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Transcript
Exterior Angles
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Printed: January 22, 2013
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C ONCEPT
Concept 1. Exterior Angles
1
Exterior Angles
Learning Objectives
• Identify the exterior angles of convex polygons.
• Find the sums of exterior angles in convex polygons.
Introduction
This lesson focuses on the exterior angles in a polygon. There is a surprising feature of the sum of the exterior angles
in a polygon that will help you solve problems about regular polygons.
Exterior Angles in Convex Polygons
Recall that interior means inside and that exterior means outside. So, an exterior angle is an angle on the outside
of a polygon. An exterior angle is formed by extending a side of the polygon.
As you can tell, there are two possible exterior angles for any given vertex on a polygon. In the figure above we only
showed one set of exterior angles; the other set would be formed by extending each side in the opposite (clockwise)
direction. However, it doesn’t matter which exterior angles you use because on each vertex their measurement will
be the same. Let’s look closely at one vertex, and draw both of the exterior angles that are possible.
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As you can see, the two exterior angles at the same vertex are vertical angles. Since vertical angles are congruent,
the two exterior angles possible around a single vertex are congruent.
Additionally, because the exterior angle will be a linear pair with its adjacent interior angle, it will always be
supplementary to that interior angle. As a reminder, supplementary angles have a sum of 180◦ .
Example 1
What is the measure of the exterior angle 6 OKL in the diagram below?
The interior angle is labeled as 45◦ . Since you need to find the exterior angle, notice that the interior angle and the
exterior angle form a linear pair. Therefore the two angles are supplementary—they sum to 180◦ . So, to find the
measure of the exterior angle, subtract 45◦ from 180◦ .
180 − 45 = 135
The measure of 6 OKL is 135◦ .
Summing Exterior Angles in Convex Polygons
By now you might expect that if you add up various angles in polygons, there will be some sort of pattern or rule.
For example, you know that the sum of the interior angles of a triangle will always be 180◦ . From that fact, you
have learned that you can find the sums of the interior angles of any polygons with n sides using the expression
180(n − 2). There is also a rule for exterior angles in a polygon. Let’s begin by looking at a triangle.
To find the exterior angles at each vertex, extend the segments and find angles supplementary to the interior angles.
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Concept 1. Exterior Angles
The sum of these three exterior angles is:
150◦ + 120◦ + 90◦ = 360◦
So, the exterior angles in this triangle will sum to 360◦ .
To compare, examine the exterior angles of a rectangle.
In a rectangle, each interior angle measures 90◦ . Since exterior angles are supplementary to interior angles, all
exterior angles in a rectangle will also measure 90◦ .
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Find the sum of the four exterior angles in a rectangle.
90◦ + 90◦ + 90◦ + 90◦ = 360◦
So, the sum of the exterior angles in a rectangle is also 360◦ .
In fact, the sum of the exterior angles in any convex polygon will always be 360◦ . It doesn’t matter how many sides
the polygon has, the sum will always be 360◦ .
We can prove this using algebra as well as the facts that at any vertex the sum of the interior and one of the exterior
angles is always 180◦ , and the sum of all interior angles in a polygon is 180(n − 2).
Exterior Angle Sum: The sum of the exterior angles of any convex polygon is 360◦
Proof. At any vertex of a polygon the exterior angle and the interior angle sum to 180◦ . So summing all of the
exterior angles and interior angles gives a total of 180 degrees times the number of vertices:
(Sum of Exterior Angles) + (Sum of Interior Angles) = 180◦ n.
On the other hand, we already saw that the sum of the interior angles was:
(Sum of Interior Angles) = 180(n − 2) = 180◦ n − 360◦ .
Putting these together we have
180n = (Sum of Exterior Angles) + (Sum of Interior Angles)
= (180n − 360) + (Sum of Exterior Angles)
360 = (Sum of Exterior Angles)
Example 2
What is m6 ZXQ in the diagram below?
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Concept 1. Exterior Angles
6
ZXQ in the diagram is marked as an exterior angle. So, we need to find the measure of one exterior angle on a
polygon given the measures of all of the others. We know that the sum of the exterior angles on a polygon must be
equal to 360◦ , regardless of how many sides the shape has. So, we can set up an equation where we set all of the
exterior angles shown (including m6 ZXQ) summed and equal to 180◦ . Using subtraction, we can find the value of
X.
