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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.

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28. In figure, find the values of x and y and then show that AB || CD.

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29. In figure, if AB || CD, CD || EF and y : z = 3 : 7, find x.

∵ AB || CD
and CD || EF
∴ AB || EF
| Lines parallel to the same line are parallel to each other
∴ x = z ...(1)
| Alternate Interior Angles
x + y = 180° ...(2)
| Consecutive interior angles on the same side of the transversal GH to parallel lines AB and CD
From (1) and (2),
z + y = 180°
y : z = 3 : 7
Sum of the ratios = 3 + 7 = 10

$therefore space space space straight y space equals space 3 over 10 cross times 180 degree space equals space 54 degree and space space straight z equals 7 over 10 cross times 180 degree equals 126 degree therefore space space space straight x space equals space straight z equals 126 degree$

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30. In figure, if AB || CD, EF ⊥ CD and ∠GED = 126°, find ∠AGE, ∠GEF and ∠FGE.

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