Download 1 7.3 Formulas Involving Polygons (p.309-313)

Geometry for Enjoyment and Challenge - Text Solutions
1
7.3 Formulas Involving Polygons (p.309-313)
1. Find the sum of the measures of the interior angles of a:
(a) Quadrilateral:
Solution: n = 4
Si = (4 − 2) · 180 = 360
(b) Heptagon
Solution: n = 7
Si = (7 − 2) · 180 = 900
(c) Octagon:
Solution: n = 8
Si = (8 − 2) · 180 = 1080
(d) Dodecagon
Solution: n = 12
Si = (12 − 2 · 180 = 1800
(e) 93-gon:
Solution: n = 93
Si = (93 − 2) · 180 = 16, 380
Ruth Doherty
2.
A
Given: m∠A = 160, m∠B = 50
m∠C = 140, m∠D = 150
B
E
Find m∠E
D
Solution:
The sum of the interior angles of a pentagon is 540◦ . So:
m∠A + m∠B + m∠C + m∠D + m∠E = 540
160 + 50 + 140 + 150 + m∠E = 540
m∠E = 40
C
3. How many diagonals can be drawn in each figure?
b.
a.
Solution:
In a polygon with n sides:
Number of diagonals =
(a) 5(5 − 3) ÷ 2 = 5
(b) 6(6 − 3) ÷ 2 = 9
(c) 4(4 − 2) ÷ 2 = 4
(d) 3(3 − 2) ÷ 2 = 0
n(n − 3)
2
c.
d.
5.
J
Given: K is a midpoint
P is a midpoint
m∠M = 70
m∠JKP = y + 15
m∠JP K = y − 10
K
Find
(a) m∠JKP
P
(b) m∠JP K
(c) m∠J
M
Solution: Because K and P are midpoints, KP ||M O by the midline theorem.
(a) m∠M = m∠JKP by P → CAC, so m∠JKP = 70
(b) m∠JKP = 70
70 = y + 15
y = 55
m∠JP K = 55 − 10 = 45
(c) m∠J + m∠JKP + m∠JP K = 180
m∠J + 70 + 45 = 180
m∠J = 65
O
10. How many sides does a polygon have if the sum of the measures of its angles is:
Solution:
(a) 900?
180(n − 2) = 900
n=7
(b) 1440?
(1440 ÷ 180) + 2 = 10
(c) 2880?
(2880 ÷ 180) + 2 = 18
(d) 180x − 720?
180x − 720
+2=x−4+2=x−2
180
(e) 436?
436 ÷ 180 + 2 =
109
90
+ 2 This is not a whole number, so it is impossible to have Si = 436
(f) Six right angles?
6 · 90 = 540 Pentagons have an Si = 540
11.
(a) In what polygon is the sum of the measures of the exterior angles, one per vertex, equal to the
sum of the measures of the angles of the polygon?
Solution:
The sum of the exterior angles of a polygon (Se ) is always 360. Only a quadrilateral has
an Si of 360.
(b) In what polygon is the sum of the measures of the angles of the polygon equal to twice the
sum of the measures of the exterior angles, one per vertex?
Solution:
The sum of the exterior angles of a polygon (Se ) is always 360. 2 · 360 = 720. Hexagons
have an Si of 720.