In the below fig, lines PQ and RS intersect each other at point O.
Question:

In the below fig, lines PQ and RS intersect each other at point O. If ∠POR: ∠ROQ = 5: 7. Find all the angles.

Solution:

Given

∠POR and ∠ROP is linear pair

∠POR + ∠ROP = 180°

Given that

∠POR: ∠ROQ = 5:7

Hence, POR = (5/12) × 180 = 75

Similarly ROQ = (7/7 + 5) × 180 = 105

Now POS = ROQ = 105°  [Vertically opposite angles]

Also, SOQ = POR = 75°   [Vertically opposite angles]

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