The ratio among the measures of the angles of a triangle is 16 to 10 to 19. Find the measure of the greatest angle of this triangle.
So we don’t know the measures of the individual angles. But we do know the ratio that exists between them. We also know key facts about the angles in triangles, which is that the angle sum of any triangle is 180 degrees.
We’ll answer this question using the concept of the sum of the parts of a ratio. By adding together the three parts of the ratio, we see that the total parts is equal to 45. As the angle sum in the triangle is 180 degrees, this means that these 45 parts are equal to 180 degrees.
All these parts are of equal size. So to find the size of one part, we can divide both sides of the ratio by 45. One part is equal to four degrees. We’re only asked to find the measure of one of the angles, the greatest angle. This is the angle with the most parts. So it’s the part of the ratio with 19 parts.
To find the value of 19 parts, we now multiply both sides of the ratio by 19. The measure of the greatest angle in the triangle is 76 degrees.
Now a sensible check here would also be to calculate the measures of the other two angles and confirm that the angle sum is indeed 180 degrees. To find the measure of the angle with 16 parts, we multiply one part by 16. 16 parts is equivalent to 64 degrees. To find the measure of the angle with 10 parts, we multiply one part by 10. 10 parts is equal to 40 degrees.
By summing the measures of the three calculated angles together, we confirm that their sum is indeed 180 degrees. And this just helps have a little bit more confidence in our answer. The measure of the greatest angle in the triangle is 76 degrees.