Choose whether the given statement is true or false.

In a quadrilateral ABCD, AO and BO are bisectors of angle A, and angle B respectively. Then $∠ AOB = \frac{1}{2} {∠ C + ∠ D}$ .

Question

Choose whether the given statement is true or false. In a quadrilateral ABCD, AO and BO are bisectors of angle A, and angle B respectively. Then ∠AOB=12{∠C+∠D}.

Accepted Answer

We draw a quadrilateral ABCD with center O. OA bisects angle A and OB bisects angle B.

https://www.vedantu.com/question-sets/5e70f802-1f7e-46c5-9063-b2ef8de71ff06452650772178244064.png

Since OA bisects ∠A

\Rightarrow ∠ OAB = ∠ OAD = \frac{1}{2} ∠ A

⇒2∠OAB=∠A .........................… (1)
Similarly, OB bisects ∠B

\Rightarrow ∠ OBA = ∠ OBC = \frac{1}{2} ∠ B

⇒2∠OBA=∠B .......................… (2)
ABCD is a quadrilateral so by property of the sum of interior angles of quadrilaterals.

\Rightarrow ∠ A + ∠ B + ∠ C + ∠ D = 360 °

Substitute values of angles A and B from equations (1) and (2) ⇒2∠OAB+2∠OBA+∠C+∠D=360∘
Divide both sides by 2

\Rightarrow \frac{2 ∠ OAB + 2 ∠ OBA + ∠ C + ∠ D}{2} = \frac{360 °}{2}

We can break the terms in LHS as

\Rightarrow \frac{2 ∠ OAB}{2} + \frac{2 ∠ OBA}{2} + \frac{∠ C + ∠ D}{2} = \frac{360 °}{2}

\Rightarrow ∠ OAB + ∠ OBA + \frac{1}{2} {∠ C + ∠ D} = 180 °

.......................… (3)
Now in △AOB apply the property of the sum of interior angles of a triangle.
⇒∠OAB+∠OBA+∠AOB=180

°

................… (4)
From (3) and (4), we can equate the LHS of both the equations.
⇒∠OAB+∠OBA+

\frac{1}{2}

{∠C+∠D}=∠OAB+∠OBA+∠AOB
Cancel same terms from both sides of the equation
⇒

\frac{1}{2}

{∠C+∠D}=∠AOB
∴ ∠AOB=

\frac{1}{2}

{∠C+∠D}
Hence, the given statement is true.

Choose whether the given statement is true or false.

In a quadrilateral ABCD, AO and BO are bisectors of angle A, and angle B respectively. Then $∠ AOB = \frac{1}{2} {∠ C + ∠ D}$ .

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