PQ is a diameter of a circle with centre If angle PQR=55 ,SPR=25 and angle PQM=50 Find QPR, QPM, and PRS

m∠QPR = 35°         m∠QPM =40°       m∠PRS = 30°

Step-by-step explanation:

ΔPRQ is a right triangle with right angle at R.  So m∠QPR = 90 - 55 = 35

ΔQPM is a right triangle with right angle at M.  So m∠QPM = 90 - 40 = 40

Arc RQ = 2(35) = 70 and arc SR = 2(25) = 50.  

So arc PS = 180 - (arc RQ + arc SR) = 180 - (70 + 50) = 180 - 120 = 60

Now arc PS is the intercepted arc for ∠PRS.  

Therefore, m∠PRS = 60/2 = 30

I used the fact that an inscribed angle has a measure 1/2 the measure of the intercepted arc several times.  Also, I used the fact that the acute angles of a right triangle are complementary.  And, finally I used the fact that an inscribed angle in a semicircle is a right angle.

I hope this helped.

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