Basic Angle Properties (Edexcel IGCSE Maths)

Spec point — 4.1B

Basic angle properties

How do I label lines segments, angles and shapes?

• The line segment AB goes from point A to point B

• The angle ABC is formed by the line segments AB and BC*

  • The angle is at point B
  • The angle is the acute or obtuse angle, not the reflex angle

• The triangle ABC is formed by the line segments AB*, BC and CA

Labels for line segments, angles and triangles

• The quadrilateral ABCD is formed by the line segments AB*, BC, CD and DA

  • The points are labelled starting at a point and following the sides
  • ABDC is not the same shape because it is formed by the line segments AB, BD, DC and CA

• The same idea is used to label other polygons

Labels for shapes

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

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

Find the values of $x$ and $y$.

Answer:

Vertically opposite angles between two intersecting lines are equal

$x=25$

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

$\begin{gathered} x + y + 98 & = 180 \\ 25 + y + 98 & = 180 \\ 123 + y & = 180 \end{gathered}$

Solve to find $y$

$y=180-123\,y=57$

$x=25,y=57$

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

Exam Hint 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 $x$.

Answer:

Label the other missing angles inside the triangle

Vertically opposite angles between two intersecting lines are equal

$y=60$

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

$\begin{gathered} z + 130 & = 180 \\ z & = 50 \end{gathered}$

Interior angles in a triangle add up to 180°

$\begin{gathered} x + 60 + 50 & = 180 \\ x + 110 & = 180 \end{gathered}$

$x=70$

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

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

Answer:

First, add together the three given angles

$97+115+85=297$

Subtract the answer from 360°

$\begin{gathered} 360 - 297 & = 63 \end{gathered}$

Add this to the diagram

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

$\begin{gathered} y + 63 & = 180 \\ y = 180 - 63 \\ y = 117 \end{gathered}$

$y=117$