Question

foundations of mathematics 11 unit 2 send-in assignment on-line cour

1. a regular polygon has 15 sides. find the measure of each angle.

13)
2 marks

1. the sum of the interior angles of a regular polygon is 3060°. find the exterior angle.

14)
2 marks

1. find the number of sides of a regular polygon with an exterior angle of 20°.

15)
2 marks

1. the following triangles are similar. write the ratios of the corresponding sides.

△sam ~ △ran
10)

Explanation:

Step1: Recall the formula for the measure of each interior angle of a regular polygon

The formula for the measure of each interior angle \(\theta\) of a regular polygon with \(n\) sides is \(\theta=\frac{(n - 2)\times180^{\circ}}{n}\)
For \(n = 15\), we substitute \(n\) into the formula: \(\theta=\frac{(15- 2)\times180^{\circ}}{15}\)

Step2: Calculate the value

First, calculate \(15 - 2=13\). Then \((13)\times180^{\circ}=2340^{\circ}\). And \(\frac{2340^{\circ}}{15}=156^{\circ}\)

Step1: Use the formula for the sum of interior angles

The sum of interior angles \(S=(n - 2)\times180^{\circ}\). Given \(S = 3060^{\circ}\), we solve for \(n\):
\((n - 2)\times180^{\circ}=3060^{\circ}\). Then \(n-2=\frac{3060^{\circ}}{180^{\circ}} = 17\), so \(n=19\)

Step2: Recall the formula for the measure of an exterior angle

The measure of an exterior angle \(\alpha\) of a regular polygon is \(\alpha=\frac{360^{\circ}}{n}\)
Substitute \(n = 19\) into the formula: \(\alpha=\frac{360^{\circ}}{19}\approx18.95^{\circ}\)

Step1: Use the formula for the measure of an exterior angle

The measure of an exterior angle \(\alpha\) of a regular polygon is \(\alpha=\frac{360^{\circ}}{n}\), where \(n\) is the number of sides.
Given \(\alpha = 20^{\circ}\), we solve for \(n\): \(n=\frac{360^{\circ}}{\alpha}\)

Step2: Calculate the value

Substitute \(\alpha=20^{\circ}\) into the formula: \(n=\frac{360^{\circ}}{20^{\circ}}=18\)

Step1: Identify corresponding sides

Since \(\triangle SAM\sim\triangle RAN\), the ratios of corresponding sides are:
\(\frac{SA}{RA}=\frac{AM}{AN}=\frac{SM}{RN}\)

Answer:

\(156^{\circ}\)

---