WAEC 2022 · Paper 2 · Q11

(a)
The exterior angles of a polygon are $42^\circ, 38^\circ, 57^\circ, x^\circ, (x + y)^\circ, (2x - 15)^\circ$ and $(3x - y)^\circ$ . If $x$ is $7^\circ$ less than $y$ , find the values of $x$ and $y$ .
Separate values with commas, e.g. 3, −2
(b)
In the diagram, $O$ is the centre of the circle $XYZ$ . $\angle ZXO = 34^\circ$ and $\angle XOY = 146^\circ$ . Find $\angle OYZ$ .

Worked solution (try it first)

(a)

The exterior angles of any polygon add up to $360^\circ$ : $42 + 38 + 57 + x + (x + y) + (2x - 15) + (3x - y) = 360$ .
The $y$ terms cancel: $122 + 7x = 360$ , so $7x = 238$ and $x = 34$ .
$x$ is $7^\circ$ less than $y$ , so $y = 34 + 7 = 41$ .

(b)

$\angle XZY$ stands on the arc $XY$ , and $\angle XOY = 146^\circ$ is the angle at the centre on the same arc: $\angle XZY = \frac12 \times 146^\circ$
$= 73^\circ$ .
$OX = OZ$ (radii), so triangle $OXZ$ is isosceles and $\angle OZX = \angle OXZ = 34^\circ$ .
So $\angle OZY = \angle XZY - \angle OZX$
$= 73^\circ - 34^\circ$
$= 39^\circ$ .
$OY = OZ$ (radii), so triangle $OYZ$ is isosceles and $\angle OYZ = \angle OZY = 39^\circ$ .

Report a problem with this question