70◦ + 60◦ + 65◦ + 40◦ + m6 ZXQ = 360◦
235◦ + m6 ZXQ = 360◦
m6 ZXQ = 360◦ − 235◦
m6 ZXQ = 125◦
The measure of the missing exterior angle is 125◦ .
We can verify that our answer is reasonable by inspecting the diagram and checking whether the angle in question
is acute, right, or obtuse. Since the angle should be obtuse, 125◦ is a reasonable answer (assuming the diagram is
accurate).
Lesson Summary
In this lesson, we explored exterior angles in polygons. Specifically, we have learned:
• How to identify the exterior angles of convex polygons.
• How to find the sums of exterior angles in convex polygons.
We have also shown one example of how knowing the sum of the exterior angles can help you find the measure of
particular exterior angles.
Review Questions
For exercises 1-3, find the measure of each of the labeled angles in the diagram.
1. x =
,y =
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2. w =
,x =
3. a =
,b =
,y =
,z =
4. Draw an equilateral triangle with one set of exterior angles highlighted. What is the measure of each exterior
angle? What is the sum of the measures of the three exterior angles in an equilateral triangle?
5. Recall that a regular polygon is a polygon with congruent sides and congruent angles. What is the measure of
each interior angle in a regular octagon?
6. How can you use your answer to 5 to find the measure of each exterior angle in a regular octagon? Draw a
sketch to justify your answer.
7. Use your answer to 6 to find the sum of the measures of the exterior angles of an octagon.
8. Complete the following table assuming each polygon is a regular polygon. Note: This is similar to a previous
exercise with more columns—you can use your answer to that question to help you with this one.
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Concept 1. Exterior Angles
TABLE 1.1:
Regular
Polygon name
Number of sides
Sum
of
measures
of
interior angles
Measure of each
interior angle
Measure of each
exterior angle
Sum
of
measures
of
exterior angles
triangle
4
5
6
7
octagon
decagon
1, 800◦
n
9. Each exterior angle forms a linear pair with its adjacent internal angle. In a regular polygon, you can use two
different formulas to find the measure of each exterior angle. One way is to compute 180◦ −(measure of each
interior angle). . . in symbols 180 − 180(n−2)
.
n
Alternatively, you can use the fact that all n exterior angles in an n−gon sum to 360◦ and find the measure of each
exterior angle with by dividing the sum by n. Again, in symbols this is 360
n
Use algebra to show these two expressions are equivalent.
Review Answers
1.
2.
3.
4.
x = 52◦ , y = 128◦
w = 70◦ , x = 70◦ , y = 110◦ , z = 90◦
a = 107.5◦ , b = 72.5◦
Below is a sample sketch.
Each exterior angle measures 120◦ , the sum of the three exterior angles is 360◦
◦
5. Sum of the angles is 180(8 − 2) = 1080◦ . So, each angle measures 1080
8 = 135
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6. Since each exterior angle forms a linear pair with its adjacent interior angle, we can find the measure of each
exterior angle with 180 − 135 = 45◦
7. 45(8) = 360◦
TABLE 1.2:
Regular
Polygon name
Number of sides
triangle
square
pentagon
hexagon
heptagon
octagon
decagon
dodecagon
n−gon
3
4
5
6
7
8
10
12
n
Sum
of
measures
of
interior angles
180◦
360◦
540◦
720◦
900◦
1, 080◦
1, 440◦
1, 800◦
180(n − 2)◦
Measure of each
interior angle
Measure of each
exterior angle
60◦
90◦
72◦
60◦
128.57◦
135◦
144◦
150◦
120◦
90◦
108◦
120◦
51.43◦
45◦
36◦
30◦
180(n−2) ◦
n
360 ◦
n
9. One possible answer.
180 −
8
180(n − 2) 180n 180(n − 2)
=
−
n
n
n
180n − 180(n − 2)
=
n
180n − 180n + 360
=
n
360
=
n
Sum
of
measures
of
exterior angles
360◦
360◦
360◦
360◦
360◦
360◦
360◦
360◦
360◦