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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.

Construction: Draw ray BL ⊥ PQ and ray CM ⊥ RS.

Proof: ∵ BL ⊥ PQ, CM ⊥ RS and PQ || RS ∴ BL || CM
∠LBC = ∠MCB    ...(1)
| Alternate Interior Angles
∠ABL = ∠LBC    ...(2)
| ∵    Angle of incidence = Angle of reflection
∠MCB = ∠MCD    ...(3)
| ∵    Angle of incidence = Angle of reflection
From (1), (2) and (3), we get
∠ABL = ∠MCD    ...(4)
Adding (1) and (4), we get
∠LBC + ∠ABL = ∠MCB + ∠MCD
⇒    ∠ABC = ∠BCD
But these are alternate interior angles and they are equal.
So, 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]