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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

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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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26. In figure, if AOB is a line, OP bisects ∠BOC and OQ bisects ∠AOC, show that ∠POQ is a right angle.

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27. In the given figure, lines XY and MN intersect each other at O. If ∠POY = 90° and a : b = 2 : 3, then find c.

∵ XOY is a line
∴ b + a + 90° = 180°

∵ XOY is a line∴ b + a + 90° = 180° a + b = 900 2

$rightwards double arrow$ a + b = 90⁰
2 + 3 = 5

$therefore space space space space space space angle straight a equals 2 over 5 cross times 90 degree equals 36 degree space space space space space space space space angle straight b space equals space 3 over 5 cross times 90 degree equals 54 degree because space space space space MON space is space straight a space line therefore space space space angle straight b plus angle straight c equals 180 degree rightwards double arrow space space space 54 degree plus angle straight c equals 180 degree space rightwards double arrow space angle straight c equals 126 degree$