Question

1. given ab∥cd

∠1=∠4
prove ∠1=∠2

1. the sum of degrees of a 15 - sided polygon is:

1. find the number of sides of a regular polygon with an interior angle of 162°.

Explanation:

Question 11

Step1: Recall the formula for the sum of interior angles

The formula for the sum of interior angles of a polygon is \(S=(n - 2)\times180^{\circ}\), where \(n\) is the number of sides.

Step2: Substitute \(n = 15\) into the formula

When \(n=15\), we have \(S=(15 - 2)\times180^{\circ}\).
First, calculate \(15-2=13\).
Then, \(S = 13\times180^{\circ}=2340^{\circ}\)

Step1: Find the measure of the exterior angle

The measure of an interior angle \(\theta_{i}\) and an exterior angle \(\theta_{e}\) of a polygon satisfy \(\theta_{i}+\theta_{e}=180^{\circ}\).
Given \(\theta_{i} = 162^{\circ}\), then \(\theta_{e}=180^{\circ}- 162^{\circ}=18^{\circ}\)

Step2: Use the formula for the number of sides of a regular polygon

The sum of exterior angles of any polygon is \(360^{\circ}\). For a regular polygon with exterior angle \(\theta_{e}\), the number of sides \(n=\frac{360^{\circ}}{\theta_{e}}\)
Substitute \(\theta_{e}=18^{\circ}\) into the formula, we get \(n=\frac{360^{\circ}}{18^{\circ}} = 20\)

Answer:

\(2340^{\circ}\)

Question 12