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End of chapter exercises

12.8 End of chapter exercises

Exercise 12.8

Calculate the size of:

  1. \(p\)
  2. \(R\hat{K}U\)
  1. \[\begin{align} 2p-55°+3p+70° &=180° &(\angle\text{s on a straight line}) \\ 5p+125°&=180° \\ 5p&=55° \\ p&=11° \end{align}\]
  2. \[R\hat{K}U = 3p =3(11^{\circ})=33^{\circ}\]

Calculate the size of \( y \)

\[\begin{align} 102°-2y &=78° & (\text{vert opp }\angle\text{s}) \\ -2y &=78°-102° \\ -2y &=-24° \\ y &= 12° \end{align}\]

Calculate the size of \( r \)

\[\begin{align} 180°-3r &=126° & (\text{vert opp }\angle\text{s}) \\ -3r &=126° - 180° \\ -3r &=-54° \\ r &=18° \end{align}\]

Find the sizes of all the angles \(\hat{1}\) to \(\hat{7}\).

\[\begin{align} \hat{1} &= 180°-154°&&(\angle\text{s on a straight line}) \\ \hat{1} &= 26° \\ \hat{2}&=154°&& (\text{vert opp }\angle\text{s}) \\ \hat{3}&=26°&& (\text{vert opp }\angle\text{s}) \\ \hat{4} &=\hat{1} &&(\text{corresp }\angle\text{s}; XY \parallel ZW) \\ \hat{4} &= 26° \\ \hat{5} &=154 °&&(\text{corresp }\angle\text{s}; XY \parallel ZW) \\ \hat{6}&=154°&& (\text{vert opp }\angle\text{s}) \\ \hat{7}&=26°&& (\text{vert opp }\angle\text{s}) \\ \end{align}\]

In the diagram, \(OK = ON\), \(KN \parallel LM\), \(KL \parallel MN\) and \(\hat{LKO}=160^{\circ}\). Calculate the value of \(x\). Give reasons for your answers.

\[\begin{align} \hat{LKN} &=140° &&(\text{opp }\angle\text{s} \text{ of parallelogram = )} \\ \hat{OKN} &=160°-140° \\ \hat{OKN} &=20° \\ \hat{ONK} &=20° &&(\angle\text{s opp = sides}) \\ \hat{ONK}+ \hat{OKN}+ x &=180°&&(\text{sum of }\angle\text{s in } \triangle ) \\ 20°+ 20° +x &=180° \\ x&=180° -20°-20° \\ x&=140° \\ \end{align}\]