ABCD is a parallelogram. The angle bisectors of ∠A and ∠D meet at O. What is the measure of ∠AOD?
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Step by Step Explanation:
- The following figure shows the parallelogram ABCD,
AO and DO are the bisectors of ∠DAB and ∠ADC respectively.
Therefore, ∠DAB = 2∠DAO ,
∠ADC = 2∠ADO
- The ∠DAB and the ∠ADC are consecutive angles of the parallelogram ABCD, we know that, the consecutive angles of a parallelogram are supplementary.
Therefore, ∠DAB + ∠ADC = 180°
⇒ 2∠DAO + 2∠ADO = 180°
⇒ 2(∠DAO + ∠ADO) = 180°
⇒ ∠DAO + ∠ADO = 90° ------(1)
- We know that, the sum of all the angles of a triangle is equal to 180°.
In ΔAOD, ∠DAO + ∠ADO + ∠AOD = 180°
⇒ 90° + ∠AOD = 180° [From equation (1), ∠DAO + ∠ADO = 90°]
⇒ ∠AOD = 180° - 90°
⇒ ∠AOD = 90°
- Hence, the measure of ∠AOD is 90°.