Transcript A polygon has 54 diagonals. How many sides does the polygon have?

Question 1:
Find the measure of  A.
A
B
44°
145°
C
Question 1: SOLUTION
The measure of an exterior angle is
the sum of the two remote interior
angles.
A
B
44°
mA  44  145
m

A

101
145°
C
101
Question 2:
Find DC if HL = 68.
H
D
S
C
L
Question 2: SOLUTION
By the Midline Theorem, DC is
half the length of HL.
H
D
S
1
1
HL   68   34
2
2
C
L
DC = 34
Question 3:
Find the sum of the exterior
angles in a regular dodecagon.
Question 3 SOLUTION:
The sum of the exterior angles
in any polygon is always 360°.
360°
Question 4:
Find the sum of the
angles in a 62-gon.
Question 4 SOLUTION:
Find the sum of the angles in a 62-gon.
Sint  n  2 180
  62  2 180   60  180
 10800
10800°
Question 5:
Find the number of
diagonals that can be
drawn in a regular
octagon.
Question 5 SOLUTION:
Find the number of diagonals that
can be drawn in a regular
octagon.
n n  3
8  8  3
d 

2
2
8  5  40


 20
2
2
20
Question 6:
How many sides does
a pentadecagon
have?
Question 6 SOLUTION:
A pentadecagon has
15 sides.
15
Question 7:
Find the number of sides in a
regular polygon whose exterior
angles each measure 7.5°.
Question 7 SOLUTION:
Find the number of sides in a regular polygon whose
exterior angles each measure 7.5°.
E 
7.5 
360
n
360
n
7.5n  360
360
n
 48
7.5
48
Question 8:
Solve x2 + 5x = 84 by factoring.
What is the greater of the two
solutions?
Question 8 SOLUTION:
Solve x2 + 5x = 84 by factoring.
What is the greater of the two solutions?
x2 + 5x – 84 = 0
(x - 7)(x + 12) = 0
x-7=0
x=7
x + 12 = 0 x = -12
7
Question 9:
The sum of the interior angles of a
polygon is 7740°. How many
sides does the polygon have?
Question 9 SOLUTION:
The sum of the interior angles of a polygon is
7740°. How many sides does the polygon
have?
Sint  n  2  180
7740  n  2  180
43  n  2
n  45
45
Question 10:
Find the measure of an
interior angle in a regular
decagon.
Question 10 SOLUTION:
Find the measure of an interior
angle in a regular decagon.
I 

n  2 180 10  2 180
n
8 180
10

10
1440

 144
10
144
Question 11:
A polygon has 54 diagonals.
How many sides does the
polygon have?
Question 11 SOLUTION:
d 
54 
n n  3
2
A polygon has 54 diagonals. How
many sides does the polygon
have?
n n  3
2
n 2  3n  108  0
n  12n  9  0
n  12  0
n 9  0
12
n  12
n  9
Question 12:
What is the measure of an
exterior angle in a
regular 72-gon?
Question 12 SOLUTION:
What is the measure of an exterior
angle in a regular 72-gon?
E 
360
n
360
E 
5
72
5
Question 13:
The measure of an angle
in a regular polygon is
140°. How many sides
does the polygon have?
Question 13 SOLUTION:
The measure of an angle in a regular polygon is
140°. How many sides does the polygon have?
I 
n  2 180
n
140 
n  2 180
n
140n  n  2180
140n  180n  360
40n  360
n 9
9
Question 14:
Find the measure of the
missing angle in the
triangle.
36°
Question 14 SOLUTION:
Find the measure of the missing angle in the triangle.
180 – 90 – 36 = 54°
36°
54
Question 15:
The ratio of an interior angle
to an exterior angle in a
regular polygon is 7:1.
How many sides does the
polygon have?
Question 15 SOLUTION:
The ratio of an interior angle to an exterior angle in a
regular polygon is 7:1. How many sides does the
polygon have?
Interior and exterior angles are supplementary.
360
1x + 7x = 180
E 
8x = 180
22.5n  360
x = 22.5
n  16
Exterior angle = 22.5°
n
22.5 
16
360
n
Question 16:
Find the measure of S.
H
D
82°
67°
L
C
S
Question 16 SOLUTION:
Find the measure of S.
H
82°
D
82°
180 – 82 – 67 = 31°
S
67°
67°
L
C
31
Question 17:
Solve for x.
(x + 2)°
(2x - 18)°
(4x-11)°
(3x + 7)°
(2x + 8)°
Question 17 SOLUTION:
Solve for x.
(x + 2)°
(2x - 18)°
Sint  n  2180  5  2180  540
(4x-11)°
(3x + 7)°
(2x + 8)°
(x+2)+(2x-18)+(4x-11)+(2x+8)+(3x+7)=540
12x – 12 = 540
12x = 552
x = 46
46
Question 18:
Find the number of diagonals
in a 16-gon.
Question 18 SOLUTION:
Find the number of diagonals in a 16-gon.
n n  3
16 16  3 
d 

2
2
16 13 208


 104
2
2
104
Question 19:
Solve x2 – 13x + 42 = 0
by factoring.
Then find the smaller of
the two solutions.
Question 19 SOLUTION:
Solve x2 – 13x + 42 = 0 by factoring.
Then find the smaller of the two solutions.
x2 – 13x + 42 = 0
(x – 7)(x – 6) = 0
x–7=0
x=7
x–6=0
x=6
6
Question 20:
Find the measure of an
angle in a regular 18-gon.
Question 20 SOLUTION:
Find the measure of an angle in a regular 18-gon.
I 

n  2 180 18  2180
n
16 180
18

18
2880

 160
18
160
Question 21:
The vertex angle in an
isosceles triangle measures
62°. Find the measure of
one base angle.
Question 21 SOLUTION:
The vertex angle in an isosceles triangle measures 62°.
Find the measure of one base angle.
x + x + 62 = 180
2x + 62 = 180
62°
2x = 118
x = 59
x°
x°
59
Question 22:
The measure of an exterior
angle of a regular polygon
is 15°. Find the number of
sides in the polygon.
Question 22 SOLUTION:
The measure of an exterior angle of a regular polygon
is 15°. Find the number of sides in the polygon.
E 
360
n
15 
360
n
15n  360
n  24
24
Question 23:
How many sides does a
polygon have if it has
275 diagonals?
Question 23 SOLUTION:
d 
n n  3
2
n n  3
275 
2
550  n n  3
n 2  3n  550  0
n  25n  22  0
How many sides does a
polygon have if it has 275
diagonals?
n  25  0 n  25
n  22  0 n  22
25
The End!