Question

use this regular pentagon to answer the questions. what is the measure of one of the central angles in the regular pentagon? what is the measure of one of the congruent base angles in triangle abc? 36 54 72 108

Explanation:

Step1: Find the measure of the central angle

The sum of central angles of a polygon is \(360^{\circ}\). For a regular pentagon (\(n = 5\)), the measure of one central angle \(\theta\) is given by the formula \(\theta=\frac{360^{\circ}}{n}\).
Substituting \(n = 5\) into the formula: \(\theta=\frac{360^{\circ}}{5}=72^{\circ}\)

Step2: Find the measure of the congruent base angles in \(\triangle ABC\)

Let the central angle \(\angle BAC = 72^{\circ}\). In \(\triangle ABC\), \(AB = AC\) (radii of the circum - circle of the regular pentagon), so \(\triangle ABC\) is isosceles. Let the base angles be \(\angle ABC=\angle ACB\).
Using the angle - sum property of a triangle (\(\angle BAC+\angle ABC+\angle ACB = 180^{\circ}\)). Let \(x=\angle ABC=\angle ACB\). Then \(72^{\circ}+x + x=180^{\circ}\).
Simplify the equation: \(2x=180^{\circ}-72^{\circ}=108^{\circ}\).
Solve for \(x\): \(x = 54^{\circ}\)

Answer:

The measure of one of the central angles is \(72^{\circ}\). The measure of one of the congruent base angles in \(\triangle ABC\) is \(54^{\circ}\)