AB is a diameter of a circle and ABCD is a cyclic quadrilateral in it. If ∠ACD…
2019
AB is a diameter of a circle and ABCD is a cyclic quadrilateral in it. If ∠ACD = 40°, then the measure of ∠BAD is
- A.
90°
- B.
50°
- C.
40°
- D.
80°
Show answer & explanation
Correct answer: B
An angle subtended by a diameter of a circle at any point on the circumference is a right angle (angle in a semicircle theorem). Also, angles subtended by the same chord at points lying on the same arc are equal (angles in the same segment).
AB is a diameter of the circle and D lies on the circle, so ∠ADB = 90° by the angle-in-a-semicircle theorem.
∠ABD and ∠ACD are angles subtended by the same chord AD from points B and C, which lie on the same arc, so ∠ABD = ∠ACD = 40° (angles in the same segment).
In triangle ABD, the three angles sum to 180°: ∠BAD + ∠ABD + ∠ADB = 180°, so ∠BAD = 180° − 90° − 40° = 50°.
AB is also a diameter, so ∠ACB = 90° (angle in a semicircle at C). So ∠BCD = ∠ACB + ∠ACD = 90° + 40° = 130°. In the cyclic quadrilateral ABCD, opposite angles sum to 180°, so ∠BAD = 180° − ∠BCD = 180° − 130° = 50°, matching the triangle calculation.
∠BAD = 50°.