Download Packet 2 for Unit 2 M2 Geo

M2 GEOMETRY PACKET 2 FOR UNIT 2 – SECTIONS 1-6 & 6-1
SUM OF ANGLES IN A POLYGON
1. Watch the video at this location, https://www.youtube.com/watch?v=ZIKHV1bvRQ4, for
a physical demonstration of the sum of the three angles in a triangle.
2. The diagonals of a polygon are segments connecting any two non-consecutive vertices.
a. Name all the diagonals of polygon DEFG shown at right.
b. Name all the diagonals of polygon HIJKL shown at right.
c. We can define the words convex and concave using diagonals. In a
convex polygons, all of the diagonals are inside the figure. What
happens with one or more of the diagonals in a concave polygon?
3. Now we’ll find a pattern between the number of sides in a convex polygon and the sum
of its angles. Divide the rows up among your group instead of doing them all yourself.
a. In the chart on the next page, write the type of polygon that has the given number of
sides in the “Type of Polygon” column.
b. Find the number of triangles that are formed by drawing all diagonals from the
labelled point in each polygon below. Then write this number in the “Number of
Triangles” column on the next page. Only the first 5 rows will be used.
B
A
C
E
D
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M2 GEOMETRY PACKET 2 FOR UNIT 2 – SECTIONS 1-6 & 6-1
c. The sum of angles in each triangle is 180 ° . Use this fact to find the sum of angles in
each polygon in part (b). Write these answers in the first 5 rows of Sum of Interior Angles
column of the chart.
d. How is the Number of Triangles related to the Number of Sides in each row of the
chart? Use this pattern to fill in the number of triangles for 8, 9, and 10 sides.
e. Write a formula for the number of triangles in a polygon with n sides. Write this answer
in the last row of the Number of Triangles column.
f. How is the Sum of Interior Angles related to the Number of Triangles in each row of the
chart? Write this as a formula in the last row of the Sum of Interior Angles column, and
use your formula to complete the rows for 8, 9, and 10 sides.
Chart for #3:
Number of Sides
Type of Polygon
3
Triangle
Number of Triangles
180°
4
5
6
7
8
9
10
n
Sum of Interior Angles
n-gon
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M2 GEOMETRY PACKET 2 FOR UNIT 2 – SECTIONS 1-6 & 6-1
The formula you found on the previous page can be used to solve problems involving angles
in a polygon. Here are 2 examples:
Ex 1: A convex polygon has 13 sides. Find
the sum of the measures of the interior
angles.
Ex 2: The measure of an interior angle of a
regular polygon is 120. Find the number of
sides.
(n – 2) ⋅ 180
Let n be the number of sides. Then there are
n angles. In a regular polygon, each angle
has the same measure (120 in this case). The
sum of the measures of all n angles is 120n.
(13 – 2) ⋅ 180
Substitute 13 for n.
11 ⋅ 180
1980
120n = (n – 2) ⋅ 180 Both describe the sum of angles.
120n = 180n – 360
Distribute.
– 60n = –360
Subtract 180n from both sides.
n=6
Divide both sides by – 60.
4. Find the sum of the measures of the interior angles of a convex polygon with 30 sides.
Show your work.
5. Find the number of sides of a regular polygon in which each angle measures 160 ° . Show
your work.
6. Find the value of x.
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M2 GEOMETRY PACKET 2 FOR UNIT 2 – SECTIONS 1-6 & 6-1
EXTERIOR ANGLES OF A POLYGON
Exterior angles of a polygon are formed by extending each side of the polygon, as shown in
the diagrams on this page. Is there a pattern in the sum of these angles? Let’s find out.
1. Use the diagram of the triangle at right to complete the proof:
Given: Triangle with interior angles measuring a°, b°, and c° and
exterior angles, ∠1, ∠2, and ∠3 .
Prove: ∠1 + ∠2 + ∠3 = ______
STATEMENTS
REASONS
1. a + b + c = 180
1.
2. a + ∠1 = 180 , __ + __ = ____, __ + __ = ____ 2.
3. a + b + c + ∠1 + ∠2 + ∠3 = 540
3.
4. 180 + ∠1 + ∠2 + ∠3 = 540
4.
