How to find the exterior angle becomes straightforward when you understand its relationship with the adjacent interior angle. In a triangle, an exterior angle is either the supplement of its adjacent interior angle or the sum of the two opposite interior angles. For a regular polygon, each exterior angle equals 360° divided by the number of sides.
Introduction
An exterior angle is formed when one side of a polygon is extended and the angle is measured between that extension and an adjacent side. At every vertex, the exterior angle and its adjacent interior angle form a straight line, so their measures add up to 180°.
Worth pausing on this one Worth keeping that in mind..
This relationship makes exterior angles useful for finding missing measurements in triangles and polygons. They also reveal an important pattern: the exterior angles of a convex polygon—one taken at each vertex in the same direction—always have a total measure of 360°.
Worth pausing on this one.
Exterior Angle Formulas
The correct formula depends on the information provided.
For a triangle
If the adjacent interior angle is known:
Exterior angle = 180° − adjacent interior angle
If the two opposite interior angles are known:
Exterior angle = opposite interior angle 1 + opposite interior angle 2
The second formula is known as the exterior angle theorem. The “opposite” angles are the two interior angles that do not touch the exterior angle Surprisingly effective..
For a polygon
If an adjacent interior angle is known:
Exterior angle = 180° − interior angle
For a regular polygon with n sides:
Each exterior angle = 360° ÷ n
For any convex polygon:
Sum of one exterior angle at each vertex = 360°
These formulas assume the usual school geometry convention that the polygon is convex and one exterior angle is selected at each vertex.
How to Find an Exterior Angle Step by Step
- Identify the polygon. Determine whether the shape is a triangle, quadrilateral, pentagon, or another polygon.
- Locate the exterior angle. Confirm that it is formed by one side and the extension of an adjacent side.
- List the known measurements. Mark every given interior or exterior angle.
- Choose the appropriate relationship. Use a supplementary pair, the triangle exterior angle theorem, the 360° polygon rule, or the regular-polygon formula.
- Substitute the known values. Place the measurements into the selected formula.
- Solve and include degrees. Write the final answer with the degree symbol, such as 65°.
- Check the result. An exterior angle and its adjacent interior angle should total 180°.
Method 1: Use the Adjacent Interior Angle
This is the simplest method when an exterior angle and an interior angle meet at the same vertex. Because they form a straight line:
Exterior angle + interior angle = 180°
Example
A triangle has an interior angle of 68° at one vertex. Find the adjacent exterior angle Simple, but easy to overlook..
[ 180^\circ - 68^\circ = 112^\circ ]
The exterior angle is 112°.
To verify the answer, add the two angles:
