What are Interior Angles?
Interior angles are the angles that exist within a polygon. Other angles form when the sides of a polygon are extended.
What are Exterior Angles?
Exterior angles like the name suggests are the angles on the outside of the polygon that are adjacent to the interior angles.
Measurement of the Interior Angles of a Triangle
The sum of a triangle’s interior angle measurements is \(180^\circ \).
\(x^\circ \) + \(y^\circ \) + \(z^\circ \) = \(180^\circ \)
Example:
Determine the interior angle’s measurement.
Solution:
We know that the sum of the interior angles of a triangle measure degree
x + 20 + 80 = 180
x + 100 = 180
x = 80
As a result, the interior angles are 80, 80, and 20 degrees.
Learn about Angle Sum Property of a Triangle in this Video
Measurement of the Exterior Angles of a Triangle
The sum of the measures of the two nonadjacent interior angles equals the measure of a triangle’s exterior angle. In other words, the exterior angle is the sum of its opposite interior angles.
\(x^\circ \) + \(y^\circ \) = \(z^\circ \)
Example:
Determine the exterior angle’s measurement.
Solution:
Exterior angle is the sum of the opposite interior angles.
2 x = (x -5) + 80
2 x = x + 75
x = 75
As a result, the exterior angle measurement is \(2(75)^\circ \) = \(150^\circ \).
Solved Examples
Example 1:
Determine the interior angle’s measurement.
x + 30 + 90 = 180
x + 120 = 180
x = 60
As a result, the interior angles are 60, 30, and 90 degrees.
Example 2:
Determine the exterior angle’s measurement.
= 50 + 30
z = 80
As a result, the exterior angle measurement is \(80^\circ \).
Example 3:
Emma is making a wooden birdhouse in the shape of a triangle. What is the measurement of the third angle if two of the interior angles measure
\(45^\circ \) and \(63^\circ \)?
Solution:
We know that the sum of a triangle’s angles is 180.
Sum of interior angles of a triangle = Angle 1 + Angle 2 + Angle 3
\(180^\circ \) = \(45^\circ\) + \(63^\circ\) + Angle 3
Angle 3 = \(180^\circ\) – (\(45^\circ\) + \(63^\circ\))
⇒ Angle 3 = \(72^\circ\)
∴ The third angle is \(72^\circ\).
A triangle has three sides and three vertices. For each vertice two exterior angles can be drawn, similarly for each side two exterior angles can be drawn. Therefore a total of 3 × 2 = 6 exterior angles can be drawn for a triangle.
Acute angles are those that are less than 90 degrees in magnitude.
An angle is obtuse when it measures greater than 90 degrees. Exterior angles are obtuse if they are adjacent to an acute interior angle. On the other hand, if they are adjacent to an obtuse interior angle, then the exterior angle is acute. Hence, no exterior angles are not always obtuse.