Advertisements
Advertisements
Question
In the following figure, AOB is a diameter of the circle and C, D, E are any three points on the semi-circle. Find the value of ∠ACD + ∠BED.
Advertisements
Solution
Since, A, C, D and E are four point on a circle, then ACDE is a cyclic quadrilateral.
∠ACD + ∠AED = 180° ...(i) [Sum of opposite angles in a cyclic quadrilateral is 180°]
Now, ∠AEB = 90° ...(ii)
We know that, diameter subtends a right angle to the circle.
On adding equations (i) and (ii), we get
(∠ACD + ∠AED) + ∠AEB = 180° + 90° = 270°
⇒ ∠ACD + ∠BED = 270°
APPEARS IN
RELATED QUESTIONS
Bisectors of angles A, B and C of a triangle ABC intersect its circumcircle at D, E and F respectively. Prove that the angles of the triangle DEF are 90°-A, 90° − `1/2 A, 90° − 1/2 B, 90° − 1/2` C.
In the given figure, ABCD is a cyclic quadrilateral. Find the value of x.
ABCD is a cyclic quadrilateral in ∠BCD = 100° and ∠ABD = 70° find ∠ADB.
Prove that the circles described on the four sides of a rhombus as diameters, pass through the point of intersection of its diagonals.
PQRS is a cyclic quadrilateral such that PR is a diameter of the circle. If ∠QPR = 67° and ∠SPR = 72°, then ∠QRS =
ABCD is a cyclic quadrilateral. M (arc ABC) = 230°. Find ∠ABC, ∠CDA, and ∠CBE.
In a cyclic quadrilaterals ABCD, ∠A = 4x, ∠C = 2x the value of x is
In the figure, ▢ABCD is a cyclic quadrilateral. If m(arc ABC) = 230°, then find ∠ABC, ∠CDA, ∠CBE.
If a pair of opposite sides of a cyclic quadrilateral are equal, prove that its diagonals are also equal.
The three angles of a quadrilateral are 100°, 60°, 70°. Find the fourth angle.
