What is the measure of an exterior angle of a triangle is greater than the measure of either remote interior angle?

From ProofWiki

The external angle of a triangle is greater than either of the opposite internal angles.

In the words of Euclid:

(The Elements: Book $\text{I}$: Proposition $16$)

Proof

Let $\triangle ABC$ be a triangle.

Let the side $BC$ be extended to $D$.

Let $AC$ be bisected at $E$.

Let $BE$ be joined and extended to $F$.

Let $EF$ be made equal to $BE$.

(Technically we really need to extend $BE$ to a point beyond $F$ and then crimp off a length $EF$.)

Let $CF$ be joined.

Let $AC$ be extended to $G$.

We have $\angle AEB = \angle CEF$ from Two Straight Lines make Equal Opposite Angles.

Since $AE = EC$ and $BE = EF$, from Triangle Side-Angle-Side Equality we have $\triangle ABE = \triangle CFE$.

Thus $AB = CF$ and $\angle BAE = \angle ECF$.

But $\angle ECD$ is greater than $\angle ECF$.

Therefore $\angle ACD$ is greater than $\angle BAE$.

Similarly, if $BC$ were bisected, $\angle BCG$, which is equal to $\angle ACD$ by Two Straight Lines make Equal Opposite Angles, would be shown to be greater than $\angle ABC$ as well.

Hence the result.

$\blacksquare$

Historical Note

This proof is Proposition $16$ of Book $\text{I}$ of Euclid's The Elements.

Sources

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.

Show Answer >

go to slidego to slidego to slide

Have questions on basic mathematical concepts?

Become a problem-solving champ using logic, not rules. Learn the why behind math with ourCuemath’s certified experts.

Book a Free Trial Class

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.