Transcript Slide 1

Section 8.2
 Find the measures of the interior angles of a polygon.
 Find the measures of the exterior angles of a polygon.
 Interior angle
 Exterior angle
 8.1 Polygon Interior Angles Sum
 8.2 Polygon Exterior Angles Sum
 To find the sum of the interior angle measures of a convex
polygon, draw all possible diagonals from one vertex of the
polygon. This creates a set of triangles. The sum of the
angle measures of all the triangles equals the sum of the
angle measures of the polygon.
Remember!
By the Triangle Sum Theorem, the sum
of the interior angle measures of a
triangle is 180°.
Polygon
Triangle
Quadrilateral
Pentagon
Hexagon
Nonagon
n - gon
# of
sides
3
# of
triangles
Sum of measures of
interior ’s
1
1●180=180
Polygon
Triangle
Quadrilateral
Pentagon
Hexagon
Nonagon
n - gon
# of
sides
3
4
# of
triangles
Sum of measures of
interior ’s
1
1●180=180
Polygon
Triangle
Quadrilateral
Pentagon
Hexagon
Nonagon
n - gon
# of
sides
3
4
# of
triangles
Sum of measures of
interior ’s
1
2
1●180=180
Polygon
Triangle
Quadrilateral
Pentagon
Hexagon
Nonagon
n - gon
# of
sides
3
4
# of
triangles
Sum of measures of
interior ’s
1
2
1●180=180
2•180°=360°
Polygon
Triangle
Quadrilateral
Pentagon
Hexagon
Nonagon
n - gon
# of
sides
3
4
5
# of
triangles
Sum of measures of
interior ’s
1
2
1●180=180
2•180°=360°
Polygon
Triangle
Quadrilateral
Pentagon
Hexagon
Nonagon
n - gon
# of
sides
3
4
5
# of
triangles
Sum of measures of
interior ’s
1
2
3
1●180=180
2•180°=360°
Polygon
Triangle
Quadrilateral
Pentagon
Hexagon
Nonagon
n - gon
# of
sides
3
4
5
# of
triangles
Sum of measures of
interior ’s
1
2
3
1●180=180
2•180°=360°
3•180°=540°
Polygon
# of
sides
Triangle
3
Quadrilateral
4
Pentagon
5
Hexagon
6
Nonagon
n - gon
# of
triangles
Sum of measures of
interior ’s
1
2
3
1●180=180
2•180°=360°
3•180°=540°
Polygon
# of
sides
Triangle
3
Quadrilateral
4
Pentagon
5
Hexagon
6
Nonagon
n - gon
# of
triangles
Sum of measures of
interior ’s
1
2
3
4
1●180=180
2•180°=360°
3•180°=540°
Polygon
# of
sides
Triangle
3
Quadrilateral
4
Pentagon
5
Hexagon
6
Nonagon
n - gon
# of
triangles
Sum of measures of
interior ’s
1
2
3
4
1●180=180
2•180°=360°
3•180°=540°
4•180°=720°
Polygon
# of
sides
Triangle
3
Quadrilateral
4
Pentagon
5
Hexagon
6
Nonagon
9
n - gon
# of
triangles
Sum of measures of
interior ’s
1
2
3
4
1●180=180
2•180°=360°
3•180°=540°
4•180°=720°
Polygon
# of
sides
Triangle
3
Quadrilateral
4
Pentagon
5
Hexagon
6
Nonagon
9
n - gon
# of
triangles
Sum of measures of
interior ’s
1
2
3
4
7
1●180=180
2•180°=360°
3•180°=540°
4•180°=720°
Polygon
# of
sides
Triangle
3
Quadrilateral
4
Pentagon
5
Hexagon
6
Nonagon
9
n - gon
# of
triangles
Sum of measures of
interior ’s
1
2
3
4
7
1●180=180
2•180°=360°
3•180°=540°
4•180°=720°
7•180°=1260°
Polygon
# of
sides
Triangle
3
Quadrilateral
4
Pentagon
5
Hexagon
6
Nonagon
9
n - gon
n
# of
triangles
Sum of measures of
interior ’s
1
2
3
4
7
1●180=180
2•180°=360°
3•180°=540°
4•180°=720°
7•180°=1260°
Polygon
# of
sides
Triangle
3
Quadrilateral
4
Pentagon
5
Hexagon
6
Nonagon
9
n - gon
n
# of
triangles
Sum of measures of
interior ’s
1
2
3
4
7
n-2
1●180=180
2•180°=360°
3•180°=540°
4•180°=720°
7•180°=1260°
Polygon
# of
sides
Triangle
3
Quadrilateral
4
Pentagon
5
Hexagon
6
Nonagon
9
n - gon
n
# of
triangles
Sum of measures of
interior ’s
1
2
3
4
7
n-2
1●180=180
2•180°=360°
3•180°=540°
4•180°=720°
7•180°=1260°
(n-2)•180°
Polygon
# of
sides
Triangle
3
Quadrilateral
4
Pentagon
5
Hexagon
6
Nonagon
9
n - gon
n
# of
triangles
Sum of measures of
interior ’s
1
2
3
4
7
n-2
1●180=180
2•180°=360°
3•180°=540°
4•180°=720°
7•180°=1260°
(n-2)•180°
In each convex polygon, the number of triangles formed is two less
than the number of sides n. So the sum of the angle measures of all
these triangles is (n — 2)180°.
