Basic Angle Properties

Mathematics (Core) revision notes · Cambridge (CIE) · 0580 · Core

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1. The basic angle facts

Four facts that start every question

  • Angles meeting at a point add to 360°
  • Angles on one side of a straight line add to 180°
  • Vertically opposite angles, formed where two straight lines cross, are equal
Two straight lines crossing, drawn twice: in the first the pair of opposite angles above and below the crossing point is shaded, and in the second the other opposite pair, left and right, is shaded, so each pair is equal
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  • The angles inside a triangle add to 180°, and the angles inside a quadrilateral add to 360°
  • Naming uses three letters with the vertex in the middle, so angle ABC is the angle at B, and it means the angle smaller than 180° rather than the reflex one
  • Diagrams are usually marked "NOT TO SCALE", so work from the numbers and the markings rather than from how the picture looks
Three straight lines crossing at one point, marked not to scale, with angles of 25 degrees and 98 degrees given and two more, x and y, to be found from the facts that angles at a point add to 360 and angles on a line add to 180
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Filling in what you can

  • Mark every angle you can work out on the diagram, even ones the question has not asked for
  • One of them is almost always the step that unlocks the angle you actually want
  • A question worth more than one mark expects the intermediate angles to appear in the working
A triangle standing on a straight line with 60 degrees and 130 degrees marked outside it and x inside at the apex, and the two angles y and z between the triangle and the line filled in as the intermediate step
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Exam tip

Name each angle you find

In an angle question worth two or three marks, a correct angle found on the way earns a mark of its own, even when the final answer is wrong. It counts only if it is clear which angle it is, so write it with three letters, as in angle BDE = 58°, or write the value in its place on the diagram.

Worked example

Crossing lines and a straight line

Three straight lines meet at a point. One angle is 112°, and x is vertically opposite it. On the straight line through that point sit x, y and an angle of 41°. Find x and y.

Three straight lines crossing at a single point. Going anticlockwise from the right-hand horizontal ray, the first angle is marked 41 degrees, the next is labelled y, and the third, up to the left-hand horizontal ray, is labelled x. Below the horizontal line, the angle vertically opposite x is marked 112 degrees.

Solution:

  • x is vertically opposite the 112° angle, and vertically opposite angles are equal
  • So x = 112
  • The three angles x, y and 41° lie on a straight line, so they add to 180°
  • 112 + y + 41 = 180
  • 153 + y = 180
  • So y = 27
  • Check: 112 + 27 + 41 = 180

2. Angles in triangles and quadrilaterals

Using the shape's own properties

  • Any triangle's three angles total 180°, and knowing two gives the third by subtraction
  • An isosceles triangle has two equal angles, and they are the ones opposite the two equal sides
    • With an apex angle of 34°, the two base angles are (180 − 34) ÷ 2 = 73° each
  • An equilateral triangle has three angles of 60°
  • Any quadrilateral's four angles total 360°
A quadrilateral with interior angles of 115, 85 and 97 degrees and one side extended, so the fourth interior angle works out at 63 degrees and the exterior angle y beside it is what the question wants
Source: Basic angle properties by Save My Exams
  • Symmetry supplies extra equal angles: a parallelogram and a rhombus have equal opposite angles, and a kite has one pair of equal opposite angles
  • A square or rectangle has four right angles
Which angles a quadrilateral's own shape makes equal: a square and a rectangle have four right angles, a parallelogram and a rhombus have equal opposite angles marked in pairs, and a kite has one pair of equal opposite angles
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Exam tip

The reason scores separately

When a question asks for a geometrical reason, the angle and the reason earn separate marks, and a question asking for two reasons gives a mark for each, so a correct value on its own loses a mark. Write the fact in full, saying which angles and what is true of them: "angles on a straight line add to 180°", "the angles in a triangle add to 180°", "an isosceles triangle's base angles are equal".

Worked example

An isosceles triangle on a straight line

LMN is a straight line. In triangle MNP, PM = PN and angle MPN = 38°. Find angle LMP, giving a reason for each step.

A straight line through L, M and N, with L on the left. Triangle MNP stands on MN with P above the middle of MN; PM and PN each carry one tick to show they are equal, and the angle at P is marked 38°. Angle LMP, outside the triangle on the left, is not marked. NOT TO SCALE.

Solution:

  • PM = PN, so triangle MNP is isosceles, and its base angles PMN and PNM are equal
  • The angles in a triangle add to 180°, so each base angle is (180 − 38) ÷ 2 = 71°
  • Angle PMN = 71°, and L, M and N lie on one straight line
  • Angles on a straight line add to 180°, so angle LMP = 180 − 71 = 109°
  • Check: 71 + 71 + 38 = 180, and 109 + 71 = 180

Worked example

The fourth angle of a quadrilateral

Three angles of a quadrilateral are 88°, 124° and 97°. Find the fourth angle.

Solution:

  • The four angles of any quadrilateral add to 360°
  • Add the three given: 88 + 124 + 97 = 309
  • Subtract from 360: 360 − 309 = 51
  • The fourth angle is 51°