Question

The diagram shows pentagon ABCDE inscribed in a circle, centre O.
Given AB=BC=CD and angle ABC =132
Calculate the value of
i Angle AEB
ii AED
iii COD
​

Answer 1

Answer:

We are given,  

ABCDE is a pentagon inscribed in a circle.

AB = BC = CD

∠ABC = 132°  

Let’s join points E & B and E & C.

Case (i): Measure of angle AEB

Since in cyclic quadrilateral ABCE, the sum of opposite angles is 180°

∴ ∠AEC + ∠ABC = 180°

⇒ ∠AEC = 180° - 132° = 48°

We have, AB = BC

So, ∠AEB = ∠BEC ….. [equal chords subtends equal angles]

∴ ∠AEB = ∠BEC = ½ * ∠AEC = ½ * 48° = 24°  

Thus, the measure of angle AEB = 24°

Case (ii): Measure of angle AED

Since we have, AB = BC = CD

∴ ∠AEB = ∠BEC = ∠CED = 24° …… [from case (i)]

Also, from the figure given, we can say,

∠AED = ∠ AEB + ∠BEC + ∠CED = 24°+ 24°+24° = 72°

Thus, the measure of angle AED = 72°

Case (iii): Measure of angle COD

From the figure, we can see that  

CD subtends ∠COD at the centre and ∠CED at the circumference of the circle.

And, we know, that the central angle is twice any inscribed angle subtended by the same chord, so,

∠COD = 2 * ∠CED = 2 * 24° = 48°

Thus, the measure of the angle COD = 48°.

Answer 2

Step-by-step explanation:

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