5.
5. Subtraction
2. Complete the proof for the sum of the exterior angles of the
quadrilateral at right:
Given: Quadrilateral with interior angles measuring a°, b°, c° , and
d° and exterior angles, ∠1, ∠2, ∠3 , and ∠4 .
Prove: ∠1 + ∠2 + ∠3 + ∠4 = ______
STATEMENTS
REASONS
1.
1. Sum of int ∠ s in a quadrilateral is 360
2.
2. Lin. pairs add up to 180
3.
3. Addition
4.
4. Substitution
5.
5. Subtraction
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M2 GEOMETRY PACKET 2 FOR UNIT 2 – SECTIONS 1-6 & 6-1
3. Follow the steps to find the sum of the exterior angles of any convex n-gon:
a. How many angles are in a polygon with n sides?
b. What is the sum of one pair of interior and exterior angles for any polygon?
c. What is the sum of all of the interior and exterior angles for a polygon with n sides?
d. What is the sum of all interior angles for a polygon with n sides?
e. Substitute your answers from (c) and (d) into the equation below. Use x for “sum of ext
angles.”
sum of all
sum of
sum of
int and ext =
+
int angles
ext angles
angles
f. Use algebra to solve for x, the sum of the exterior angles of any polygon.
g. Complete this statement:
The sum of the exterior angles of any convex polygon is ______.
4. Use your rule to find the measure of each exterior angle of a regular polygon with 20
sides.
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M2 GEOMETRY PACKET 2 FOR UNIT 2 – SECTIONS 1-6 & 6-1
PRACTICE FOR 1-6 & 6-1
Name each polygon by its number of sides and then classify it as convex or concave and regular or
irregular. (See p. 1-2 in this packet.)
1.
2.
Find the sum of the measures of the interior angles of each convex polygon. (See p. 4-5 of this packet.)
3. nonagon
4. heptagon
The measure of an interior angle of a regular polygon is given. Find the number of sides in the polygon.
(See p. 4-5 of this packet.)
5. 108 °
6. 150 °
Classify the polygon by number of sides, and find the measure of each interior angle. (See p. 4-5 of this
packet.)
7.
8.
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M2 GEOMETRY PACKET 2 FOR UNIT 2 – SECTIONS 1-6 & 6-1
Find the measures of each interior angle of each regular polygon. (See p. 4-5 in this packet.)
9. pentagon
10. Dodecagon (12 sides)
Find the measures of each exterior angle of each regular polygon. (See p. 7 in this packet.)
11. octagon
12. nonagon
13. Write an equation, and solve to find the measures of the interior angles
in the figure at right. Then find the measures of each exterior angle.
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M2 GEOMETRY PACKET 2 FOR UNIT 2 – SECTIONS 1-6 & 6-1
Review of 1-6 & 6-1
Name each polygon by its number of sides and then classify it as convex or concave and regular or
irregular.
1.
2.
Find the sum of the measures of the interior angles of each convex polygon.
3. 11-gon
4. 14-gon
The measure of an interior angle of a regular polygon is given. Find the number of sides in the polygon.
5. 144 °
6. 156 °
Find the measure of each interior angle.
7.
8.
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M2 GEOMETRY PACKET 2 FOR UNIT 2 – SECTIONS 1-6 & 6-1
Find the measure of the angle labeled “x” in each diagram.
9.
10.
Find the measures of one exterior angle and one interior angle given the number of sides of each regular
polygon. Round to the nearest tenth, if necessary.
11. 16 sides
12. 40 sides
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M2 GEOMETRY PACKET 2 FOR UNIT 2 – SECTIONS 1-6 & 6-1
Answers to review problems:
1. Hexagon, concave, irregular
2. Nonagon, convex, regular
3. 1620 °
4. 2160 °
5. 10 sides
6. 15 sides
7. ∠J = 115°, ∠K = 130°, ∠M = 50°, ∠N = 65°
8. ∠R = 128°, ∠T = 128°, ∠S = 52°, ∠U = 52°
9. 78 °
10. 150 °
11. Exterior 22.5 ° , interior 157.5 °
12. Exterior 9 ° , interior 171 °
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