Bisectors of adjacent angles on a straight line
Bisectors of adjacent angles on a straight line
IP 1
7 marks
\(AOB\) is a straight line and \(C\) is a point on one side of it. \(OP\) bisects \(\angle AOC\) and \(OQ\) bisects \(\angle COB\). The diagram is not drawn to scale.
- Given that \(\angle AOC=64^\circ\), find \(\angle POQ\).[2]
- Prove that \(\angle POQ=90^\circ\) for every position of \(C\).[3]
- Now two rays \(OC\) and \(OD\) are drawn on the same side of \(AOB\), in the order \(A\), \(C\), \(D\), \(B\). \(OP\) bisects \(\angle AOC\) and \(OQ\) bisects \(\angle DOB\). Show that \(\angle POQ=90^\circ+\dfrac12\angle COD\).[2]
Solution:
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Answer:(a) \(90^\circ\) (b) \(\angle POQ=\dfrac12(\angle AOC+\angle COB)=\dfrac12(180^\circ)\) (c) \(\angle POQ=\dfrac12(180^\circ-\angle COD)+\angle COD\)
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