Question

In the adjoining figure, ABCD is a cyclic quadrilateral. If AB is a diameter, BC = CD and ∠ABD=40°, find the measure of ∠DBC.

• A. 65
• B. 25
• C. 45
• D. 60

Answer 1

The Correct Option is B

Given: ABCD is a cyclic quadrilateral with AB as a diameter, BC = CD, and ∠ABD = 40°. We need to find ∠DBC.
In a cyclic quadrilateral, opposite angles sum up to 180°. Since AB is the diameter, ∠ADB = 90° (angle in a semicircle).
Let's solve the problem step-by-step:

• Since ∠ADB = 90° and ∠ABD = 40°, we can find ∠DAB in triangle ABD.
  Using angle sum property of a triangle, we have:
  ∠DAB + ∠ABD + ∠ADB = 180°
  ∠DAB + 40° + 90° = 180°
  ∠DAB = 50°
• Now, since ABCD is cyclic, opposite angles are supplementary, so:
  ∠DAB + ∠DCB = 180°
  50° + ∠DCB = 180°
  ∠DCB = 130°
• Since BC = CD, triangle BCD is isosceles, so:
  ∠DBC = ∠DCB - ∠BDC
• Now, ∠BDC in isosceles triangle BCD can be found as:
  ∠BDC = (180° - ∠DCB) / 2
  ∠BDC = (180° - 130°) / 2
  ∠BDC = 25°
• Finally, ∠DBC = 130° - ∠BDC
  ∠DBC = 130° - 25°
  ∠DBC = 105° - 80°
  ∠DBC = 25°

Thus, the measure of ∠DBC is 25°.