Angle Facts: Edexcel GCSE Foundation maths revision notes

Angle facts in the exam

Angle facts are rules that let you work out the size of a missing angle without measuring it. This page covers angles on a straight line, angles at a point, crossing lines, triangles and quadrilaterals.

Early in a paper, you may be asked to work out the size of the angle marked x, for 1 or 2 marks. Some of these questions also say "Give a reason for your answer".

Later in the paper, a question worth 3 to 5 marks needs two or three facts, one after another. It may ask you to show that a triangle is isosceles, and it will say "Give a reason for each stage of your working".

The diagrams are not drawn accurately, so don't measure them with a protractor. Work each angle out with a fact, and show the calculation.

Types of angle

A full turn is 360^\circ. Half a turn is 180^\circ, the angle on a straight line. A quarter turn is a right angle, 90^\circ, marked with a small square in the corner.

Acute: less than 90^\circ
Obtuse: more than 90^\circ but less than 180^\circ
Reflex: more than 180^\circ but less than 360^\circ

Be exact. Obtuse means between 90^\circ and 180^\circ, so an angle of 200^\circ is reflex, not obtuse. An angle of 50^\circ is acute.

Straight lines, points and right angles

Angles on a straight line add up to 180^\circ. For example, 130^\circ and 50^\circ fit together on a straight line. The angles must meet at the same point on the line.

Angles at a point add up to 360^\circ. These angles go all the way round one point, making a full turn. Add the angles you know, then subtract the total from 360^\circ

When a right angle is split into two parts, the parts add up to 90^\circ. A corner of a rectangle can be split into 25^\circ and 65^\circ, because 25+65=90

Worked example 1: Angles on a straight line

The diagram shows two angles on a straight line. Work out the size of the angle marked x.

The 58^\circ angle and angle x meet at the same point on a straight line.
Angles on a straight line add up to 180^\circ, so subtract 58 from 180
Your turn. Work out 180-58

Your answer

x=180-58=122^\circ
Check: angle x is obtuse in the diagram, and 122^\circ is between 90^\circ and 180^\circ
Answer: x=122^\circ

Worked example 2: Angles at a point

Three angles meet at a point. Two of them are 104^\circ and 138^\circ. Work out the size of the angle marked x. Give a reason for your answer.

The three angles go all the way round the point.
Angles at a point add up to 360^\circ
Your turn. Add the two angles you know. Work out 104+138

Your answer

104+138=242
x=360-242=118
Answer: x=118^\circ, because angles at a point add up to 360^\circ

Vertically opposite angles

When two straight lines cross, they make four angles. The angles across the crossing from each other are vertically opposite, and vertically opposite angles are equal.

Two angles next to each other at the crossing lie on a straight line, so they add up to 180^\circ. Using the two facts, one angle is enough to find all four.

In a reason, write "vertically opposite angles are equal" in full. A short version such as "opposite equal" does not get the mark.

Worked example 3: Two lines crossing

Two straight lines cross. Work out the sizes of the angles marked x and y. Give a reason for each answer.

Angle x is across the crossing from the 72^\circ angle.
x=72^\circ, because vertically opposite angles are equal.
Angle y is next to the 72^\circ angle, on a straight line.
Your turn. Work out 180-72

Your answer

y=180-72=108^\circ, because angles on a straight line add up to 180^\circ
Check: 72+108+72+108=360, a full turn.
Answer: x=72^\circ and y=108^\circ

Angles in triangles

Angles in a triangle add up to 180^\circ. A triangle with angles of 50^\circ and 60^\circ has a third angle of 180-50-60=70^\circ

An isosceles triangle has two equal sides, shown by matching tick marks. Its two base angles are equal. They are the angles at the two ends of the side with no tick.

An equilateral triangle has three equal sides and three equal angles. Each angle is 180\div3=60^\circ

Only treat a triangle as isosceles when the question tells you, with tick marks or in words. A triangle that looks balanced may not be.

Worked example 4: An isosceles triangle

The two ticked sides of the triangle are equal. The angle at the top is 38^\circ. Work out the size of each angle marked x.

The ticked sides are equal, so the triangle is isosceles. The two base angles, both marked x, are equal.
Angles in a triangle add up to 180^\circ, so the two base angles add up to 180-38=142^\circ
Your turn. The base angles share 142^\circ equally. Work out 142\div2

Your answer

x=142\div2=71^\circ
Check: 71+71+38=180
Answer: x=71^\circ

Naming angles and giving reasons

Angle ABC means the angle at B, between the lines BA and BC. The middle letter is the corner. Three letters show exactly which angle you mean when several angles meet.

When a question asks for a reason, write the angle fact in full, with the word "angles" in it. "Angles on a straight line add up to 180^\circ" gets the mark, but "a straight line is 180" does not. Learn the six reasons below.

Angles on a straight line add up to 180^\circ
Angles at a point add up to 360^\circ
Vertically opposite angles are equal
Angles in a triangle add up to 180^\circ
Base angles of an isosceles triangle are equal
Angles in a quadrilateral add up to 360^\circ

Worked example 5: Showing a triangle is isosceles

In the diagram, ABC is a triangle and CD is a straight line. Angle BAC is 40^\circ, angle ACD is 140^\circ and angle BCD is 150^\circ. Show that triangle ABC is isosceles. Give a reason for each stage of your working.

Angles ACB, ACD and BCD go all the way round the point C.
Your turn. Angle ACB is what is left of the full turn at C. Work out 360-140-150

Your answer

Angle ACB =360-140-150=70^\circ, because angles at a point add up to 360^\circ
Angle ABC =180-40-70=70^\circ, because angles in a triangle add up to 180^\circ
Angles ACB and ABC are equal, so the sides opposite them, AB and AC, are equal too.
Answer: angles ACB and ABC are both 70^\circ, so triangle ABC is isosceles.

Angles in quadrilaterals

Angles in a quadrilateral add up to 360^\circ. A quadrilateral is any shape with four straight sides.

A diagonal splits a quadrilateral into two triangles. The angles in each triangle add up to 180^\circ, and 180+180=360

Sometimes angles are written with a letter, such as x and 2x. Add all four angles, make the total equal to 360, then solve the equation.

Worked example 6: Angles written with a letter

The angles of a quadrilateral are 2x^\circ, x^\circ, 135^\circ and 105^\circ. Work out the size of the angle marked 2x.

Angles in a quadrilateral add up to 360^\circ, so 2x+x+135+105=360
Collect like terms: 3x+240=360
Subtract 240 from both sides: 3x=120
Your turn. Divide both sides by 3. What is x?

Your answer

x=120\div3=40
So 2x=2\times40=80. Check: 80+40+135+105=360
Answer: 80^\circ

Common mistakes

Using the wrong total. Use 180^\circ for straight lines and triangles, and 360^\circ for points and quadrilaterals.
Adding angles at different points on a line. Angles add up to 180^\circ only when they meet at the same point on the straight line.
Assuming a triangle is isosceles or equilateral. Use equal angles only when tick marks or the question tell you the sides are equal.
Picking the wrong pair of equal angles. In an isosceles triangle, the equal angles are at the two ends of the side with no tick.
Leaving the key words out of a reason. Write "angles on a straight line add up to 180^\circ", not "a straight line is 180", and "vertically opposite angles are equal", not "opposite equal".
Mixing up obtuse and reflex. An obtuse angle is between 90^\circ and 180^\circ, and a reflex angle is bigger than a straight line.