 From the previous slide, we have discovered that the
sum of the measures of the interior angles of a convex
n - gon is
(n – 2) ∙ 180
 This relationship can be used to find the measure of
each interior angle in a regular n - gon because the
angles are all congruent.
 Theorem 8.1 Polygon Interior Angles Sum
If a convex polygon has n sides and S is the sum of
its interior angles, then S = (n – 2)180.
B
A
E
D
Example:
m∠A+m∠B+m∠C+m∠D+m∠E = (5 – 2)180 = 540
C
Example 1
Use the Polygon Interior Angles Theorem
Find the sum of the measures of the interior
angles of a convex heptagon.
SOLUTION
A heptagon has 7 sides. Use the Polygon Interior Angles
Theorem and substitute 7 for n.
(n – 2) · 180° = (7 – 2) · 180°
ANSWER
Substitute 7 for n.
= 5 · 180°
Simplify.
= 900°
Multiply.
The sum of the measures of the interior
angles of a convex heptagon is 900°.
Example 2
Find the Measure of an Interior Angle
Find the measure of A in the diagram.
SOLUTION
The polygon has 6 sides, so the sum of
the measures of the interior angles is:
(n – 2) · 180° = (6 – 2) · 180° = 4 · 180° = 720°.
Add the measures of the interior angles and set the sum
equal to 720°.
136° + 136° + 88° + 142° + 105° + mA = 720° The sum is 720°.
607° + mA = 720° Simplify.
mA = 113° Subtract 607°.
ANSWER The measure of A is 113°.
Example 3
Interior Angles of a Regular Polygon
Find the measure of an interior angle
of a regular octagon.
SOLUTION
The sum of the measures of the interior
angles of any octagon is:
(n – 2) · 180° = (8 – 2) · 180° = 6 · 180° = 1080°.
Because the octagon is regular, each angle has the same
measure. So, divide 1080° by 8 to find the measure of one
interior angle.
1080°
= 135°
8
The measure of an interior angle of a
ANSWER
regular octagon is 135°.
Your Turn:
In Exercises 1–3, find the measure of G.
1.
ANSWER
72°
2.
ANSWER
66°
ANSWER
130°
ANSWER
150°
3.
4. Find the measure of an
interior angle of a regular
polygon with twelve sides.
Example 4A: Finding Interior Angle
Measures and Sums in Polygons
Find the measure of each interior angle of a regular
16-gon.
Step 1 Find the sum of the interior angle measures.
(n – 2)180°
Polygon  Sum Thm.
(16 – 2)180° = 2520°
Substitute 16 for n and
simplify.
Step 2 Find the measure of one interior angle.
The int. s are , so divide by 16.
Example 4B: Finding Interior Angle
Measures and Sums in Polygons
Find the measure of each
interior angle of pentagon
ABCDE.
(5 – 2)180° = 540°
Polygon  Sum Thm.
mA + mB + mC + mD + mE = 540° Polygon  Sum Thm.
35c + 18c + 32c + 32c + 18c = 540
135c = 540
c=4
Substitute.
Combine like terms.
Divide both sides by 135.
Example 4B Continued
mA = 35(4°) = 140°
mB = mE = 18(4°) = 72°
mC = mD = 32(4°) = 128°
Your Turn
Find the sum of the interior angle measures of a convex
15-gon.
(n – 2)180°
Polygon  Sum Thm.
(15 – 2)180°
A 15-gon has 15 sides, so
substitute 15 for n.
2340°
Simplify.
Your Turn
Find the measure of each interior angle of a regular
decagon.
Step 1 Find the sum of the interior angle measures.
(n – 2)180°
Polygon  Sum Thm.
(10 – 2)180° = 1440°
Substitute 10 for n and
simplify.
Step 2 Find the measure of one interior angle.
The int. s are , so divide by 10.
Find the measure of each interior angle of
parallelogram RSTU.
Step 1
Find x.
Since
the sum of the measures of the interior
angles is
Write an equation to express
the sum of the measures of the interior angles
of the polygon.
Sum of measures
of interior angles
Substitution
Combine like terms.
Subtract 8 from
each side.
Divide each side
by 32.
