MATH SOLVE

4 months ago

Q:
# Help with inscribed polygons. Me and my daughter do not understand this.

Accepted Solution

A:

Answer:Part 1) a.1) The central angle of pentagon is 72°a.2) The central angle of hexagon is 60°a.3) The central angle of decagon is 36°a.4) The central angle of dodecagon is 30°b.1) The measure of each interior angle of pentagon is 108°b.2) The measure of each interior angle of hexagon is 120°b.3) The measure of each interior angle of decagon is 144°b.4) The measure of each interior angle of dodecagon is 150°Part 2) The central angle and the interior angle are supplementary anglesPart 3) As the number of sides increases, the central angle decreases and the interior angle increases.Step-by-step explanation:Part 1. For each polygon, include the following information in the paragraph box below:
a) What was the central angle you used to locate the vertices? Show your calculation.
we know thatTo find the central angle divide 360 degrees by the number of sides of the polygoncase a.1) PentagonThe pentagon has 5 sidessoThe central angle is equal to360°/5=72°case a.2) HexagonThe pentagon has 6 sidessoThe central angle is equal to360°/6=60°case a.3) DecagonThe pentagon has 10 sidessoThe central angle is equal to360°/10=36°case a.4) DodecagonThe pentagon has 12 sidessoThe central angle is equal to360°/12=30°b) What is the measure of each interior angle of the polygon? Show your calculationwe know thatThe sum of the interior angle of the polygon is equal toS=(n-2)*180°wheren is the number of sidesTo find each the measure of each interior angle, divide the sum of the interior angles by the number of sidescase b.1) PentagonThe pentagon has 5 sidessoS=(n-2)*180°S=(5-2)*180°=540°Divide by the number of sidesThe measure of each interior angle is equal to540°/5=108°case b.2) HexagonThe hexagon has 6 sidessoS=(n-2)*180°S=(6-2)*180°=720°Divide by the number of sidesThe measure of each interior angle is equal to720°/6=120°case b.3) DecagonThe hexagon has 10 sidessoS=(n-2)*180°S=(10-2)*180°=1,440°Divide by the number of sidesThe measure of each interior angle is equal to1,440°/10=144°case b.4) DodecagonThe hexagon has 12 sidessoS=(n-2)*180°S=(12-2)*180°=1,800°Divide by the number of sidesThe measure of each interior angle is equal to1,800°/12=150°Part 2. What is the relationship between the central angle and the interior angle?we know thatThe sum of the central angle plus the interior angle is equal to 180 degreesthereforeThe central angle and the interior angle are supplementary anglesVerifyPentagon72°+108°=180° Hexagon60°+120°=180° Decagon36°+144°=180° Dodecagon30°+150°=180° Part 3. As the number of sides increases, how do the angles change?we know thatAs the number of sides increases, the central angle decreases and the interior angle increases.