Calculate the size of \(x\) and \(y\).
12.5 Finding unknown angles on parallel lines
Worked example 12.8: Working out unknown angles on parallel lines
Work out the sizes of the unknown angles, \(x\), \(y\) and \(z\). Give reasons for your answers.
Remember that you can use abbreviations when you give your reasons within your workings. For example, for ‘alternate angles of parallel lines AB and CD are equal’ you write (alt \(\angle\)s ; \(AB \parallel CD\)).
Identify all the angles vertically opposite the given angle.
\[\begin{align} y &= 74^{\circ} &&(\text{vert opp }\angle) \end{align}\]Identify any angles that are alternate or corresponding to the given angle.
\[\begin{align} x &= 74^{\circ} &&(\text{alt}\angle\text{s; } AB \parallel CD) \end{align}\]Identify any angles that are adjacent to the given angle on a straight line.
\[\begin{align} z =180^{\circ}-74^{\circ}= 106^{\circ} &&(\angle\text{s on a straight line}) \\ z = 106^{\circ} \end{align}\]Alternatively, we could have also solved the problem like this:
\[\begin{align} y &= 74^{\circ} && (\text{vert opp }\angle\text{s}) \\ x &= y &&(\text{corresp }\angle\; AB \parallel CD) \\ x &= 74^{\circ} \\ z +x &= 180^{\circ} &&(\text{co-int }\angle\; AB \parallel CD) \\ z +74^{\circ} &= 180 \\ z &= 180^{\circ}-74^{\circ} \\ z &= 106^{\circ} \end{align}\]Calculate the size of \(p\), \(q\) and \(r\).
Calculate the size of \(a\), \(b\), \(c\) and \(d\).
In the diagram, \(MP \parallel XY\) and \(JK \parallel RS\). Find the sizes of the marked angles.
Calculate the size of \(x\), \(y\) and \(z\).
Calculate the size of \(x\).
Calculate the size of \(a\) and \(C\hat{E}P\).