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Angles on a straight line

12.2 Angles on a straight line

Let us look at angles formed on one side of a straight line.

In this diagram, line segment \(AB\) meets line segment \(DC\). The angle at the vertex, \(C\), where they meet, is now split into two angles: \(\hat{C_1}\) and \(\hat{C_2}\).

\(\hat{C_1}\) is the name for the angle at vertex \(C\) labelled "1" (or \(A\hat{C}D\)).

The sum of the angles formed on a straight line

The sum of the angles that are formed on a straight line is always \(180^{\circ}\).

We can shorten this property as: \(\angle\)s on a straight line.

  • Two angles that add up to \(180^{\circ}\) are called supplementary angles.
  • Angles that share a vertex and a common side are said to be adjacent angles. \(\hat{C_1}\) and \(\hat{C_2}\) are supplementary angles.
  • Hence, \(\hat{C_1}\) and \(\hat{C_2}\) are called adjacent supplementary angles.
supplementary angles
two angles that add up to \(180^{\circ}\)
adjacent angles
angles that share a vertex and a common side

You can have more than one line meeting at the same point on a straight line. Here are a few examples of angles on a straight line.

The sum of the angles on perpendicular lines

When two lines are perpendicular, the adjacent supplementary angles are both equal to \(90^{\circ}\).

In the diagram, \(\hat{M_1}=\hat{M_2} =90^{\circ}\).

A right angle is shown by forming a square at one of the right angles, like this: ⦜.

Finding unknown angles on straight lines

Worked example 12.1: Calculating unknown angles on a straight line

Calculate the size of \(x\).

In the diagram, we have two angles that are on the same side of the straight line. The first angle is \(100^{\circ}\) and the second angle is unknown (\(x\)). We need to calculate the size of \(x\).

We know that the two angles have a sum of \(180^{\circ}\), so we can say that:

\(100^{\circ}+x=180^{\circ}\) (\(\angle\)s on a straight line)

Now we can solve this equation.

\[\begin{align} x&=180^{\circ}-100^{\circ} \\ x&=80^{\circ} \end{align}\]

Worked example 12.2: Calculating unknown angles on a straight line

Calculate the size of \(x\).

Notice that there are three angles on the same side of the straight line. We have \(x\), an angle of \(29^{\circ}\) and an angle of \(90^{\circ}\). (Remember that the ⦜ symbol on the diagram indicates a \(90^{\circ}\) angle.) These three angles have a sum of \(180^{\circ}\), so we can say that:

\(x+29^{\circ}+90^{\circ}=180^{\circ}\) (\(\angle\)s on a straight line)

Now we can solve this equation.

\[\begin{align} x+29^{\circ}+90^{\circ}&=180^{\circ} \\ x+119^{\circ}&=180^{\circ} \\ x&=180^{\circ}-119^{\circ} \\ x&=61^{\circ} \end{align}\]

There is a simpler way to solve for \(x\). It is given that we have a perpendicular line. Adjacent angles on a perpendicular line are both equal to \(90^{\circ}\). So, we have a different equation to solve.

\[\begin{align} x+29^{\circ}&=90^{\circ} \\ x&=90^{\circ}-29^{\circ} \\ x&=61^{\circ} \end{align}\]

Worked example 12.3: Calculating unknown angles on a straight line

Calculate the size of \(y\).

Notice that we have three angles on the same side of the straight line. We have \(2y\), an angle of \(48^{\circ}\) and an angle of \(52^{\circ}\). These three angles have a sum of \(180^{\circ}\), so we can say that:

\(2y+48^{\circ}+52^{\circ}=180^{\circ}\) (\(\angle\)s on a straight line)

Now we can solve this equation.

\[\begin{align} 2y+48^{\circ}+52^{\circ}&=180^{\circ} \\ 2y+100^{\circ}&=180^{\circ} \\ 2y&=180^{\circ}-100^{\circ} \\ 2y&=80^{\circ} \\ y&=40^{\circ} \end{align}\]
Exercise 12.1

Calculate the size of \(a\).

\[\begin{align} a+63^{\circ}&=180^{\circ} &(\angle\text{s on a straight line}) \\ x&=180^{\circ}-63^{\circ} \\ x&=117^{\circ} \end{align}\]

Calculate the size of:

  1. \(x\)
  2. \(\hat{ECB}\)
  1. \[\begin{align} x+3x+2x&=180^{\circ} &(\angle\text{s on a straight line}) \\ 6x&=180^{\circ} \\ x&=30^{\circ} \end{align}\]
  2. \[\begin{align} \hat{ECB}&=2x \\ &=2(30^{\circ})\\ &=60^{\circ} \end{align}\]

Calculate the size of:

  1. \(x\)
  2. \(\hat{GEH}\)
  1. \[\begin{align} (x+30^{\circ})+(x+40^{\circ}) +(2x+10^{\circ}) &=180^{\circ} &(\angle\text{s on a straight line}) \\ 4x+80^{\circ}&=180^{\circ} \\ 4x&=100^{\circ} \\ x&=25^{\circ} \end{align}\]
  2. \[\begin{align} \hat{GEH}&=x+40^{\circ} \\ &=25^{\circ}+40^{\circ} \\ &=65^{\circ} \end{align}\]

Hint: Remember that the matching curved lines “))” indicate that the angles are equal.

Calculate the size of:

  1. \(k\)
  2. \(\hat{TYP}\)
  1. \[\begin{align} (2k)+(k+65^{\circ}) +(2k) &=180^{\circ} &&(\angle\text{s on a straight line}) \\ 5k+65^{\circ}&=180^{\circ} \\ 5k &=115^{\circ} \\ &=23^{\circ} \end{align}\]
  2. \[\begin{align} \hat{TYP}&=k+65^{\circ} \\ &=23^{\circ}+65^{\circ} \\ &=88^{\circ} \end{align}\]