What are the 3 exterior angles?

So, we all know that a triangle is a 3-sided figure with three interior angles. But there exist other angles outside the triangle, which we call exterior angles.

We know that the sum of all three interior angles is always equal to 180 degrees in a triangle.

Similarly, this property holds for exterior angles as well. Also, each interior angle of a triangle is more than zero degrees but less than 180 degrees. The same goes for exterior angles.

In this article, we will learn about:

• Triangle exterior angle theorem,
• exterior angles of a triangle, and,
• how to find the unknown exterior angle of a triangle.

What is the Exterior Angle of a Triangle?

The exterior angle of a triangle is the angle formed between one side of a triangle and the extension of its adjacent side.

In the illustration above, the triangle ABC’s interior angles are a, b, c, and the exterior angles are d, e, and f. Adjacent interior and exterior angles are supplementary angles.

In other words, the sum of each interior angle and its adjacent exterior angle is equal to 180 degrees (straight line).

Triangle Exterior Angle Theorem

The exterior angle theorem states that the measure of each exterior angle of a triangle is equal to the sum of the opposite and non-adjacent interior angles.

Remember that the two non-adjacent interior angles opposite the exterior angle are sometimes referred to as remote interior angles.

For example, in triangle ABC above;

⇒ d = b + a

⇒ e = a + c

⇒ f = b + c

Properties of exterior angles

• An exterior angle of a triangle is equal to the sum of the two opposite interior angles.
• The sum of exterior angle and interior angle is equal to 180 degrees.

⇒ c + d = 180°

⇒ a + f = 180°

⇒ b + e = 180°

• All exterior angles of a triangle add up to 360°.

Proof: 

⇒ d + e + f = b + a + a + c + b + c

⇒ d +e + f = 2a + 2b + 2c

= 2(a + b + c)

But, according to triangle angle sum theorem,

a + b + c = 180 degrees

Therefore, ⇒ d +e + f = 2(180°)

= 360°

How to Find the Exterior Angles of a Triangle?

Rules to find the exterior angles of a triangle are pretty similar to the rules to find the interior angles. It is because wherever there is an exterior angle, there is an interior angle with it, and both add up to 180 degrees.

Let’s take a look at a few example problems.

Example 1

Given that for a triangle, the two interior angles 25° and (x + 15) ° are non-adjacent to an exterior angle (3x – 10) °, find the value of x.

Solution

Apply the triangle exterior angle theorem:

⇒ (3x − 10) = (25) + (x + 15)

⇒ (3x − 10) = (25) + (x +15)

⇒ 3x −10 = x + 40

⇒ 3x – 10 = x + 40

⇒ 3x = x + 50

⇒ 3x = x + 50

⇒ 2x = 50

x =25

Hence, x = 25°

Substitute the value of x into the three equations.

⇒ (3x − 10) = 3(25°) – 10°

= (75 – 10) ° = 65°

⇒ (x+15) = (25 + 15) ° = 40°

Therefore, the angles are 25°, 40° and 65°.

Example 2

Calculate values of x and y in the following triangle.

Solution

It is clear from the figure that y is an interior angle and x is an exterior angle.

By Triangle exterior angle theorem.

⇒ x = 60° + 80°

x = 140°

The sum of exterior angle and interior angle is equal to 180 degrees (property of exterior angles). So, we have;

⇒ y + x = 180°

⇒ 140° + y = 180°

subtract 140° from both sides.

⇒ y = 180° – 140°

y = 40°

Therefore, the values of x and y are 140° and 40°, respectively.

Example 3

The exterior angle of a triangle is 120°. Find the value of x if the opposite non-adjacent interior angles are (4x + 40) ° and 60°.

Solution

Exterior angle = sum of two opposite non-adjacent interior angles.

⇒120° =4x + 40 + 60

Simplify.

⇒ 120° = 4x + 100°

Subtract 120° from both sides.

⇒ 120° – 100° = 4x + 100° – 100°

⇒ 20° = 4x

Divide both sides by to get,

x = 5°

Therefore, the value of x is 5 degrees.

Verify the answer by substitution.

120°= 4x + 40 + 60

120° = 4° (5) + 40° + 60°

120° = 120° (RHS = LHS)

Example 4

Determine the value of x and y in the figure below.

Solution

Sum of interior angles = 180 degrees

y + 41° + 92° = 180°

Simplify.

y + 133° = 180°

subtract 133° from both sides.

y = 180° – 133°

y = 47°

Apply the triangle exterior angle theorem.

x = 41° + 47°

x = 88°

Hence, the value of x and y is 88° and 47°, respectively.

Solution: Using the Exterior Angle Theorem 145 = 80 + x

x = 65

Now, if you forget the Exterior Angle Theorem, you can still get the answer by noticing that a straight angle has been formed at the vertex of the 145º angle. See Example 2.

The exterior angle theorem states that when a triangle's side is extended, the resultant exterior angle formed is equal to the sum of the measures of the two opposite interior angles of the triangle. The theorem can be used to find the measure of an unknown angle in a triangle. To apply the theorem, we first need to identify the exterior angle and then the associated two remote interior angles of the triangle.

What is Exterior Angle Theorem?

