The sum of the measures of the interior angles of a convex polygon is given. Find the number of sides of each polygon. 6) 7020° 7) 1980° 8) 6120° 9) 1800° 10) 3420°
Answers
Answer:5
Step-by-step explanation: Consider a triangle, a polygon with three sides. The sum of the interior angle measures is
180
˚
.
Consider any quadrilateral, a polygon with four sides. The sum of the interior angles measures
360
˚
. We can therefore deduce that for each polygon with an additional side has
180
˚
more than the previous figure.
This forms an arithmetic series. Note: An arithmetic series is a sequence of numbers where a common difference is added or subtracted from previous terms to give the next terms. For example, 2, -1, -4 forms an arithmetic series, with a common difference of 3.
The general term of an arithmetic series is given by
t
n
=
a
+
(
n
−
1
)
d
.
We know
t
n
, which is
720
˚
, and
a
, which is
0
˚
( a figure with one line would have an angle measure of
0
˚
), and
d
is
180
.
720
=
0
+
(
n
−
1
)
180
720
+
180
=
180
n
900
=
180
n
5
=
n
Since the figure with angles measuring
0
˚
is 1 lines, then the figure with interior angles of
720
˚
has
1
+
5
=
6
sides.