A triangle is a geometrical figure with three sides and three angles. It is a fundamental shape used in numerous applications such as architecture, engineering, and design. In triangle geometry, each angle should add up to 180 degrees. Therefore, given the measurements of two angles in a triangle, one can easily calculate the third angle using simple mathematical formulas. In this article, we will discuss the different ways to calculate the third angle of a triangle.

Method 1: Using the Sum of Angles Formula

The sum of the angles of a triangle is 180 degrees. Therefore, to calculate the third angle, we can simply add up the other two angles and subtract the result from 180 degrees. This method is the simplest and most straightforward way of finding the third angle in a triangle.

For example, if we know that the first angle of the triangle is 50 degrees and the second angle is 70 degrees, we can use the following formula to calculate the third angle:

Third angle = 180 – (50 + 70)
Third angle = 60 degrees

Therefore, the third angle of this triangle is 60 degrees.

Method 2: Using the Law of Cosines

The law of cosines is a mathematical formula used to find an unknown side or angle of a triangle. It is often used in trigonometry and is based on the cosine function. The formula states that:

a² = b² + c² – 2bc cos(A)

Where a, b, and c are the sides of a triangle, and A is the angle opposite to the side a.

Using the Law of Cosines, we can find the measure of the third angle if we know the measure of two sides and the included angle. Here is the step-by-step process:

Step 1: Write down the formula for the law of cosines, making sure that you understand the variables;

Step 2: Identify the values of the known variables: a, b, c, and A;

Step 3: Solve the formula for the desired variable, in this case, A.

For example, let’s say we know the lengths of the two sides of a triangle are 5 cm and 7 cm, respectively, and the measure of the included angle between these sides is 60 degrees. Using the law of cosines, we can calculate the measure of the third angle as follows:

a² = b² + c² –2bc cos(A)

a² = 5² + 7² – 2(5)(7) cos(60)

a² = 25 + 49 – 70 x 0.5

a² = 48

a = √48

a = 4√3

Finally, we can find the measure of the third angle by using the same formula from Method 1:

Third angle = 180 – (50 + 70)
Third angle = 60 degrees

Therefore, the third angle of this triangle is 60 degrees.

In conclusion, calculating the third angle of a triangle is a simple process that requires the knowledge of the sum of the angles formula or the law of cosines. It is essential to understand the formula for each method to ensure that you can apply them to any given situation. These methods have broad applications in various fields, including engineering, architecture, and design.

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