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

Given: l || m
To Prove: ∠1 + ∠2 - ∠3 = 180°
Construction: Through C, draw CF || l || m

Given: l || mTo Prove: ∠1 + ∠2 - ∠3 = 180°Construction: Throug

Proof: ∵ l || CF by construction and a transversal BC intersects them
∴ ∠1 + ∠FCB = 180°
| ∵ Sum of consecutive interior angles on the same side of a transversal is 180°
⇒ ∠1 + ∠2 - ∠FCD = 180° ...(1)
But ∠FCD = ∠3 ...(2)
| Alternate interior angles
∴ From (1) and (2),
∠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.

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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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