The angle subtended by a diameter/semicircle on any point on the circumference of a circle is 90°.
Therefore, ∠RSP = 90°
The Tangent-Chord Theorem, states that: The angle measure between a chord of a circle and a tangent through any of the chord's endpoints is equal to the measure of an angle in the alternate segment.
Using the Tangent-Chord Theorem, it follows that:
[tex]\angle APQ=\angle PSQ[/tex]Notice that:
[tex]\angle APQ=3x[/tex]and
[tex]\angle PSQ=54^{\circ}[/tex]Therfore, we have that:
[tex]3x=54^{\circ}[/tex]Divide both sides of the equation by 3:
[tex]\begin{gathered} \frac{3x}{3}=\frac{54^{\circ}}{3} \\ x=18^{\circ} \end{gathered}[/tex]Notice that the interior angles of triangle PRS are:
[tex]\angle RSP,x,\text{ and },y[/tex]Since the sum of the interior angles of a triangle is 180°.
Therefore, we must have that:
[tex]\begin{gathered} \angle RSP+x+y=180^{\circ} \\ \text{ Substitute }\angle RSP=90^{\circ}\text{ and }x=18^{\circ}\text{ into the equation:} \\ 90^{\circ}+18^{\circ}+y=180^{\circ} \\ y+90^{\operatorname{\circ}}+18^{\operatorname{\circ}}=180^{\operatorname{\circ}} \\ \text{ Hence} \\ y=180^{\operatorname{\circ}}-90^{\operatorname{\circ}}-18^{\operatorname{\circ}} \\ y=72^{\operatorname{\circ}} \end{gathered}[/tex]The Same Segment Theorem states that the angles at the circumference subtended by the same arc are equal. More simply, angles in the same segment are equal.
Since angles y and z are both subtended by arc SP, it follows from the Same Segment Theorem that:
[tex]\begin{gathered} z=y \\ \text{ Since }y=72^{\circ},\text{ it follows that:} \\ z=72^{\circ} \end{gathered}[/tex]Therefore, the answer is:
(i) x = 18°, (ii)y = 72°, and (iii) z = 72°
