Basic Angle Properties (Cambridge O Level Maths)

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Basic Angle Properties

What are the basic angle properties?

  • Angles that meet at a point add up to 360°
  • Angles that meet at a point on a straight line add up to 180°
  • Vertically opposite angles are equal
    • Vertically opposite angles occur when two lines intersect
    • They are the angles either side of the intersection that are opposite to each other

Opposite-Angles

Exam Tip

  • Find all the missing angles that you can using the angles that are given to you in a question
    • They might not seem to help you straight away but having more angles will lead you to find the angle you need

Worked example

The diagram below shows three straight lines intersecting at a point.

cie-igce-core-rn-basic-angle-properties-we-diagram-1Find the values of x and y.

Vertically opposite angles between two intersecting lines are equal.

x space equals space 25

Angles that meet on a straight line add up to 180°.

table row cell x space plus space y space plus space 98 space end cell equals cell space 180 space end cell row cell 25 space plus space y space plus space 98 space end cell equals cell space 180 end cell row cell 123 space plus space y space end cell equals cell space 180 end cell end table

Solve to find y.

table row cell y space end cell equals cell space 180 space minus space 123 space end cell row cell y space end cell equals cell space 57 end cell end table

bold italic x bold space bold equals bold space bold 25 bold space bold comma bold space bold italic y bold space bold equals bold space bold 57

Angles in Triangles

What are the angle properties with triangles?

  • The three interior angles inside any triangle add up to 180°
  • If the triangle is isosceles then two angles will be equal
    • These will be the two angles opposite the two sides of equal length
  • If the triangle is equilateral then the three angles will be equal
    • Each angle will equal 60°
  • A right-angled triangle has one 90° angle

MPfz1HV0_cie-igcse-core-rn-angles-in-triangles-diagram-2---types-of-triangles

  • The exterior angle of a triangle is equal to the sum of the two opposite interior angles

xmBt3Rwl_cie-igcse-core-rn-angles-in-triangles-diagram-1---exterior-angle

Exam Tip

  • Find all the missing angles that you can using the angles that are given to you in a question
    • They might not seem to help you straight away but having more angles will lead you to find the angle you need

Worked example

The diagram below is formed by three straight lines. Find the value of x.

3-2-1-cie-igcse-basic-angle-we-diagram

Label the other missing angles inside the triangle.
 

3-2-1-cie-igcse-basic-angle-we-solution-diagram

Vertically opposite angles between two intersecting lines are equal.

y space equals space 60

Angles that meet on a straight line add up to 180°.

table row cell z space plus space 130 space end cell equals cell space 180 space end cell row cell z space end cell equals cell space 50 end cell end table

Interior angles in a triangle add up to 180°.

table row cell x space plus space 60 space plus space 50 space end cell equals cell space 180 space end cell row cell x space plus space 110 space end cell equals cell space 180 end cell end table

bold italic x bold space bold equals bold space bold 70

Angles in Quadrilaterals

What are the angle properties with quadrilaterals?

  • The four interior angles inside any quadrilateral add up to 360°
  • If the quadrilateral is a square or a rectangle then all the angles are equal
    • Each angle will equal 90°
  • You can use any symmetries of the quadrilateral to identify other equal angles
    • For a parallelogram or rhombus opposite angles are equal
    • For a kite one pair of opposite angles are equal

3CPFtt75_cie-igcse-core-rn-angles-in-quadrilaterals-diagram

Exam Tip

  • Find all the missing angles that you can using the angles that are given to you in a question
    • They might not seem to help you straight away but having more angles will lead you to find the angle you need

Worked example

The diagram below shows an irregular quadrilateral. Find the value of y.

cie-igce-core-rn-basic-angle-properties-angles-in-quads-we-diagram

Find the missing angle inside the quadrilateral using the rule 'angles in a quadrilateral add up to 360°'.

First, add together the three given angles.

97 space plus space 115 space plus space 85 space equals space 297

Subtract the answer from 360°.

table attributes columnalign right center left columnspacing 0px end attributes row cell 360 space minus space 297 space end cell equals cell space 63 space end cell end table

Add this to the diagram.

cie-igce-core-rn-basic-angle-properties-angles-in-quads-we-solution-1

Angles on a straight line add up to 180°, so subtract the answer from 180°.

table row cell y space plus space 63 space end cell equals cell space 180 space end cell row cell y space end cell equals cell space 180 space minus space 63 end cell row blank equals cell space 117 end cell end table

bold italic y bold space bold equals bold space bold 117

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Amber

Author: Amber

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.