QQuestion
In the given diagram, chords AC and BC are equal. If ∠ACD = 120°, then ∠AEC is:
(a) 30°
(b) 60°
(c) 90°
(d) 120°
✓Answer
(d) 120°
Explanation:
∠ACD and ∠ACB form a linear pair because BD is a straight line.
∠ACD + ∠ACB = 180°
⇒ 120° + ∠ACB = 180°
⇒ ∠ACB = 60°
In △ABC, AC = BC.
Therefore, ∠ABC = ∠BAC.
Let ∠ABC = ∠BAC = x.
x + x + 60° = 180°
⇒ 2x = 120°
⇒ x = 60°
Since ABCE is a cyclic quadrilateral, its opposite angles are supplementary.
∠ABC + ∠AEC = 180°
⇒ 60° + ∠AEC = 180°
⇒ ∠AEC = 120°
Related Questions
More ICSE Moderate level questions
In the given diagram, PS and PT are tangents to the circle. SQ ∥ PT and ∠SPT = 80°. The value of ∠QST is: (a) 140° (b) 90° (c) 80° (d) 50°
In the given diagram, RT is a tangent touching the circle at S. If ∠PST = 30° and ∠SPQ = 60°, then ∠PSQ is equal to: (a) 40° (b) 30° (c) 60° (d) 90°
The figure shows a circle of radius 9 cm with O as the centre. The diameter AB produced meets the tangent PQ at P. If PA = 24 cm, find the length of tangent PQ.
In the given diagram, an isosceles △ABC is inscribed in a circle with centre O. PQ is a tangent to the circle at C. OM is perpendicular to chord AC and ∠COM = 65°. Find: (a) ∠ABC (b) ∠BAC (c) ∠BCQ
In the given figure, O is the centre of the circle. PQ is a tangent to the circle at T. Chord AB produced meets the tangent at P. AB = 9 cm, BP = 16 cm, ∠PTB = 50° and ∠OBA = 45°. Find: (a) the length of PT (b) ∠BAT (c) ∠BOT (d) ∠ABT
In the given figure, O is the centre of the circle. CE is a tangent to the circle at A. If ∠ABD = 26°, find: (a) ∠BDA (b) ∠BAD (c) ∠CAD (d) ∠ODB
In the given diagram, O is the centre of the circle. PR and PT are two tangents drawn from the external point P and touching the circle at Q and S respectively. MN is a diameter of the circle. Given ∠PQM = 42° and ∠PSM = 25°. Find: (a) ∠OQM (b) ∠QNS (c) ∠QOS (d) ∠QMS
Assertion (A): If sin² A + sin A = 1, then cos⁴ A + cos² A = 1. Reason (R): 1 − sin² A = cos² A. (a) A is true, R is false. (b) A is false, R is true. (c) Both A and R are true, and R is the correct reason for A. (d) Both A and R are true, and R is not the correct reason for A.
A man invested in a company paying 12% dividend on its shares. If the percentage return on his investment is 10%, then the shares are: (a) at par (b) below par (c) above par (d) cannot be determined
The factor common to the two polynomials x² − 4 and x³ − x² − 4x + 4 is: (a) (x + 1) (b) (x − 1) (c) (x − 2) (d) (x − 4)
The solution set for 0 < − < 2, x ∈ Z is: (a) {−5, −4, −3, −2, −1} (b) {−6, −5, −4, −3, −2, −1} (c) {−5, −4, −3, −2, −1, 0} (d) {−6, −5, −4, −3, −2, −1, 0} bb
In the given diagram, △ABC ∼ △EFG. If ∠ABC = ∠EFG = 60°, then the length of side FG is: (a) 15 cm (b) 20 cm (c) 25 cm (d) 30 cm