Question

In the figure, arc AXB is a semicircle; m∠PAB = 40°. Hence, find the values of


(1) m∠APB
(2) m(arc PYB)
(3) m(arc AZP)

Answer 1

(1) arc AXB and arc APB are semi-circular arcs.
The measure of the semicircular arc AXB is 180°.
∠APB is inscribed in the arc APB, arc AXB is an arc intercepted by it.
∴ m∠APB = $\frac{1}{2}$ × m (arc AXB)
= $\frac{1}{2}$ × 180 = 90°
∴ m∠APB = 90°

(2) Measure of intercepted arc = 2 × Measure of inscribed angle
Measure of inscribed angle in arc PYB = 40°
⇒ Measure of intercepted arc = 80°
∴ m(arc PY) = 80°

(3) Measure of intercepted arc = 2 × Measure of inscribed angle
Measure of inscribed angle in arc AZP = 50°
⇒ Measure of intercepted arc = 100°
∴ m( arc AZP) = 100°