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Choose whether given statement is true or false:

In a regular pentagon ABCDE. If the bisector of angle A of the pentagon meets the side CD in point M. Then $∠ A M C = 90 °$ .

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True

False

answer is A.

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Detailed Solution

Given that, a pentagon ABCDE. The line segment AM is the bisector of the

∠ A

In pentagon ABCDE,
As we know the measure of each interior angle of a regular pentagon is

108 °

∴ ∠ B A M = 12 × 108 ° = 54 °

Applying the angle sum property of quadrilateral that is the sum of angles of quadrilateral is

360 ∘

.
In quadrilateral ABCM,

∠ B A M + ∠ A B C + ∠ B C M + ∠ A M C = 360 ∘ 54 ∘ + 108 ∘ + 108 ∘ + ∠ A M C = 360 ∘ ∠ A M C = 360 ∘ − 270 ∘ ∠ A M C = 90 ∘

So,

∠ A M C = 90 °

.
Hence, the given statement is true and option 1 is correct.

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