Counters PQ and RS run parallel along a market aisle. A cable goes from A on PQ to B, then to C on RS. The dashed line BX is parallel to both. Angle QAB is ...
Question 1 Report
Counters PQ and RS run parallel along a market aisle. A cable goes from A on PQ to B, then to C on RS. The dashed line BX is parallel to both. Angle QAB is 47° and angle SCB is 68°.
- Write down angle ABX. (1)
- Write down angle CBX. (1)
- Work out angle ABC. (2)
Answer Details
Drawing \(BX\) parallel to both counters is the standard construction for a bent path between parallel lines. It converts one awkward angle at \(B\) into two ordinary alternate-angle pairs.
(a) \(PQ\) and \(BX\) are parallel, and \(AB\) is a transversal crossing them. Angles \(QAB\) and \(ABX\) are alternate angles, so they are equal:
Angle \(ABX = 47\)°. [1]
(b) \(RS\) and \(BX\) are parallel, and \(CB\) is a transversal crossing them. Angles \(SCB\) and \(CBX\) are alternate angles, so they are equal:
Angle \(CBX = 68\)°. [1]
(c) \(BX\) lies inside angle \(ABC\), so the two parts add:
\[\text{angle } ABC = 47 + 68\][1]
\[= 115^{\circ}\][1]
The reason \(BX\) can be drawn at all is that if a line is parallel to one of two parallel lines it is parallel to the other, so a single construction line serves both halves of the figure. Without it, the angle at \(B\) has no parallel line to be measured against.
If the bend at \(B\) went the other way, so that the two counters lay on the same side of \(B\), the parts would be subtracted rather than added. Checking whether \(BX\) lies inside or outside the angle decides between the two.
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