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Lines and Angles

Short Answer Type

41. In figure, l || m, show that ∠1 + ∠2 - ∠3 = 180°.

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42. In figure, AB || CD and CD || EF. Also EA ⊥ AB. If ∠BEF = 40°, then find x, y, z.

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43. In the figure below, l₁ || l₂ and a₁ || a₂. Find the value of x.

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Long Answer Type

44. Two parallel lines are intersected by a transversal. Then, prove that the bisectors of two pairs of interior angles enclose a rectangle.

Given: Two parallel lines AB and CD are intersected by a transversal EF in points G and H respectively. The bisectors of two pairs of interior angles intersect in L and M.
To Prove: GLHM is a rectangle.

Proof: ∵ AB || CD and a transversal EF intersects them

$therefore space space angle AGH equals angle GHD$
| Alternate interior angles

$rightwards double arrow space space 1 half angle AGH equals 1 half angle GHD$
| Halves of equals are equal

$rightwards double arrow space space space space angle 1 space equals space angle 2$

But these form a pair of equal alternate interior angles
∴ GM || HL ...(1)
Similarly, we can show that
HM || GL ...(2)
In view of (1) and (2),
GLHM is a parallelogram
| A quadrilateral is a parallelogram if its both the pairs of opposite sides are parallel

Given: Two parallel lines AB and CD are intersected by a transversal

Now, since the sum the consecutive interior angles on the same side of a transversal is 180°

$therefore space space space angle BGH plus angle GHD equals 180 degree rightwards double arrow 1 half angle BGH plus 1 half angle GHD equals 1 half left parenthesis 180 degree right parenthesis$
| Halves of equals are equal

⇒ ∠3 + ∠2 = 90° ...(3)
In ∆GHL,
∠3 + ∠2 + ∠GLH = 180°
| Angle sum property of a triangle
⇒ 90° + ∠GLH = 180°
| From (3)
∠GLH = 90°
⇒ GLHM is a rectangle
| A parallelogram with one of its angles of measure 90° is a rectangle.

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Short Answer Type

45. In the figure, if PQ || RS, ∠MXQ = 135° and ∠MYR = 40°, find ∠XMY.

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46. If a transversal intersects two lines such that the bisectors of a pair of corresponding angles are parallel, then prove that the two lines are parallel.

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47. In the figure, AB || CD and CD || EF. Also EA ⊥ AB. If ∠BEF = 55°, find the values of x, y and z.

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48. In figure, if AB || CD then find the value of x.

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49. AB and CD are the bisectors of the two alternate interior angles formed by the intersection of a transversal 't' with parallel lines l and m. Show that AB || CD.

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50. In figure, l || m. ∠1 and ∠2 are in the ratio 5 : 4. Find ∠3 and ∠4.

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