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CEMC Cayley, 2020

Geometry Difficulty 2.2 Junior Find the answer Canada

In the diagram, point $Q$ lies on $PR$ and point $S$ lies on $QT$ .

What is the value of $x$ ?

$10$
$30$
$50$
$40$
$20$

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Since $QST$ is a straight angle, then $\angle QSP = 180^\circ - \angle TSP = 180^\circ - 50^\circ = 130^\circ$ .

Now $\angle RQS$ is an exterior angle for $\triangle QSP$ .

This means that $\angle RQS = \angle QSP + \angle SPQ$ .

Using the information that we know, $150^\circ = 130^\circ + x^\circ$ and so $x = 150 - 130 = 20$ .

(Alternatively, we could have noted that $\angle RQS$ and $\angle SQP$ are supplementary and then used the sum of the angles in $\triangle QSP$ .)

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