Solving for the Angles of a Triangle Given the Sum and Difference

In this article, we'll explore how to solve for the angles of a triangle when given the sum and the difference of two of its angles. We'll work through a detailed example to illustrate the process. The triangle angle sum property will be a core concept in our journey.

Understanding the Problem

Consider a triangle with angles A, B, and C. We are given that the sum of A and B is 116 degrees and their difference is 24 degrees. We need to find all three angles of the triangle.

Solving the Equations

To solve this problem, we can set up a system of equations based on the given information:

A B 116 degrees A - B 24 degrees

First, let's add the two equations to eliminate B:

2A  140

Divide both sides by 2 to solve for A:

A  70

Now, substitute A 70 into the first equation to solve for B:

70   B  116B  116 - 70B  46

So, we have A 70 degrees and B 46 degrees. The remaining angle C can be found using the triangle angle sum property, which states that the sum of all angles in a triangle is 180 degrees:

C  180 - (A   B)C  180 - (70   46)C  180 - 116C  64

Therefore, the three angles of the triangle are 70 degrees, 46 degrees, and 64 degrees.

Alternative Methods

We can also use another method to solve for the angles:

Let p and q represent the two angles of the triangle. We have: p q 116 …1 p - q 24 …2

Adding the two equations to eliminate q, we get:

2p  140p  70

Substitute p 70 into the first equation to find q:

70   q  116q  116 - 70q  46

The third angle is 180 - 116 64 degrees. Therefore, the angles are 70 degrees, 46 degrees, and 64 degrees.

Conclusion

The process of finding the angles of a triangle using the given sum and difference of two angles is straightforward once we set up the appropriate equations. The angle sum property is a key concept in solving such problems. Let's summarize the solution:

The three angles of the triangle are 70 degrees 46 degrees and 64 degrees.

Thank you for reading this article. We hope you found it helpful in understanding how to solve for triangle angles using the given sum and difference. If you have any questions or need further clarification, feel free to ask!