Question 1. In the adjoining figure, AB  CD. If ∠ APQ = 54° and ∠ PRD = 126°, then find x and y.
Solution: ∵ AB  CD and PQ is a transversal, then interior alternate angles are equal.
⇒ ∠ APQ = ∠ PQR
⇒ 54° = x [∵ ∠ APQ = 54° (Given)]
Again, AB  CD and PR is a transversal, then ∠ APR = ∠ PRD [Interior alternate angles]
But ∠ PRD = 126° [Given]
∴ ∠ APR = 126°
Now, exterior∠ PRD = x + y
⇒ 126° = 54° + y
⇒ y = 126°  54° = 72°
Thus, x = 54° and y = 72°.
Question 2. In the adjoining figure AB  CD  EG, find the value of x.
Solution: Through E, let us draw FEG  AB  CD.
Now, since FE  AB and BE is a transversal.
∴ ∠ ABE + ∠ BEF = 180°
[Interior opposite angles]
⇒ 127° + ∠ BEF = 180°
⇒ ∠ BEF = 180° ∠ 127° = 53°
Again, EG  CD and CE is a transversal.
∴ ∠ DCE + ∠ CEG = 180° [Interior opposite angles]
⇒ 108° + ∠ CEG = 180° ⇒ ∠ CEG = 180°  108° = 72°
Since FEG is a straight line, then
⇒ ∠BEF + ∠BEC + ∠CEG = 180°
[Sum of angles at a point on the same side of a line = 180°]
⇒ 53° + x + 72°
= 180°
⇒ x = 180°  53°  72°
= 55°
Thus, the required measure of x = 55°.
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