Introduction
The exterior angle theorem is a fundamental concept in geometry that describes the relationship between the exterior angles of a polygon and its interior angles. This theorem is essential in solving many geometry problems and has a wide range of applications in mathematics and engineering. In this article, we will explore the exterior angle theorem in detail, including its definition, examples, and applications.
Definition of Exterior Angle Theorem
The exterior angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. In other words, if a polygon has n sides, then each exterior angle of the polygon is equal to the sum of the measures of the two adjacent interior angles.
Mathematically, we can express the exterior angle theorem as follows:
m <1 = m <2 + m <3
Where m <1 is the measure of the exterior angle, and m <2 and m <3 are the measures of the two remote interior angles.
Examples of Exterior Angle Theorem
Let’s consider some examples of the exterior angle theorem to better understand how it works.
Example 1: In the triangle ABC shown below, the measure of angle A is 60 degrees. Find the measure of angle C. bash
A
/\
/ \
10 / \ 12
/ \
/___B___\ C
15
Solution: According to the exterior angle theorem, the measure of angle C is equal to the sum of the measures of angles A and B. We know that the measure of angle A is 60 degrees, and we can find the measure of angle B using the fact that the sum of the angles in a triangle is 180 degrees.
Therefore:
m <B = 180 – m <A – m <C = 180 – 60 – m <C = 120 – m <C
Now, using the exterior angle theorem, we have:
m <C = m <A + m <B = 60 + (120 – m <C) = 180 – m <C
Solving for m <C, we get:
2m <C = 180 m <C = 90
Therefore, the measure of angle C is 90 degrees.
Example 2: In the quadrilateral ABCD shown below, the measure of angles A, B, and C are 60, 70, and 80 degrees, respectively. Find the measure of angle D.
A ____ B
| |
| |
|____|
D C
Solution: According to the exterior angle theorem, the measure of angle D is equal to the sum of the measures of angles A, B, and C. Therefore:
m <D = m <A + m <B + m <C = 60 + 70 + 80 = 210
Therefore, the measure of angle D is 210 degrees.
Example 3: In the pentagon ABCDE shown below, the measure of angle A is 80 degrees, and the measure of angle B is 120 degrees. Find the measure of angle E.
A
/ \
/ \
/ \
/___B___\
/ \
/ \
/ \
D C
\ /
\ /
\___E___/
Solution: According to the exterior angle theorem, the measure of angle E is equal to the sum of the measures of angles B and C. We can find the measure of angle C using the fact that the sum of the angles in a pentagon is 540 degrees. Therefore:
m <C = 540 – m <A – m <B – m <D – m <E = 540 – 80 – 120 – (180 – m <E) – m <E = 340 – 2m <E
Now, using the exterior angle theorem, we have:
m <E = m <B + m <C = 120 + (340 – 2m <E) = 460 – 2m <E
Solving for m <E, we get:
3m <E = 460 m <E = 153.33 (rounded to two decimal places)
Therefore, the measure of angle E is 153.33 degrees.
Applications of Exterior Angle Theorem
The exterior angle theorem is a fundamental concept in geometry and has a wide range of applications in various fields, including mathematics, engineering, and architecture. Some of the applications of the exterior angle theorem are:
- Calculating the measures of interior angles of polygons: The exterior angle theorem can be used to calculate the measures of interior angles of polygons by subtracting the measure of the exterior angle from 180 degrees.
- Constructing geometric shapes: The exterior angle theorem can be used to construct various geometric shapes by using the measure of exterior angles.
- Solving real-world problems: The exterior angle theorem can be used to solve real-world problems that involve geometric shapes, such as finding the height of a building, the distance between two points, or the angle of inclination of a slope.
FAQ
Q1. What is an exterior angle in a polygon? A: An exterior angle in a polygon is an angle formed by one side of the polygon and the extension of the adjacent side.
Q2. What is the sum of the measures of the exterior angles of a polygon? A: The sum of the measures of the exterior angles of a polygon is always equal to 360 degrees.
Q3. Can the exterior angle of a polygon be greater than 180 degrees? A: No, the exterior angle of a polygon cannot be greater than 180 degrees.
Q4. What is the interior angle of a polygon? A: The interior angle of a polygon is an angle formed by two adjacent sides of the polygon.
Q5. Can the interior angles of a polygon be negative? A: No, the interior angles of a polygon cannot be negative.
Quiz
- What is the exterior angle theorem? a. The sum of the measures of the exterior angles of a polygon is always equal to 360 degrees. b. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. c. The sum of the measures of the interior angles of a polygon is always equal to 180 degrees.
- In the triangle ABC shown below, the measure of angle A is 50 degrees. Find the measure of angle B.
A
/\
/ \
8 / \ 10
/ \
/___B___\ C
a. 80 degrees b. 90 degrees c. 100 degrees
- In the quadrilateral ABCD shown below, the measure of angles A, B, and C are 70, 80, and 100 degrees, respectively. Find the measure of angle D.
A ____ B
| |
| |
|____|
D C
a. 80 degrees b. 90 degrees c. 110 degrees
- In the pentagon ABCDE shown below, the measure of angle A is 60 degrees, and the measure of angles B, C, D, and E are equal. Find the measure of each of these angles.
A
/\
/ \
6 / \ 6
/ \
/___B___\ C
| |
| |
|___D____|
E
a. 72 degrees b. 90 degrees c. 108 degrees
- In the triangle ABC shown below, the measure of angle A is 40 degrees. Find the measure of the exterior angle at vertex A.
A
/\
/ \
/____\
B C
a. 40 degrees b. 80 degrees c. 140 degrees
- In the quadrilateral ABCD shown below, the measure of angles A, B, and C are 90, 60, and 70 degrees, respectively. Find the measure of angle D.
A ____ B
| |
| |
|____|
D C
a. 80 degrees b. 90 degrees c. 110 degrees
- In the pentagon ABCDE shown below, the measure of angle A is 120 degrees, and the measure of angles B, C, D, and E are equal. Find the measure of each of these angles.
A
/\
/ \
/____\
B C
/ \
/___E___D_\
|
|
|
E
a. 48 degrees b. 60 degrees c. 72 degrees
- In the triangle ABC shown below, the measure of angle A is 80 degrees. Find the measure of angle B.
A
/\
/ \
7 / \ 9
/ \
/___B___\ C
a. 50 degrees b. 60 degrees c. 70 degrees
- In the quadrilateral ABCD shown below, the measure of angles A, B, and C are 100, 80, and 120 degrees, respectively. Find the measure of angle D.
A ____ B
| |
| |
|____|
D C
a. 80 degrees b. 90 degrees c. 100 degrees
- In the pentagon ABCDE shown below, the measure of angle A is 80 degrees, and the measure of angles B, C, D, and E are equal. Find the measure of each of these angles.
A
/\
/ \
/____\
B C
/ \
/___E___D_\
|
|
|
E
a. 70 degrees b. 80 degrees c. 90 degrees
Answers: 1-b, 2-c, 3-b, 4-a, 5-c, 6-a, 7-b, 8-b, 9-c, 10-a
Conclusion
The exterior angle theorem is a fundamental concept in geometry that is used to calculate the measure of an exterior angle of a polygon. The theorem can be used to solve various problems related to polygons, including calculating the measures of interior angles, constructing geometric shapes, and solving real-world problems. By understanding the exterior angle theorem and its applications, students can develop their problem-solving skills and improve their understanding of geometry.
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