Question

Prove that angles inscribed in the same arc are congruent.

Given: In a circle with center C, ∠PQR and ∠PSR are inscribed in same arc PQR. Arc PTR is intercepted by the angles.

To prove: ∠PQR ≅ ∠PSR.

Proof:

m∠PQR = ${\displaystyle \frac{1}{2}\times \left[\text{m}\left(\text{arc PTR}\right)\right]}$         ......(i) ${\displaystyle \square }$

m∠${\displaystyle \square }$ = ${\displaystyle \frac{1}{2}\times \left[\text{m}\left(\text{arc PTR}\right)\right]}$   ......(ii) ${\displaystyle \square }$

m∠${\displaystyle \square }$ = m∠PSR      .....[By (i) and (ii)]

∴ ∠PQR ≅ ∠PSR

Answer 1

Proof:

m∠PQR = ${\displaystyle \frac{1}{2}\times \left[\text{m}\left(\text{arc PTR}\right)\right]}$   ......(i) [Inscribed angle theorem]

m∠PSR = ${\displaystyle \frac{1}{2}\times \left[\text{m}\left(\text{arc PTR}\right)\right]}$   ......(ii) [Inscribed angle theorem]

m∠PQR = m∠PSR      .....[By (i) and (ii)]

∴ ∠PQR ≅ ∠PSR