Exterior Angle Theorem of a Triangle

The Exterior Angle Theorem of a triangle is a fundamental concept in geometry that describes the relationship between an exterior angle of a triangle and its two non-adjacent interior angles. An exterior angle is formed when one side of the triangle is extended beyond a vertex. The theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent (remote) interior angles.

Formulae

Let’s consider a triangle ABC. Extend side BC to a point D, forming exterior angle $\angle ACD$. Let the interior angles of the triangle be $\angle A$, $\angle B$, and $\angle C$. Then, the Exterior Angle Theorem states:

• $m\angle ACD = m\angle A + m\angle B$

Examples

Let’s look at some examples:

Example-1

In triangle XYZ, angle X is 60 degrees, and angle Y is 40 degrees. Extend side XY to form an exterior angle at vertex X. What is the measure of this exterior angle?

Solution: The exterior angle is equal to the sum of the two non-adjacent interior angles, which are angle Y and angle Z. First, find $\angle Z = 180 – \angle X – \angle Y = 180 – 60 – 40 = 80$ degrees. Then, the exterior angle is $60 + 40 = 100$ degrees (also equals $180 – 80 = 100$ degrees).

Example-2

In triangle PQR, the exterior angle at vertex R is 110 degrees. Angle P is 30 degrees. What is the measure of angle Q?

Solution: Let the exterior angle at R be $\angle PRS$. We know that $m\angle PRS = 110^\circ$ and $m\angle P = 30^\circ$. According to the Exterior Angle Theorem, $m\angle PRS = m\angle P + m\angle Q$. Therefore, $110^\circ = 30^\circ + m\angle Q$. Solving for $m\angle Q$, we get $m\angle Q = 110^\circ – 30^\circ = 80^\circ$.

Theorem with Proof

Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.

Proof:

1. Consider a triangle ABC. Let’s extend side BC to form exterior angle $\angle ACD$.
2. By the Angle Sum Property of a triangle: $m\angle A + m\angle B + m\angle ACB = 180^\circ$ (Equation 1)
3. Also, $\angle ACB$ and $\angle ACD$ form a linear pair, which means they are supplementary. Therefore, $m\angle ACB + m\angle ACD = 180^\circ$ (Equation 2).
4. From Equations 1 and 2: $m\angle A + m\angle B + m\angle ACB = m\angle ACB + m\angle ACD$.
5. Subtract $m\angle ACB$ from both sides: $m\angle A + m\angle B = m\angle ACD$.
6. Therefore: $m\angle ACD = m\angle A + m\angle B$, which proves the Exterior Angle Theorem.

Common mistakes by students

• Confusing the exterior angle with an interior angle: Students sometimes incorrectly use the measure of an interior angle in place of the exterior angle.
• Adding the adjacent interior angle: Students may accidentally add the adjacent interior angle to the other two interior angles, instead of using the two remote interior angles.
• Incorrectly identifying remote interior angles: Difficulty in correctly identifying the two non-adjacent interior angles. This often stems from a misunderstanding of the term “remote”.

Real Life Application

The Exterior Angle Theorem has practical applications in various fields:

• Architecture and Engineering: Used to calculate angles in building designs, truss construction, and bridge design.
• Navigation: Helps determine angles for plotting courses and calculating distances.
• Surveying: Used for land surveying to determine angles and boundaries.

Fun Fact

The Exterior Angle Theorem is a cornerstone in Euclidean geometry and provides a direct link to proving other geometric properties of triangles, such as the relationship between the angles and sides.

Recommended YouTube Videos for Deeper Understanding

Q.1 In $\triangle ABC$, if $\angle A = 50^\circ$ and the exterior angle at $B$ is $110^\circ$, then what is the measure of $\angle C$?

Check Solution

Ans: B

The exterior angle at $B$ is equal to $\angle A + \angle C$. Therefore, $110^\circ = 50^\circ + \angle C$. Thus, $\angle C = 110^\circ – 50^\circ = 60^\circ$.

Q.2 In a triangle, one interior angle is $40^\circ$, and an exterior angle is $100^\circ$. What is the measure of the other interior angle?

Check Solution

Ans: A

The exterior angle is equal to the sum of the two non-adjacent interior angles. Let the other interior angle be $x$. Therefore, $100^\circ = 40^\circ + x$. So, $x = 100^\circ – 40^\circ = 60^\circ$.

Q.3 If the exterior angles of a triangle are in the ratio $2:3:4$, what is the measure of the smallest interior angle?

Check Solution

Ans: D

The sum of the exterior angles is $360^\circ$. The exterior angles are $2x, 3x, 4x$. So, $2x + 3x + 4x = 360^\circ$, which simplifies to $9x = 360^\circ$. Therefore, $x = 40^\circ$. The exterior angles are $80^\circ, 120^\circ, 160^\circ$. The corresponding interior angles are $180^\circ – 80^\circ = 100^\circ$, $180^\circ – 120^\circ = 60^\circ$, and $180^\circ – 160^\circ = 20^\circ$. The smallest interior angle is $20^\circ$.

Q.4 In $\triangle PQR$, the exterior angle at $Q$ is $130^\circ$. If $\angle P = 50^\circ$, what is the measure of $\angle R$?

Check Solution

Ans: D

The exterior angle at $Q$ is equal to $\angle P + \angle R$. Thus, $130^\circ = 50^\circ + \angle R$. Therefore, $\angle R = 130^\circ – 50^\circ = 80^\circ$.

Q.5 The sides of a triangle are extended to form exterior angles. If one interior angle is $x$ degrees and one exterior angle is $3x$ degrees, find the value of $x$.

Check Solution

Ans: B

The interior angle and exterior angle at a vertex are supplementary, which means they add up to $180^\circ$. So, $x + 3x = 180^\circ$. This simplifies to $4x = 180^\circ$, and therefore $x = 45^\circ$.

Next Topic: Congruence of Triangles: Concept