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In the last lesson we used the Law of Cosines to find the length of the third side of a triangle when you know the length of two sides and the angle in between. This law can also be used to find the measure of an angle when the three sides of a triangle are given. To use the Law of Cosines to find an unknown angle: 1. Identify c as the side across from the angle you are trying to find. 2. Substitute the values into the Law of Cosines.
In the diagram shown above, we see triangle ABC and the dotted line through A that is parallel to BC. Since angle A, from the triangle, and angles 1 and 2 form a straight (dotted) line, the sum of their measures is 180Âº. We also know that since the dotted line is parallel to BC that by alternate interior angles. This gives us: Substituting for angles 1 and 2 we have This result is true for all triangles: Summarizing: Theorem: The sum of the three angles in a triangle is always 180Âº. Let's Practice: If two of the three angles in a triangle have measure 34Âº and 72Âº, what is the measure of the third angle? We have 34Âº + 72Âº = 106Âº. Therefore the third angle has a measure of 180Âº - 106Âº = 74Âº . In a triangle ABC, we have .