Question

if each interior angle of a regular polygon is p times as large as each exterior angle , prove that the number of sides is 2(p+1).

Answer 1

Working out:

This is a polygon and as usual it is having interior and exterior angles. But it is not provided the exact number of sides of polygon. So,

Let,

The number of sides of the polygon be n

We know,

Formula for finding the measure of each interior and exterior angle of a regular polygon:

• Interior angle = (n - 2)180° / n
• Exterior angle = 360°/ n

Here, n is divided by the total sum of angle measure of the polygon because it's a regular polygon$.$

According to question,

➝ Interior angle = n × Exterior angle

➝ (n - 2) 180°/n = p × 360°/n

n is in the denominator of fractions in both sides of the equation, so we will cancel them to get:

➝ (n - 2)180 = 360p

Opening the parentheses,

➝ 180n - 360 = 360p

➝ 180n = 360p + 360

➝ n = 360p + 360/180

➝ n = 2p + 2

➝ n = 2(p + 1)

Hence, proved !!

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Answer 2

Answer:

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