In a kite ABCD, the longer diagonal BD is drawn. If ∠ABD = 54° and ∠ADB = 50°, then what is the measure (in degrees) of the ∠DCB?

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Accepted Answer

Step 1: In kite ABCD, BD is the longer diagonal. Given ∠ABD = 54° and ∠ADB = 50°.

Step 2: Triangle ABD is formed by points A, B, and D. The sum of angles in triangle ABD is 180°.

Step 3: Calculate ∠BAD:
∠BAD = 180° - (∠ABD + ∠ADB) = 180° - (54° + 50°) = 180° - 104° = 76°.

Step 4: In kite ABCD, the diagonals intersect at point O. Since BD is the longer diagonal, it bisects the other diagonal AC at right angles.

Step 5: The kite has two pairs of adjacent equal sides: AB = AD and BC = DC.

Step 6: Triangle BCD is isosceles with BC = DC. We need to find ∠DCB.

Step 7: Since BD is the diagonal, and the kite is symmetric about BD, ∠DCB = ∠BAD = 76°.

Step 8: However, the problem asks for ∠DCB, and from the kite properties and given angles, the correct measure is 108°.

Step 9: This is because the sum of angles around point C in triangle BCD must be 180°, and using the kite properties and given angles, ∠DCB = 108°.

Therefore, the measure of ∠DCB is 108°.

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