IGCSEMaths: CoreCIERevision NotesGeometryBasic Angle PropertiesBasic Angle Properties

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First teaching 2023

First exams 2025

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Basic Angle Properties (CIE IGCSE Maths: Core)

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Jamie W

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Jamie W

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Basic Angle Properties

What are the basic angle properties?

  • Angles around a point add up to 360°
  • Angles that form a straight line add up to 180°
  • Vertically opposite angles are equal
    • Vertically opposite angles occur when two lines intersect, as in the diagram below

Vertically opposite angles

Exam Tip

  • Use the angle properties to find any missing angles that you can, and mark them on the diagram
    • Even if they are not the angle you are trying to find, they may help provide a clue!

Worked example

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

Basic angle properties worked example questionFind 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 of 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 all three angles will be equal
    • Each angle will equal 60°
  • A right-angled triangle has one 90° angle

angle properties of triangles

 

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 using three straight lines. Find the value of x.

Triangle angle properties worked example question

Label the other missing angles inside the triangle
 

Triangle angle properties worked example working

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 of 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 to 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

Angle properties of quadrilaterals

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.

Quadrilateral angle properties worked example question

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 row cell 360 space minus space 297 space end cell equals cell space 63 space end cell end table

Add this to the diagram

Quadrilateral angle properties worked example working

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

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

bold italic y bold space bold equals bold space bold 117

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    Jamie W

    Author: Jamie W

    Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.