Answer:
7. The sum of the measures of the interior angles of this polygon is 900°.
8. The sum of the measures of the interior angles of this polygon is 540°.
9. The polygon has 10 sides.
10. The polygon has 5 sides.
Explanation:
7. Vertices: A, B, C, D, E, F and G
Number of vertices: n=7
Sum of the measures of the interior angles of the polygon: S=?
S=180°(n-2)
Substituting n by 7 in the formula above:
S=180°(7-2)
S=180°(5)
S=900°
8. Vertices: A, B, C, D and E
Number of vertices: n=5
Sum of the measures of the interior angles of the polygon: S=?
S=180°(n-2)
Substituting n by 5 in the formula above:
S=180°(5-2)
S=180°(3)
S=540°
9. Each interior angle of a regular polygon (i) measures 144°:
i=144°
How many sides does the polygon have?
n=?
i=180°(n-2)/n
Substituting i by 144° in the formula above:
144°=180°(n-2)/n
Solving for n: Cross multiplication:
144°n=180°(n-2)
Eliminating the parentheses on the right side of the equation applying the distributive property:
144°n=180°(n)-180°(2)
Multiplying
144°n=180°n-360°
Grouping n's on the right side of the equation: Subtracting 144°n both sides of the equation:
144°n-144°n=180°n-360°-144°n
Subtracting:
0=36°n-360°
Adding 360° both sides of the equation:
0+360°=36°n-360°+360°
Adding:
360°=36°n
Dividing both sides of the equation by 36°:
360°/36°=36°n/36°
10=n
n=10
The polygon has 10 sides.
10. Each interior angle of a regular polygon (i) measures 108°:
i=108°
How many sides does the polygon have?
n=?
i=180°(n-2)/n
Substituting i by 108° in the formula above:
108°=180°(n-2)/n
Solving for n: Cross multiplication:
108°n=180°(n-2)
Eliminating the parentheses on the right side of the equation applying the distributive property:
108°n=180°(n)-180°(2)
Multiplying
108°n=180°n-360°
Grouping n's on the right side of the equation: Subtracting 108°n both sides of the equation:
108°n-108°n=180°n-360°-108°n
Subtracting:
0=72°n-360°
Adding 360° both sides of the equation:
0+360°=72°n-360°+360°
Adding:
360°=72°n
Dividing both sides of the equation by 72°:
360°/72°=36°n/72°
5=n
n=5
The polygon has 5 sides.