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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.

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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

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

∵ Ray OR is perpendicular to line PQ.
∴ ∠QOR = ∠POR = 90°    ...(1)
∠QOS = ∠QOR + ∠ROS    ...(2)
∠POS = ∠POR - ∠ROS    ...(3)
From (2) and (3),
∴ ∠QOS - ∠POS = (∠QOR - ∠POR) + 2∠ROS = 2∠ROS | Using (1)

$rightwards double arrow space angle ROS equals 1 half left parenthesis angle QOS minus angle POS right parenthesis$

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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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