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 angles
Part 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 that
To find the central angle divide 360 degrees by the number of sides of the polygon
case a.1) Pentagon
The pentagon has 5 sides
so
The central angle is equal to
360°/5=72°
case a.2) Hexagon
The pentagon has 6 sides
so
The central angle is equal to
360°/6=60°
case a.3) Decagon
The pentagon has 10 sides
so
The central angle is equal to
360°/10=36°
case a.4) Dodecagon
The pentagon has 12 sides
so
The central angle is equal to
360°/12=30°
b) What is the measure of each interior angle of the polygon? Show your calculation
we know that
The sum of the interior angle of the polygon is equal to
S=(n-2)*180°
where
n is the number of sides
To find each the measure of each interior angle, divide the sum of the interior angles by the number of sides
case b.1) Pentagon
The pentagon has 5 sides
so
S=(n-2)*180°
S=(5-2)*180°=540°
Divide by the number of sides
The measure of each interior angle is equal to
540°/5=108°
case b.2) Hexagon
The hexagon has 6 sides
so
S=(n-2)*180°
S=(6-2)*180°=720°
Divide by the number of sides
The measure of each interior angle is equal to
720°/6=120°
case b.3) Decagon
The hexagon has 10 sides
so
S=(n-2)*180°
S=(10-2)*180°=1,440°
Divide by the number of sides
The measure of each interior angle is equal to
1,440°/10=144°
case b.4) Dodecagon
The hexagon has 12 sides
so
S=(n-2)*180°
S=(12-2)*180°=1,800°
Divide by the number of sides
The measure of each interior angle is equal to
1,800°/12=150°
Part 2. What is the relationship between the central angle and the interior angle?
we know that
The sum of the central angle plus the interior angle is equal to 180 degrees
therefore
The central angle and the interior angle are supplementary angles
Verify
Pentagon
72°+108°=180°
Hexagon
60°+120°=180°
Decagon
36°+144°=180°
Dodecagon
30°+150°=180°
Part 3. As the number of sides increases, how do the angles change?
we know that
As the number of sides increases, the central angle decreases and the interior angle increases.
