You can find the missing measures of any triangle by using the ? if you know the measures of two angles and a side.
Let's assume that we have a triangle △ABC with the side lengths a, b, and c.
If we know the measures of two angles and one side, we will consider how to find the missing measures. In this type of situation, we can use the Law of Sines. This law states that the ratio of the sine of an angle to its opposite side is constant.
sin A/a=sin B/b=sin C/c
Let's think about exactly how each missing measure would be found.
Finding the Missing Angle Measure
If we know the measures of any two angles, let's say ∠ A and ∠ B, we can find the measure of the missing angle ∠ C using the Triangle Angle Sum Theorem. This tells us that the sum of the angles of a triangle is equal to 180^(∘).
m∠ A + m∠ B + m∠ C = 180^(∘) ⇕ m∠ C = 180^(∘) - ( m∠ A + m∠ B)
Finding the Missing Side Measures
Now that we have all of the angle measures, we can find the missing side lengths. It does not matter which side length of the triangle is known. Let's assume that we know the measure of a.
Using the Law of Sines, we can find the measures of b and c where a, m∠A, m∠B, and m∠C are known.
sin A/a=sin B/b [1em]
sin A/a=sin C/c
Conclusion
Now, we are able to complete the sentence.
You can find the missing measures of any triangle by using the Law of Sines if you know the measures of two angles and a side.
