In the given figure, two lines AB and CD intersect at a point O. If ∠AOC + ∠BOD = 70°, find the measure of ∠AOD.
Lines & Angles
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Question

In the given figure, two lines AB and CD intersect at a point O. If ∠AOC + ∠BOD = 70°, find the measure of ∠AOD.

KnowledgeBoat · Accepted Answer

In the given figure, two straight lines AB and CD intersect at point O.

When two lines intersect, the angles opposite to each other are equal:

∴ ∠AOC = ∠BOD $\quad$ [Vertically opposite angles]

The question states that:

∠AOC + ∠BOD = 70°

Since they are equal, we can replace ∠BOD with ∠AOC in the equation:

⇒ ∠AOC + ∠AOC = 70°

⇒ 2∠AOC = 70°

⇒ ∠AOC = $\dfrac{70^{\circ}}{2}$

⇒ ∠AOC = 35°

So, ∠AOC = 35° and ∠BOD = 35°.

Since CD is a straight line, the angles ∠AOC and ∠AOD form a linear pair and must sum to 180°.

∴ ∠AOC + ∠AOD = 180°

⇒ 35° + ∠AOD = 180°

⇒ ∠AOD = 180° - 35°

⇒ ∠AOD = 145°

The measure of ∠AOD is 145°.

Mathematics