Finding the Third Angle of a Triangle Given the Sum of Two Angles
The sum of the internal angles of any triangle, whether it be a right-angled triangle or an equilateral triangle, isalways 180 degrees. This fundamental property of triangles is a cornerstone of geometry and forms the basis for solving problems involving the angles of triangles.
Understanding the Concept
Given that the sum of the three internal angles in a triangle is 180 degrees, if the sum of two angles is known, finding the third angle is straightforward. For example, if the sum of two angles in a triangle is 114 degrees, the third angle can be found as follows:
Third angle 180° - 114° 66°
This principle applies to all triangles, regardless of the shape or size. Let's take another example where the sum of two angles is 120 degrees:
Examples for Different Sum Values
If the sum of two angles is 120 degrees, the third angle is:
Third angle 180° - 120° 60°
No matter what the values of the two angles are, as long as their sum is less than 180 degrees, subtracting this sum from 180 degrees will give the measure of the third angle.
Piecewise Example Solutions
Example 1: Sum of 114° and Unknown Third Angle
Given: Two angles sum to 114° To find: The third angle
Solution:
Third angle 180° - 114° 66°
Example 2: Sum of 120° and Unknown Third Angle
Given: Two angles sum to 120° To find: The third angle
Solution:
Third angle 180° - 120° 60°
Example 3: Sum of 135° and Unknown Third Angle
Given: Two angles sum to 135° To find: The third angle
Solution:
Third angle 180° - 135° 45°
General Formula
If the sum of two angles of a triangle is given as S, the third angle can be found by the following formula:
Third angle 180° - S
This formula is derived directly from the fact that the sum of all interior angles of a triangle is always 180 degrees.
Conclusion
Understanding the sum of the angles in a triangle is crucial for solving a variety of geometric problems. Whether it's calculating the missing angle or verifying the angles of a triangle, knowing this fundamental property allows for easy computation and problem-solving.