Download Unit #5 Lesson #1: Polygons and Their Angles. Polygons are figures

Unit #5 Lesson #1: Polygons and Their Angles.
Polygons are figures created from segments that do not intersect at any points other than their
endpoints.
A polygon is convex if all of the interior angles have measure less than 180°.
A polygon that is not convex is concave.
Naming Polygons
Number of Sides
3
4
5
6
7
8
9
10
n
Name
triangle
quadrilateral
pentagon
hexagon
heptagon
octagon
nonagon
decagon
n-gon
Theorem: The sum of the interior angles of a convex polygon with n sides is (n – 2)180.
Theorem: The sum of the exterior angles of a convex polygon is 360°.
In a regular polygon all sides are congruent and all angles are congruent.
Exploration:
1. Draw a polygon with congruent sides, but not congruent angles.
2. Draw a polygon with congruent angles, but not congruent sides.
Examples:
1. Find the measure of the largest angle in the polygon below.
(9x)°
(12x + 21)°
(13x - 10)°
(5x + 33)°
(8x - 6)°
(15x)°
2. Find the measure of an interior angle of a regular pentagon.
3. Find the measure of an exterior angle in a polygon where the ratio of interior to exterior
angles is 3:1.
Problem Set #1:
1. Find the sum of the interior angles of each polygon described below.
a. Quadrilateral
b. Hexagon
c. Decagon
2. Find the sum of the exterior angles of each polygon described below.
a. Quadrilateral
b. Pentagon
c. Decagon
3. Find the measure of one exterior angle of a regular
a. Quadrilateral
b. Hexagon
c. Octagon
4. Find the measure of one interior angle of a regular
a. Quadrilateral
b. Hexagon
c. Octagon
5. A pentagon has angles measuring 84°, 112°, 134°, and 92°. What is the measure of the
fifth angle?
6. Find the measure of the smallest angle in a polygon having angles measuring 4x – 1, 12x,
9x + 6, and 8x + 25.
7. How many sides does a regular polygon have if an exterior angle measures 120°?
8. How many sides does a regular polygon have if an interior angle measures 120°?
9. How many sides does a polygon have if the sum of the interior angles is 2340°?
10. In a regular polygon the measure of an interior angle is five times the measure of an
exterior angle.
a. What is the measure of an interior angle?
b. How many sides does the polygon have?
11. The ratio of interior to exterior angles in a regular polygon is 3:2.
a. Find the measure of an exterior angle.
b. Name the polygon.
12. A regular polygon’s interior and exterior angles have the same measure. What is the
most specific name for that polygon?
13. Polygon ABCDEF is a regular hexagon. Find the value of x.
B
C
A
x°
F
D
E
14. True or False. If the statement is true, give an example or explain why it is true. If the
statement is false, provide a counterexample.
a. A triangle that is equiangular must be equilateral.
b. A quadrilateral that is equiangular must be equilateral.
c. A regular polygon can have an interior angle that measures 100°.
d. As the number of sides of a polygon increases the number of exterior angles
increases.
e. If the measure of an interior angle of a regular polygon is equal to the measure of the
exterior angle, then the polygon must be a pentagon.
f. When a new polygon is formed by joining the consecutive midpoints of the sides of
an original polygon, the sum of the measures of the angles of the new polygon is
equal to the sum of the measures of the angles of the original polygon.
15.
16.
17.
18.
19.
Questions 20 & 21: Given regular pentagon ABCDE with diagonals AC and BE intersecting at
F.
20. Prove AC  DE .
21. Prove AFE is isosceles.
22. Prove that the opposite sides of a regular quadrilateral are parallel.
23. Prove that the opposite sides of a regular hexagon are parallel.
24. Complete the table below.
# of sides in a
3
4
5
6
7
8
9
10
polygon
# of angles
in the polygon
a. Graph the points putting the number of sides on the x-axis and the number of angles
on the y-axis.
b. Write an equation for the line that goes through the points you graphed.
c. Graph the line.
25. Complete the table below.
# of sides in a
3
4
5
6
7
8
9
10
polygon
sum of the
interior angles
in the polygon
a. Graph the points putting the number of sides on the x-axis and the number of angles
on the y-axis.
b. Write an equation for the line that goes through the points you graphed.
c. Graph the line.
d. What does the slope of the line represent?
e. Does the y-intercept have meaning? Why or why not?
PS #5:
1. a. 360° b. 720° c. 1440°
2. a. 360° b. 360° c. 360°
3. a. 90° b. 60° c. 45°
4. a. 90° b. 120° c. 135°
5. 118°
6. 39°
7. 3
8. 6
9. 15
10. a. 150° b. 12
11. a. 36° b. decagon
12. square
13. 60°
14. a. T b. F c. F d. T e. F f. T
15. g = 105° and h = 82°
16. j = 120° and k = 38°
19. a = 116°, b = 64°, c = 90°, d = 82°, e = 99°, f = 88°, g = 150°, h = 56°, j = 106°, k = 74°,
m = 136°, n = 118°, p = 99°
24. b. y = x
25. b. y  180 x  360 (or y  ( x  2)180 ) d. It is the number of degrees the sum increases by
each time the number of sides increases by one.