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Lines and Angles

Short Answer Type

31.

In figure, if PQ || ST, ∠PQR = 110° and ∠RST = 130°, find ∠QRS.
[Hint. Draw a line parallel to ST through point R.]

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32. In figure, if AB II CD, ∠APQ = 50° and ∠PRD = 127°, find x and y.

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33. In figure, PQ and RS are two mirrors placed parallel to each other. An incident ray AB strikes the mirror PQ at B, the reflected ray moves along the path BC and strikes the mirror RS at C and again reflects back along CD. Prove that AB || CD.

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34.

In figure, if x = y and a = b, prove that r || n.

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35. If two lines are perpendicular to the same line, prove that they are parallel to each other.

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36.

In figure, EF is a transversal to two parallel lines AB and CD, GM and HL are the bisectors of the corresponding angles EGB and EHD. Prove that GM || HL.

[Hint. First prove that ∠EGM = ∠GHL]

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37. In figure, if AB || CD then find the value of y.

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38. In figure, is line I parallel to m? Justify your answer.

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39.
In figure, PQ || RS and T is any point as shown in the figure then show that

∠PQT + ∠QTS + ∠RST = 360°.

Given: PQ || RS and T is any point.
To Prove: ∠PQT + ∠QTS + ∠RST = 360°
Construction: Through T, draw TU || PQ || RS

Proof: ∵ PQ || UT | By construction and a transversal QT intersects then
∴ ∠PQT + ∠QTU = 180°    ...(1)
| Sum of consecutive interior angles on the same sides of a transversal is 180°
∵ UT || RS    | By construction and a transversal TS intersects them
∴ ∠UTS + ∠RST = 180°    ...(2)
| Sum of consecutive interior angles on the same side of a transversal is 180°
Adding (1) and (2), we get
∠PQT + (∠QTU + ∠UTS) + ∠RST = 360°
⇒     ∠PQT + ∠QTS + ∠RST = 360°

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