22. ABCDE is a pentagon. If the vertices of the quadrilateral ABCD and BCDE, both lie on a
circle, prove that the vertices of the pentagon ABCDE also lie on a circle.
Answers
Step-by-step explanation:
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A regular pentagon = ABCDE (Given)
Quadrilateral = ABCD and BCDE (Given)
In Pentagon ABCDE
Sides AB = CB = DC = DE = AE ( As angle of a regular pentagon is 108°)
Sum of angles in a regular pentagon = 540°
Thus,
Each interior angle will be = 540/5
= 108°
Now, In ΔADE,
AE = DE ( alternate sides)
∠ADE = ∠DAE ( The angles opposite to equal sides are also equal)
∠ADE + ∠DAE +∠AED = 180°
∠ADE + ∠DAE + 108° = 180°
2∠ADE = 180° - 108°
2∠ADE = 72°
∠ADE = 36°
Therefore,
= ∠ADE = ∠DAE = 36°
= ∠DAB = 108° – 36° = 72°
Considering the quadrilateral ABCD
=∠DAB + ∠C = 72° + 108°
= ∠DAB + ∠C = 180°
Since the sum of the opposite angles of a quadrilateral is supplementary, thus ABCDE lies on the circle.