Question

In a given figure, PQ is the diameter of the circle. If ∠PQR=65∘,∠RPS=25∘ and ∠QPT=60∘. Find the measure of - (i) ∠QPR (ii) ∠PRS
ASAP

Answer 1

Answer:

(i) ∠QPR ∴ PQ is a diameter ∴ ∠PRQ = 900 [Angle in a semi-circle is 900] In ΔPQR, ∠QPR + ∠PRQ + ∠PQR = 1800 [Angle Sum Property of a triangle] => ∠QPR + 900 + 650= 1800 => ∠QPR + 1550 = 1800 => ∠QPR = 1800 - 1550 => ∠QPR = 250. (ii) ∠PRS ∴ PQRS is a cyclic quadrilateral ∴ ∠PSR + ∠PQR = 1800 [∴ Opposite angles of a cyclic quadrilateral are supplementary] => ∠PSR + 650 = 1800 => ∠PSR = 1800- 650 => ∠PSR = 1150 In DPSR, ∠PSR + ∠SPR + ∠PRS = 1800 [Angles Sum Property of a triangle] => 1150 + 400 + ∠PRS = 1800 => 1150 + ∠PRS = 1800 => ∠PRS = 1800 - 1550 => ∠PRS = 250 (iii) ∠QPM ∴ PQ is a diameter ∴ ∠PMQ = 900 [∴ Angle in a semi - circle is 900] In ΔPMQ, ∠PMQ + ∠PQM + ∠QPM = 1800 [Angle sum Property of a triangle] => 900 + 500 + ∠QPM = 1800 => 1400 + ∠QPM = 1800 => ∠QPM = 1800 - 1400 => ∠QPM = 400.

Step-by-step explanation:

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