Sum of Exterior Angles of Triangle
A triangle is a three-sided polygon with three sides, three vertices, and three edges. The exterior angle of a triangle is defined as the angle formed between one of its sides and its adjacent extended side. The sum of exterior angles of a triangle is equal to 360 degrees. Let us understand more about the sum of exterior angles of triangle by looking into its proof.
What is the Sum of Exterior Angles of Triangle?
The sum of exterior angles of a triangle is equal to 360°. The exterior angle of a triangle is the angle that is formed between one of the sides of a triangle and its adjacent extended side. There are 3 exterior angles in a triangle. Let us read about the formula for the sum of the exterior angles of a triangle.
Sum of Exterior Angles of a Triangle Formula
The formula for the sum of exterior angles of a triangle can be understood by observing the figure shown below.
From the figure, we can observe that ∠1, ∠2, and ∠3 are the exterior angles of triangle ABC. We know that the sum of all the exterior angles of any polygon is equal to 360°.
Thus, the sum of exterior angles of a triangle formula is stated as follows,
Sum of all the exterior angles of a triangle = 360°
Therefore in triangle ABC the sum of exterior angles,
∠1 + ∠2 + ∠3 = 360°
Relationship between the Exterior and Interior Angle of a Triangle
In a triangle, an interior angle and its corresponding exterior angle are supplementary. Therefore, the relationship between the exterior and the interior angle is expressed as,
Exterior angle + Interior angle = 180°
(or) Exterior angle = 180° - Interior angle
Sum of Exterior Angles of a Triangle Proof
Let’s analyze the diagram shown below to understand the proof of the sum of exterior angles of a triangle.
From the above figure, we see that R and Y are the interior and exterior angles of the triangle respectively.
Therefore, Y and R form a linear pair.
Y + R = 180°
Y = 180° - R
Therefore,
(Y + R) + (Y + R) + (Y + R) = 180° + 180° + 180°
3Y + 3R = 540° --------- (1)
The sum of interior angles of the triangle is as follows,
R + R + R = 180° [Using angle sum property of a triangle]
3R = 180° --------- (2)
Substituting the value from equation(2) in equation(1) we get,
3Y + 180° = 540°
3Y = 540° - 180°
3Y = 360°
Therefore, the sum of exterior angles = 360°
Hence, we have proved that the sum of exterior angles of a triangle is equal to 360°.
Sum of Exterior Angles of Equilateral Triangle
The sum of exterior angles of an equilateral triangle is 360°. Let us understand the working behind it. Consider an equilateral triangle ABC as shown below.
We know that all the sides of an equilateral triangle are equal in length and each interior angle measures 60°. We also know that in a triangle, the interior angle and its corresponding exterior angle form a linear pair, i.e., Exterior angle = 180° - Interior angle.
Therefore, the exterior angle of an equilateral triangle = 180° - 60° = 120°
Now, the sum of exterior angles of an equilateral triangle = 120° + 120° + 120° = 360°
Hence, the sum of exterior angles of an equilateral triangle is equal to 360 degrees.
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Sum of Exterior Angles of Triangle Examples
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Example 1: Find the measures of x, y, z, and k in the given figure.
Solution:
We know that in a triangle the interior angle and its corresponding exterior angle form a linear pair, i.e., Exterior angle = 180° - Interior angle
Thus, y + 56° = 180°
y = 180° - 56° = 124°
Similarly, x + 144° = 180°
x = 180° - 144° = 36°
Now,
x + y + z = 180° [Sum of interior angles of a triangle]
Using the values of x and y in the above equation we get,
36° + 124° + z = 180°
z = 180° - (36° + 124°)
z = 20°
Now the value of k can be found by using the sum of exterior angles of a triangle,
56° + 144° + k = 360°
k = 360° - (56° + 144°)
k = 160°
Thus, the values of x, y, z, and k are 36°, 124°, 20°, and 160° respectively.
FAQs on Sum of Exterior Angles of Triangle
What is the Sum of Exterior Angles of a Triangle?
The sum of exterior angles of a triangle is equal to 360°. We know that there are 3 exterior angles in a triangle. The exterior angle of a triangle is the angle that is formed between one of the sides of a triangle and its adjacent extended side.
What is the Formula for the Sum of the Exterior Angles of a Triangle?
We know that the sum of all the exterior angles of a polygon adds up to 360°. Thus, the formula for the sum of exterior angles of a triangle is, Sum of all the exterior angles of a triangle = 360°
What is the Relationship between the Exterior and Interior Angles of a Triangle?
In a triangle, the interior angle and its corresponding exterior angle form a linear pair. Thus, Exterior Angle + Interior Angle = 180°. According to the exterior angle theorem of a triangle, the external angle is equal to the sum of the interior opposite angles of a triangle.
How to find the Sum of the Exterior Angles of a Triangle?
For a triangle, if the interior angles are known, the corresponding exterior angles can be calculated by subtracting the interior angles from 180°. Thus, the obtained values can be added to find the exterior angles of a triangle.
What is the Sum of the Exterior Angles of an Equilateral Triangle?
In an equilateral triangle, each interior angle is equal to 60°. Thus, every exterior angle will be 180° - 60° = 120°. Hence, the sum of the exterior angles of an equilateral triangle is 120° + 120° + 120° = 360°.
How to Prove that the Sum of the Exterior Angles of a Triangle is 360?
In a triangle, the interior angle and its corresponding exterior angle form a linear pair. Since there are three such pairs, the sum of all the interior angles and the exterior angles is equal to (3 × 180°). Now, the sum of all the interior angles is 180°. Using the relationship between the interior and exterior angles of a triangle,
Sum of exterior angles of the triangle = (3 × 180°) - sum of interior angles of the triangle
= (3 × 180°) - 180°
= 540° - 180° = 360°. Therefore, it is proved that the sum of the exterior angles of a triangle is 360°.