The adjoining figure shows a pentagon inscribed in a circle with center O.
Given AB = BC = CD.
Angle ABC = 130. Find
(I) angle AEB
(II) angle AED
(III) angle COD?

Question

The adjoining figure shows a pentagon inscribed in a circle with center O.Given AB = BC = CD.Angle ABC = 130. Find (I) angle AEB (II) angle AED (III) angle COD?

Answer

Answer

1 ...25 degree
2... 75 degree
3... 50 degree
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Answer

Given Information
- A pentagon is inscribed in a circle with center O.
- Sides: AB = BC = CD
- Angle ABC = 130°
Finding Angles
Angle AEB
- Angle AEB is an inscribed angle that subtends arc AB.
- The measure of an inscribed angle is half the measure of the central angle subtending the same arc.
- Central angle AOB = Angle ABC + Angle BOC.
- Since AB = BC, angle BOC = angle AEB.
- Therefore, angle AEB = 90° (as inscribed angles subtending the same arc are equal).
Angle AED
- Angle AED subtends arc AD.
- Angle ABC is given as 130°.
- Since AB = BC = CD, angle ACD = angle ABC = 130°.
- Angle AOD, the central angle for arc AD, is obtained by adding angles ABC and ACD.
- Therefore, angle AED = half of angle AOD = (130° + 130°) / 2 = 130°.
Angle COD
- Angle COD subtends arc CD.
- Since sides are equal, angle AOB = angle BOC = angle COD.
- Angle AOB = 180° - (angle ABC + angle ACD).
- Therefore, angle COD = central angle for arc CD = 180° - (130° + 130°) = 20°.
Summary of Angles
- Angle AEB = 90°
- Angle AED = 130°
- Angle COD = 20°
These relationships rely on the properties of inscribed angles and equal chords in a circle.

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