ABCD is a parallelogram. The angle bisectors of ∠A and ∠D meet at O. What is the measure of ∠AOD?


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Answer:

90°

Step by Step Explanation:
  1. 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
  2. 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)
  3. 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°
  4. Hence, the measure of ∠AOD is 90°.

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