Question

example 12
in the diagram, p, q, r and s are points on the circle. ∠qpr = 48°, ∠psq = 35° and ∠prs = 31°. calculate
i) ∠pqr;
ii) ∠qrs.

Explanation:

Step1: Recall the angle - in - the - same - segment property

Angles in the same segment of a circle are equal. $\angle PQR$ and $\angle PSR$ are angles in the same segment. Given $\angle PSQ = 35^{\circ}$, and $\angle PQR$ and $\angle PSQ$ subtend the same arc $PR$. So $\angle PQR=\angle PSQ = 35^{\circ}$.

Step2: Find $\angle QPR$ in $\triangle PQR$

We know $\angle QPR = 48^{\circ}$ and we just found $\angle PQR = 35^{\circ}$. In $\triangle PQR$, the sum of interior angles of a triangle is $180^{\circ}$. Let's first find $\angle PRQ$. $\angle PRQ=180^{\circ}-\angle QPR - \angle PQR=180^{\circ}-48^{\circ}-35^{\circ}=97^{\circ}$.

Step3: Calculate $\angle QRS$

We know $\angle PRS = 31^{\circ}$. $\angle QRS=\angle PRQ+\angle PRS$. So $\angle QRS=97^{\circ}+31^{\circ}=128^{\circ}$.

Answer:

i) $35^{\circ}$
ii) $128^{\circ}$