Introduction to Exterior Angles
Exterior angles are angles that are parallel to the inner angles of a polygon but lie on the outside of it. The measure of an exterior angle is equal to the sum of the two internal opposite angles.
In the given image ‘a’ and ‘b’ are interior angles and ‘d’ is the exterior angle.
In this case, we can write ∠d = ∠a + ∠b
There are two external angles that are equivalent at each vertex.
However, it is typical for each vertex to display just one exterior angle.
Exterior Angles of a Triangle
The angles that are created outside of a triangle are its external angles. In other terms, the angle created between one of the sides of a triangle and its adjacent extended side is the exterior angle of a triangle. Hence, the angle created by any extended side of a triangle and its adjacent side is referred to as the external angle of a triangle.
A triangle has three external angles. It should be noticed that every exterior angle and the corresponding interior angle constitute a linear pair.
Properties of Exterior Angles of a Triangle
Exterior angles of a triangle have three fundamental characteristics:
- Each exterior angle and corresponding interior angle of a triangle make up a linear pair of angles. Accordingly, the interior and exterior angles add up to \(180^\circ\).
- The sum of the two opposing internal angles determines the measure of the exterior angle of a triangle. Another name for this characteristic is the exterior angle theorem.
- The total sum of exterior angles of a triangle is \(360^\circ\).
Exterior Angle of Triangle Formula
The following formulas can be used to determine the exterior angles of a triangle based on the characteristics of exterior angles. The following triangle is provided so that the formulas may be better understood.
- Each exterior angle = \(180^\circ\) – corresponding interior angle
- Exterior angle =sum of interior opposite angles. Here, ∠d = ∠a +∠b .
- The total sum of the exterior angles of a triangle is \(360^\circ\).
Solved Examples
Example 1: Find the exterior angle of a triangle in the following image.
Solution:
According to the exterior angle theorem,
Exterior angle = Sum of Interior Opposite Angles
In this case, the exterior angle,
∠QRX = ∠RPQ + ∠RQP
∠QRX = \(49^\circ~+~80^\circ\) [Substituting the value of interior angles in the formula]
∠QRX = \(129^\circ\) [Add]
Therefore, the exterior angle of the triangle, ∠QRX = \(129^\circ\).
Example 2: Find the exterior angle of a triangle in the following diagram.
Solution:
According to an exterior angle theorem,
Exterior angle = Sum of Interior opposite angles
In this case, the exterior angle,
∠VUX = ∠UTV + ∠UVT
∠VUX = \(53^\circ~+~103^\circ\) [Substituting the value of interior angles]
∠VUX = \(156^\circ\) [Add]
Therefore, the exterior angle of the triangle, ∠VUX = \(156^\circ\)
Example 3: Charlie was looking for an external angle of nacho chips in a packet of nachos. Find the external angle measurement of a nacho chip from the image.
Solution:
According to an exterior angle theorem,
Exterior angle = Sum of Interior Opposite Angles
In this case, the exterior angle,
∠JKX = ∠KLJ + ∠KJL
∠JKX = \(93^\circ~+~44^\circ\) [Substituting the value of interior angles]
∠JKX = \(137^\circ\) [Add]
Therefore, the exterior angle of the nacho chip, ∠JKX = \(137^\circ\).
The angle created by one side of a triangle and the subsequent extended side is known as the external angle of a triangle. A triangle has three exterior angles, and the sum of those three external angles is always 360 degrees.
Yes, the sum total of the measures of exterior angles of a triangle are always equal to 360 degrees.
The exterior angle theorem states that the sum of the inner opposite angles(remote interior angles) determines the measure of an external angle. This means that if the remote interior angles of a triangle are known and we need to determine the exterior angle of the triangle, the value of the exterior angle will be equal to the sum of the two remote interior opposite angles.
The sum total of the exterior angles of a triangle is always equal to 360 degrees. The outer angles of all polygons add up to 360 degrees since this property holds true for all polygons.