In the given figure, ABCD is a kite whose diagonals intersect at O. If ∠DAB = 54° and ∠BCD = 76°, calculate :

(i) ∠ODA

(ii) ∠OBC.

(i) In kite ABCD,

AB = AD [Adjacent sides of kite are equal]

In triangle ABD,

⇒ ∠BDA = ∠ABD [Angles opposite to equal sides in a triangle]

In △ADB,

⇒ ∠BDA + ∠ABD + ∠DAB = 180° [∵ Angle sum property]

⇒ 2∠BDA + 54° = 180° [∵ ∠ODA = ∠OBA]

⇒ 2∠ODA = 180° - 54°

⇒ 2∠ODA = 126°

⇒ ∠ODA = 63°.

Hence, ∠ODA = 63°.

(ii) DC = CB [Adjacent sides of kite are equal]

∠BDC = ∠CBD [Angles opposite to equal sides in a triangle are equal]

In △CDB,

⇒ ∠BDC + ∠DCB + ∠CBD = 180°

⇒ 2∠CBD + 76° = 180°

⇒ 2∠OBC = 180° - 76°

⇒ 2∠OBC = 104°

⇒ ∠OBC = 52°.

Hence, ∠OBC = 52°.