The exterior angle theorem is one of the most fundamental theorems of triangles. Before we begin the discussion, let us have a look at what a triangle is. A polygon is defined as a plane figure bounded by a finite number of line segments to form a closed figure. Triangle is the polygon bounded by a least number of line segments, i.e. three. It has three edges and three vertices. Figure 1 below represents a triangle with three sides AB, BC, CA, and three vertices A, B and C. ∠ABC, ∠BCA and ∠CAB are the three interior angles of ∆ABC.
Fig. 1 Triangle ABC
One of the basic theorems explaining the properties of a triangle is the exterior angle theorem. Let us discuss this theorem in detail.
Statement: If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles.
Fig. 2 Exterior Angle Theorem
The above statement can be explained using the figure provided as:
According to the Exterior Angle property of a triangle theorem, the sum of measures of ∠ABC and ∠CAB would be equal to the exterior angle ∠ACD.
General proof of this theorem is explained below:
Proof:
Consider a ∆ABC as shown in fig. 2, such that the side BC of ∆ABC is extended. A line, parallel to the side AB is drawn as shown in the figure.
Fig. 3 Exterior Angle Theorem
S. No | Statement | Reason |
1. | ∠CAB = ∠ACE ⇒∠1=∠x | Pair of alternate angles[BA || CE] and [AC] is the transversal] |
2. | ∠ABC = ∠ECD ⇒∠2 = ∠y | Corresponding angles [BA] ||[CE] and [BD] is the transversal] |
3. | ⇒∠1+∠2 = ∠x+∠y | From statements 1 and 2 |
4. | ∠x+∠y = ∠ACD | From fig. 3 |
5. | ∠1+∠2 = ∠ACD | From statements 3 and 4 |
Thus, from the above statements, it can be seen that the exterior ∠ACD of ∆ABC is equal to the sum of two opposite interior angles i.e. ∠CAB and ∠ABC of the ∆ABC.
Hence proved.
To know more about triangles and the properties of triangles, download BYJU’S-The Learning App from Google Play Store.
Frequently Asked Questions – FAQs
The below formulas can be stated from the exterior angle theorem. According to the exterior angle inequality theorem, the measure of an exterior angle of a triangle is greater than either of its interior opposite angles. If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles. Yes, the sum of exterior angles in a polygon is always added up to 360 degrees. The sum of the measures of the exterior angles of any triangle is 360 degrees.What is the exterior angle theorem formula?
The measure of exterior angle = Sum of two opposite interior angles’ measureWhat is the exterior angle Inequality Theorem?
What is
the exterior angle property?
Do all polygon exterior angles add up to 360?
What is the sum of the measures of the exterior angles of any triangle?
Image credit: Desmos
Before we cover the exterior angle theorem, let's review a few definitions.
- Adjacent angles: angles that share a side and a vertex [ex., BCA and DCA]
- Supplementary angles: two angles that add to 180°
- Interior angles: the angles inside a triangle
- Exterior angles: angles formed between a side of a shape and a line that extends from the next side
We'll use the above triangle to demonstrate the exterior angle theorem's principles:
- An exterior angle should equal the sum of the remote interior angles of a triangle. In the triangle above, the exterior angle of the triangle, angle ACD, will equal the sum of the measures of interior angles BAC and ABC.
- An exterior angle and its adjacent interior angle are supplementary angles, so they add to 180°. Above, BCA plus ACD add to 180°.
Breaking Down the Exterior Angle Theorem
Let's look at how the exterior angle theorem works. First, let’s review the angle sum theorem, which states that the interior angles of a triangle equal 180°.
Image credit: Desmos
In the above triangle ECD, the exterior angle of DEF and its adjacent interior angle CED are linear pairs. That means together, they form a straight line and equal 180°.
Because these two adjacent angles add to 180° and the interior measures of the angles of a triangle also equal 180°, the sum of the remote interior angles ECD and CDE must equal the measure of exterior angle DEF.
Next, we'll use this knowledge to find angle measurements.
Applying the Exterior Angle Theorem
Let's use the exterior angle theorem in the triangle below:
Image credit: Desmos
Since we know that the angle EST = 125° and the adjacent interior angle TSU is its supplementary angle, let's solve for the measure of this interior angle:
Now let's use the second part of the exterior angle theorem: The exterior angle equals the sum of the remote interior angles. We'll follow this logic and find the remote interior angle TUS by subtracting STU from EST:
Understanding Exterior and Interior Angles
The exterior angle theorem states that:
- The measure of an exterior angle of a triangle is supplementary to its adjacent interior angle.
- The sum of the remote interior angles must equal the measure of the exterior angle of the triangle.
This theorem can help you solve for missing angles and understand the relationship between exterior and interior angles within a triangle.
More Math Homework Help:
- Vertical Angle Theorem: What It Is and How to Use It
- How to Find the Measure of an Angle in a Triangle: 3 Methods
- How to Recognize Adjacent Supplementary Angles