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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.

x + y = w + z ...(1) | Given
∵ The sum of all the angles round a point is equal to 360°.
$therefore$ x + y + w + z = 360⁰
$rightwards double arrow$ x + y + x + y = 360⁰ | Using (1)
$rightwards double arrow$ 2(x + y) = 360⁰

$rightwards double arrow space space space space space space space space space space space space space straight x plus straight y equals fraction numerator 360 degree over denominator 2 end fraction$
$rightwards double arrow$ x + y = 180⁰

∴ AOB is a line.

| If the sum of two adjacent angles is 180°, then the non-common arms of the angles form 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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