The exterior angle theorem states that the measure of an exterior angle is equal to the sum of the measures of the two opposite(remote) interior angles of the triangle. Let us recall a few common properties about the angles of a triangle: A triangle has 3 internal angles which always sum up to 180 degrees. It has 6 exterior angles and this theorem gets applied to each of the exterior angles. Note that an exterior angle is supplementary to its adjacent interior angle as they form a linear pair of angles. Exterior angles are defined as the angles formed between the side of the polygon and the extended adjacent side of the polygon.

We can verify the exterior angle theorem with the known properties of a triangle. Consider a Δ ABC.

The three angles a + b + c = 180 (angle sum property of a triangle) ----- Equation 1

c= 180 - (a+b) ----- Equation 2 (rewriting equation 1)

e = 180 - c----- Equation 3 (linear pair of angles)

Substituting the value of c in equation 3, we get

e = 180 - [180 - (a + b)]

e = 180 - 180 + (a + b)

e = a + b

Hence verified.

Proof of Exterior Angle Theorem

Consider a ΔABC. a, b and c are the angles formed. Extend the side BC to D. Now an exterior angle ∠ACD is formed. Draw a line CE parallel to AB. Now x and y are the angles formed, where, ∠ACD = ∠x + ∠y

Hence proved that the exterior angle of a triangle is equal to the sum of the two opposite interior angles.

Exterior Angle Inequality Theorem

The exterior angle inequality theorem states that the measure of any exterior angle of a triangle is greater than either of the opposite interior angles. This condition is satisfied by all the six external angles of a triangle.

Related Articles

Check out a few interesting articles related to Exterior Angle Theorem.

Important notes

• The exterior angle theorem states that the measure of an exterior angle is equal to the sum of the measures of the two remote interior angles of the triangle.
• The exterior angle inequality theorem states that the measure of any exterior angle of a triangle is greater than either of the opposite interior angles.
• The exterior angle and the adjacent interior angle are supplementary. All the exterior angles of a triangle sum up to 360º.

1. Example 1: Find the values of x and y by using the exterior angle theorem of a triangle.

   

   Solution:

   ∠x is the exterior angle.

   ∠x + 92 = 180º (linear pair of angles)

   ∠x = 180 - 92 = 88º

   Applying the exterior angle theorem, we get, ∠y + 41 = 88

   ∠y = 88 - 41 = 47º

   Therefore, the values of x and y are 88º and 47º respectively.

2. Example 2: Find ∠BAC and ∠ABC.

   

   Solution:

   160º is an exterior angle of the Δ ABC. So, by using the exterior angle theorem, we have, ∠BAC + ∠ABC = 160º

   x + 3x = 160º

   4x = 160º

   x = 40º

   Therefore, ∠BAC = x = 40º and ∠ABC = 3xº = 120º

3. Example 3: Find ∠ BAC, if ∠CAD = ∠ADC

   

   Solution:

   Solving the linear pair at vertex D, we get ∠ADC + ∠ADE = 180º

   ∠ADC = 180º - 150º = 30º

   Using the angle sum property, for Δ ACD,

   ∠ADC + ∠ACD + ∠CAD = 180º

   ∠ACD = 180 - ∠CAD -∠ADC

   180º - ∠ADC -∠ADC (given ∠CAD= ∠ADC)

   180º - 2∠ADC

   180º - 2 × 30º

   ∠ACD = 180º - 60º = 120º

   ∠ACD is the exterior angle of ∠ABC

   Using the exterior angle theorem, for Δ ABC, ∠ACD = ∠ABC + ∠BAC

   120º = 60º + ∠BAC

   Therefore, ∠BAC = 120º - 60º = 60º.

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FAQs on Exterior Angle Theorem

The exterior angle theorem states that the measure of an exterior angle is equal to the sum of the measures of the two remote interior angles of the triangle. The remote interior angles are also called opposite interior angles.

How do you use the Exterior Angle Theorem?

To use the exterior angle theorem in a triangle we first need to identify the exterior angle and then the associated two remote interior angles of the triangle. A common mistake of considering the adjacent interior angle should be avoided. After identifying the exterior angles and the related interior angles, we can apply the formula to find the missing angles or to establish a relationship between sides and angles in a triangle.

What are Exterior Angles?

An exterior angle of a triangle is formed when any side of a triangle is extended. There are 6 exterior angles of a triangle as each of the 3 sides can be extended on both sides and 6 such exterior angles are formed.

What is the Exterior Angle Inequality Theorem?

The measure of an exterior angle of a triangle is always greater than the measure of either of the opposite interior angles of the triangle.

What is the Exterior Angle Property?

An exterior angle of a triangle is equal to the sum of its two opposite non-adjacent interior angles. The sum of the exterior angle and the adjacent interior angle that is not opposite is equal to 180º.

What is the Exterior Angle Theorem Formula?

The sum of the exterior angle = the sum of two non-adjacent interior opposite angles. An exterior angle of a triangle is equal to the sum of its two opposite non-adjacent interior angles.

Where Should We Use Exterior Angle Theorem?

Exterior angle theorem could be used to determine the measures of the unknown interior and exterior angles of a triangle.

Do All Polygons Exterior Angles Add up to 360?

The exterior angles of a polygon are formed when a side of a polygon is extended. All the exterior angles in all the polygons sum up to 360º.