Q4 of 34 Page 116

In the figure, QT ⊥ PR, ∠TQR = 40° and ∠SPR = 30°. Find ∠TRS and ∠PSQ.

Given: ∠TQR = 40° and ∠SPR = 30°.


∠ QTR = 90° (right angle)


As we know, using theorem (1), in any triangle, sum of the three interior angles is 180 °.


So,


∠TQR + ∠QTR + ∠ TRQ = 180°


⇒ 40° + 90° + ∠ TRQ = 180°


⇒ ∠ TRQ = 180° - 40° - 90°


⇒ ∠ TRQ = 50°


⇒ ∠ TRQ =∠ TRS = 50°


Also,


∠SPR + ∠PRS + ∠ RSP = 180°


∠SPR + ∠TRS + ∠ RSP = 180°


⇒ 30° + 50° + ∠ RSP = 180°


⇒ ∠ RSP = 180° - 30° - 50°


⇒ ∠ RSP = 100°


As R, S and Q lie on the same line.


So,


∠ RSP + ∠ PSQ = 180°


⇒ ∠ PSQ + 100° = 180°


⇒ ∠ PSQ = 80°


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