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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.

∠POQ = ∠POC + ∠COQ

equals 1 half angle BOC plus 1 half angle AOC space space space space space space space space space space space space space left enclose table row cell because space OP space bisects space angle AOC end cell row cell therefore space angle POC equals 1 half angle BOC end cell row cell because space OQ space bisects space angle AOC end cell row cell therefore angle COQ equals 1 half angle AOC end cell end table end enclose equals 1 half left parenthesis angle BOC plus angle AOC right parenthesis equals space 1 half left parenthesis 180 degree right parenthesis space space space space space vertical line space L i n e a r space P a i r space A x i o m equals space 90 degree space space space space space space space space

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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.

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

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