} \]
What's happening?
  • 105° is outside the shape, so first turn it into the interior angle at \(D\) using angles on a straight line.
  • Now all four interior angles are known apart from \(x\), and they add up to 360°.
  • \(x = 83^{\circ}\).

💡 Example 4: forming an equation

The four angles of a quadrilateral are \(x^{\circ}\), \((2x + 10)^{\circ}\), \((3x - 25)^{\circ}\) and \((2x + 15)^{\circ}\). Work out \(x\), then the largest angle.

\[ \begin{array}{rcl} 8x + 10 - 25 + 15 &=& 360 \\ 8x &=& 360 \\ x &=& 45 \end{array} \] \[ \begin{array}{rcl} 3x - 25 &=& 3 \times 45 - 25 \\ &=& 110 \end{array} \]
What's happening?
  • Add the four expressions and set the total equal to 360. The \(x\) terms give \(8x\), and the numbers give \(10 - 25 + 15\).
  • The numbers cancel to zero here, which is a useful check that you collected them correctly.
  • The question also asked for an angle, so substitute \(x = 45\) back in. The largest angle is 110°.

💡 Example 5: parallel lines and an isosceles triangle together

The two lines marked with arrows are parallel, and \(PQ = PR\). The angle between the top line and \(PQ\) is 64°. Work out angle \(QPR\).

64° x q r P Q R
\[ \begin{array}{rcl} q &=& 64 \\ r &=& 64 \\ x &=& 180 - 64 - 64 \\ &=& 52 \end{array} \]
What's happening?
  • \(q = 64^{\circ}\) because it is alternate to the 64° angle: the two lines are parallel and \(PQ\) crosses both.
  • \(r = q\) because \(PQ = PR\), so the triangle is isosceles and the base angles are equal.
  • The angle sum of the triangle finishes it: angle \(QPR\) is 52°.
  • Three facts, three reasons. That is exactly what the mark scheme is looking for.

💡 Example 6: when there are two possible triangles

An isosceles triangle has one angle of 36°. Work out the two possible sizes of the angle between the two equal sides.

36° Case 1 36° 36° x Case 2
\[ \begin{array}{rcl} x &=& 36 \\ \text{or} \quad x &=& 180 - 36 - 36 \\ &=& 108 \end{array} \]
What's happening?
  • Case 1: the 36° angle IS the angle between the equal sides, so \(x = 36^{\circ}\) and the two base angles are 72° each.
  • Case 2: the 36° angle is one of the two equal base angles, so the other base angle is also 36° and \(x = 108^{\circ}\).
  • Nothing in the question fixes which case it is, so both answers are needed. Room 3 of the practice below gives you two boxes for exactly this reason.
  • The only value that would not give two answers is 60°, because that triangle is equilateral.

🔑 Key points

  • The three angles of a triangle add up to 180°, and the four angles of a quadrilateral add up to 360°.
  • Equal sides give equal angles, and each equal angle sits opposite one of the equal sides.
  • An exterior angle of a triangle equals the sum of the two interior angles it does not touch.
  • The exterior angles of a triangle add up to 360°, and so do the exterior angles of a quadrilateral: you turn through one full turn walking round either.
  • Turn an outside angle into an inside one first, using angles on a straight line (180°) or angles round a point (360°).
  • Write a reason beside every line of working. On the longer questions the reasons carry marks of their own.

⚠️ Common pitfalls

  • Assuming a triangle is isosceles because it looks like it. Only the tick marks or the wording tell you.
  • Measuring the diagram. Exam sketches are not accurately drawn, and the numbers printed on them are the only ones you can trust.
  • Using the exterior angle rule with the wrong pair. The two angles you add are the ones at the OTHER two corners, never the one next to it.
  • Forgetting that "an isosceles triangle has an angle of 40°" usually describes two different triangles.
  • Stopping at \(x\) when the question asked for an angle. Substitute back in.
  • Adding the four angles of a quadrilateral to 180° out of habit. Split it into two triangles if you need reminding why it is 360°.
⇩ Practice Questions ⇩

Once triangles and quadrilaterals feel comfortable, the same walking-round-a-shape idea extends to pentagons, hexagons and beyond, where the angle sum becomes a formula.

Next: Angles in Polygons →

Angles in Triangles and Quadrilaterals: Practice Rooms

Practise angles in triangles and quadrilaterals in these free IGCSE Maths rooms: the angle sum of a triangle, isosceles and equilateral triangles, exterior angles, the special quadrilaterals and their angle sum, and the multi-step problems that mix in parallel lines and algebra. Difficulty rises from left to right: Starter, Builder, Challenger, Master. Every question carries a sketch, and the sketches are not accurately drawn, so work from the marked values and never from measuring. Type the number only; adding ° after it is also fine. Where a question has two possible answers you get two boxes, and the card is marked once both are filled.

✓ Correct 0
✗ Re-attempts 0
🔥 Streak 0
🏆 Best 0

Difficulty increases left to right across the columns and down each column. Room 1 works inside triangles and on their exterior angles, Room 2 covers the quadrilaterals, Room 3 builds the multi-step and algebraic problems, and Room 4 mixes every question type on this page, one family per column.