Exterior Angle Of A Triangle

What Is an Exterior Angle of a Triangle?

An exterior angle of a triangle is an angle formed by extending one of the triangle's sides. It is the angle between that extended side and the adjacent side of the triangle. Think of it as an angle on the 'outside' of the shape.

Imagine a triangle with vertices labeled A, B, and C. If you take the side connecting B and C and extend it past C to a point D, you create a new angle, ∠ACD. This angle, ∠ACD, is an exterior angle. The interior angle right next to it, ∠ACB, is called the adjacent interior angle. Together, they form a straight line, which means they are supplementary and their measures add up to 180°. So, m∠ACD + m∠ACB = 180°.

The other two angles inside the triangle, ∠A and ∠B, have a special name in relation to ∠ACD. They are called the remote interior angles because they are 'remote,' or far away from, the exterior angle. These remote interior angles are the key to unlocking the most important property of exterior angles.

It's important to know that every triangle has six exterior angles. There is one at each vertex, formed by extending a side. Since vertical angles are equal, there are technically two equal exterior angles at each vertex, forming three pairs of equal angles.

What Is the Exterior Angle Theorem?

The Exterior Angle Theorem is a fundamental rule in geometry that provides a shortcut for finding angle measures. The theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.

This is an incredibly useful property. Instead of having to calculate the third interior angle of a triangle to then figure out its supplementary exterior angle, you can simply add the two remote interior angles together. It saves a step and reduces the chance for calculation errors.

Let's use our triangle ABC with the exterior angle ∠ACD. The remote interior angles are ∠A and ∠B. According to the theorem:

This simple formula is the heart of the theorem. It creates a direct relationship between the angle on the outside and the two angles furthest from it on the inside.

How Can We Prove the Exterior Angle Theorem?

You don't have to just take our word for it! The Exterior Angle Theorem can be proven easily using two other facts you already know about angles: the sum of interior angles in a triangle is 180°, and angles on a straight line also sum to 180°.

Let's walk through the logical steps of the proof. Consider a triangle ΔABC. We extend side BC to a point D, which creates the exterior angle ∠ACD.

1. Sum of Interior Angles: We know that the three interior angles of any triangle add up to 180°. For ΔABC, this means:
   $m\angle A+m\angle B+m\angle ACB=180°$
2. Angles on a Straight Line (Linear Pair): The exterior angle ∠ACD and its adjacent interior angle ∠ACB lie on the straight line segment BD. Therefore, they are supplementary and add up to 180°.
   $m\angle ACB+m\angle ACD=180°$
3. Set the Equations Equal: Since both expressions are equal to 180°, we can set them equal to each other:
   $m\angle A+m\angle B+m\angle ACB=m\angle ACB+m\angle ACD$
4. Isolate the Key Angles: Notice that m∠ACB is on both sides of the equation. We can subtract m∠ACB from both sides to simplify.
   $m\angle A+m\angle B=m\angle ACD$

And there you have it! We have just proven that the measure of the exterior angle ( ∠ACD) is equal to the sum of the measures of the two remote interior angles ( ∠A and ∠B). This logical proof shows why the theorem works for every single triangle.

How Do You Find a Missing Exterior Angle?

Using the Exterior Angle Theorem to find a missing exterior angle is straightforward. All you need are the measures of the two remote interior angles.

Here are the steps:

1. Identify the exterior angle you need to find.
2. Locate the two interior angles that are remote (not adjacent) to it.
3. Add the measures of these two remote interior angles together.
4. The sum is the measure of your exterior angle.

Let's see this in action with an example.

How Do You Find a Missing Interior Angle?

You can also use the Exterior Angle Theorem to work backward and find a missing remote interior angle if you know the exterior angle and the other remote interior angle.

The steps are just as simple:

1. Identify the known exterior angle and the known remote interior angle.
2. Set up the formula: Exterior Angle = (Known Remote Angle) + (Missing Remote Angle).
3. Subtract the measure of the known remote interior angle from the measure of the exterior angle.
4. The result is the measure of the missing remote interior angle.

Let's work through an example.

How Can We Use Algebra with the Exterior Angle Theorem?

In more advanced problems, you won't be given simple numbers for angle measures. Instead, you'll see expressions with variables like x. The process is the same: you just need to use your algebra skills to solve for the variable first.

The key is to set up the equation correctly based on the Exterior Angle Theorem: (Remote Angle 1) + (Remote Angle 2) = Exterior Angle. Once you solve for the variable, be sure to substitute it back into the expressions to find the actual angle measures if the question asks for them.

What Are Common Mistakes to Avoid?

The Exterior Angle Theorem is straightforward, but there are a few common pitfalls students fall into. Being aware of them can help you avoid making these errors yourself.

• Adding the Wrong Angles: The most frequent mistake is adding the adjacent interior angle to one of the remote angles. Remember, the theorem only involves the two remote (opposite) interior angles. The adjacent one is not part of the sum.
• Confusing Interior and Exterior: Some students mistakenly set the sum of the two remote angles equal to the third interior angle. Always double-check that you are setting the sum equal to the exterior angle.
• Algebra Errors: When variables are involved, simple algebra mistakes can lead to the wrong answer. Be careful when combining like terms and isolating the variable. Always check your solution.
• Stopping After Finding x: In algebra problems, a common error is solving for x and forgetting to complete the problem. If the question asks for the measure of an angle, you must substitute the value of x back into the expression to get the final answer in degrees.

Quick Summary Reference

Here is a quick table to summarize the most important concepts related to the exterior angle of a triangle.