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Lines and Angles

Short Answer Type

21. In figure, lines XY and MN intersect at O. If ∠POY = 90° and a : b = 2 : 3, find c.

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22. In figure, ∠PQR = ∠PRQ, then prove that ∠PQS = ∠PRT.

∵ Ray QP stands on line ST
∴ ∠PQS + ∠PQR = 180°    ...(1)
| Linear Pair Axiom
∵ Ray RP stands on line ST
∴ ∠PRQ + ∠PRT = 180°    ...(2)
| Linear Pair Axiom
From (1) and (2), we obtain
∠PQS + ∠PQR = ∠PRQ + ∠PRT
⇒    ∠PQS = ∠PRT.
| ∵ ∠PQR = ∠PRQ (Given)

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23. In figure, if x + y = w + z, then prove that AOB is a line.

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24. In figure, POQ is a line. Ray OR is perpendicular to line PQ. OS is another ray lying between rays OP and OR. Prove that

angle ROS equals 1 half left parenthesis angle QOS minus angle POS right parenthesis

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25. It is given that ∠XYZ = 64° and XY is produced to point P. Draw a figure from the given information. If ray YQ bisects ∠ZYP, find ∠XYQ and reflex ∠QYP.

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