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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.

because space space

Ray OP stands on line Xy

therefore

angle POX plus angle POY equals 180 degree

| Liner Pair Axiom

rightwards double arrow space space space space space angle POX space plus space 90 degree space equals space 180 degree space space space space space space space space space space space space space space space space space space space space space space space space space space vertical line space because space angle POY equals 90 degree space left parenthesis Given right parenthesis rightwards double arrow space space space space space space angle POX equals 180 degree minus 90 degree rightwards double arrow space space space space space space space space angle POX equals 90 degree rightwards double arrow space space space angle POM plus angle XOM equals 90 degree

rightwards double arrow

a + b = 90⁰ .....(1)

a : 2 = 2 : 3

rightwards double arrow

straight a over straight b equals 2 over 3

rightwards double arrow

straight a over 2 equals b over 3 equals k left parenthesis s a y right parenthesis

rightwards double arrow

a = 2k
b = 3k

Putting the values of a and b in (1), we get

2k + 3k = 90⁰

open table attributes columnalign right end attributes row cell therefore space space space space space space straight a equals 2 straight k equals 2 left parenthesis 18 degree right parenthesis equals 36 degree end cell row cell b equals 3 k equals 3 left parenthesis 18 degree right parenthesis equals 54 degree end cell end table close curly brackets space space space space space space space space space space space space space space.... left parenthesis 2 right parenthesis

∵ Ray OX stands on line MN
∴ ∠XOM + ∠XON = 180°
| Linear Pair Axiom
⇒    b + c = 180°
⇒    54° + c = 180° | Using (2)
⇒    c = 180° - 54°
⇒    c = 126°.

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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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