Two straight lines PQ and RS intersect each other at O as shown in the figure. If ∠POT = 75°, find the values of a, b and c.

Given: ∠POT = 75°

Observe that ∠ROP, ∠POT, and ∠TOS form a linear pair on line RS at point O.

⇒ ∠ROP + ∠POT + ∠TOS = 180°

From the figure, we have ∠ROP = 4b, ∠POT = 75°, ∠TOS = b

⇒ 4b + 75° + b = 180°

⇒ 5b + 75° = 180°

⇒ 5b = 105°

Now, observe that ∠QOS and ∠ROP form a pair of vertically opposite angles.

⇒ ∠QOS = ∠ROP

From the figure, we have ∠QOS = a, ∠ROP = 4b

⇒ a = 4b

But, we found b = 21°

⇒ a = 4 × 21°

∴ a = 84°

We also have ∠ROQ and ∠QOS forming a linear pair on line RS at point O.

⇒ ∠ROQ + ∠QOS = 180°

From the figure, we have ∠ROQ = 2c, ∠QOS = a

⇒ 2c + a = 180°

But, we found a = 84°

⇒ 2c + 84° = 180°

⇒ 2c = 96°

Thus, a = 84°, b = 21° and c = 48°.