Step 2
Use the value of x to find the measure of
each angle.
m  R = 5x
= 5(11) or 55
m  S = 11x + 4
= 11(11) + 4 or 125
m  T = 5x
= 5(11) or 55
m  U = 11x + 4
= 11(11) + 4 or 125
Answer: mR = 55, mS = 125, mT = 55,
mU = 125
Find the value of x.
A. x = 7.8
B. x = 22.2
C. x = 15
D. x = 10
The measure of an interior angle of a regular
polygon is 150. Find the number of sides in the
polygon.
Use the Interior Angle Sum Theorem to write an
equation to solve for n, the number of sides.
S = 180(n – 2)
(150)n = 180(n – 2)
150n = 180n – 360
0 = 30n – 360
Interior Angle Sum
Theorem
S = 150n
Distributive Property
Subtract 150n from each
side.
360 = 30n
Add 360 to each side.
12 = n
Divide each side by 30.
Answer: The polygon has 12 sides.
The measure of an interior angle of a regular
polygon is 144. Find the number of sides in the
polygon.
A. 12
B. 9
C. 11
D. 10
 Exterior angle is an angle formed by one side of a
polygon and the extension of another side.
3
2
1
∠1, ∠2 and ∠3 are
exterior angles.
Interestingly, the measures of the exterior angles of a polygon
is an even easier formula. Let’s look at the following example
to understand it.
In the polygons below, an exterior angle has been measured
at each vertex.
Notice that in each case, the sum of the exterior angle
measures is 360°.
Theorem 8.2 Polygon Exterior
Angles Sum
If a polygon is convex, then the
sum of the measures of the
exterior angles, one at each
vertex, is 360°.
Example:
m∠A+m∠B+m∠C+m∠D+m∠E+
m∠F+m∠G+m∠H+m∠J = 360˚
Example 7
Find the Measure of an Exterior Angle
Find the value of x.
SOLUTION
Using the Polygon Exterior Angles
Theorem, set the sum of the measures
of the exterior angles equal to 360°.
95° + 85° + 2x° + x° = 360°
180 + 3x = 360
3x = 180
x = 60
ANSWER
Polygon Exterior Angles Theorem
Combine like terms.
Subtract 180 from each side.
Divide each side by 3.
The value of x is 60.
Your turn:
Find the value of x.
1.
2.
3.
ANSWER
91
ANSWER
21
ANSWER
125
ANSWER
30
4.
Example 8A: Finding Exterior Angle
Measures and Sums in Polygons
Find the measure of each exterior angle of a regular
20-gon.
A 20-gon has 20 sides and 20 vertices.
sum of ext. s = 360°.
measure of one ext.  =
Polygon  Sum Thm.
A regular 20-gon has 20
 ext. s, so divide the
sum by 20.
The measure of each exterior angle of a regular 20-gon is 18°.
Example 8B: Finding Interior Angle
Measures and Sums in Polygons
Find the value of b in polygon
FGHJKL.
Polygon Ext.  Sum Thm.
15b° + 18b° + 33b° + 16b° + 10b° + 28b° = 360°
120b = 360
b=3
Combine like terms.
Divide both sides by 120.
Your Turn
Find the measure of each exterior angle of a regular
dodecagon.
A dodecagon has 12 sides and 12 vertices.
sum of ext. s = 360°.
measure of one ext.
Polygon  Sum Thm.
A regular dodecagon has
12  ext. s, so divide the
sum by 12.
The measure of each exterior angle of a regular dodecagon is 30°.
Your Turn
Find the value of r in polygon JKLM.
4r° + 7r° + 5r° + 8r° = 360°
24r = 360
r = 15
Polygon Ext.  Sum Thm.
Combine like terms.
Divide both sides by 24.
Find the value of x in the diagram.
Use the Polygon Exterior Angles Sum Theorem to write
an equation. Then solve for x.
5x + (4x – 6) + (5x – 5) + (4x + 3) + (6x – 12) + (2x + 3) +
(5x + 5) = 360
(5x + 4x + 5x + 4x + 6x + 2x + 5x) + [(–6) + (–5) + 3 +
(–12) + 3 + 5] = 360
31x – 12 = 360
31x = 372
x = 12
Answer:
x = 12
Find the measure of each exterior angle of a regular
decagon.
A regular decagon has 10 congruent sides and
10 congruent angles. The exterior angles are also
congruent, since angles supplementary to congruent
angles are congruent. Let n = the measure of each
exterior angle and write and solve an equation.
10n = 360
Polygon Exterior Angle
Sum Theorem
n = 36
Divide each side by 10.
Answer:
The measure of each exterior angle of a
regular decagon is 36.
A. Find the value of x in
the diagram.
A. 10
B. 12
C. 14
D. 15
B. Find the measure of each exterior angle of a
regular pentagon.
A. 72
B. 60
C. 45
D. 90
 Pg. 420 – 423 #1 – 25 odd, 26 – 28 all, 29 – 43 odd