Question

The angle inscribed in the semicircle is a right angle. Prove the result by completing the following activity.

Given: ∠ABC is inscribed angle in a semicircle with center M.

To prove: ∠ABC is a right angle.

Proof: Segment AC is a diameter of the circle.

∴ m(arc AXC) = `square`

Arc AXC is intercepted by the inscribed angle ∠ABC

∠ABC = `square`   ...[Inscribed angle theorem]

= `1/2 xx square`

∴ m∠ABC = `square`

∴ ∠ABC is a right angle.

Answer 1

Given: ∠ABC is inscribed angle in a semicircle with center M.

To prove: ∠ABC is a right angle.

Proof: Segment AC is a diameter of the circle.

∴ m(arc AXC) = \[\boxed{180°}\]   ...(i) [Measure of semi circular arc is 180°]

Arc AXC is intercepted by the inscribed angle ∠ABC

∠ABC = \[\boxed{\frac{1}{2} \text{m(arc AXC)}}\]   ...[Inscribed angle theorem]

= \[\frac{1}{2} \times \boxed{180°}\]   ...[From (i)]

∴ m∠ABC = \[\boxed{90°}\]

∴ ∠ABC is a right angle.