In right triangle ABC what is the arithmetic mean (average), in degrees, of the measures of the two smallest angles?
Explain how the answer is 45 degrees.
45
To find the arithmetic mean or average of the measures of the smallest angles in a right triangle, we need to first understand the properties of a right triangle.
In a right triangle, one angle measures 90 degrees and is referred to as the right angle. The two remaining angles are acute angles, meaning they are less than 90 degrees.
Since we are looking for the arithmetic mean of the two smallest angles, we can assume that one of the acute angles is x degrees.
By the definition of a right triangle, the sum of the measures of all three angles must be 180 degrees. So, we can express the measures of the angles as follows:
Right angle = 90 degrees
Acute angle 1 = x degrees
Acute angle 2 = 180 - (90 + x) = 90 - x degrees
To find the arithmetic mean, we need to calculate the average of the two acute angles:
Average of acute angles = (Acute angle 1 + Acute angle 2) / 2
Substituting the values from above:
Average of acute angles = (x + (90 - x)) / 2
Average of acute angles = (90 + 90) / 2
Average of acute angles = 180 / 2
Average of acute angles = 90 degrees
Therefore, the arithmetic mean of the measures of the two smallest angles in a right triangle is 90 degrees.
To find the arithmetic mean of the measures of the two smallest angles in a right triangle ABC, we need to first determine the measures of these angles.
In a right triangle, one of the angles is a right angle, which measures 90 degrees. Therefore, we are interested in finding the measures of the other two angles.
The sum of all angles in any triangle is always 180 degrees. In a right triangle, since one angle is 90 degrees, the sum of the other two angles must be equal to 90 degrees as well. This is because the sum of angles in a triangle is equal to 180 degrees, and if one angle is 90 degrees, the other two angles must add up to the remaining 90 degrees.
Therefore, the two remaining angles in the right triangle ABC must have equal measures in order to add up to 90 degrees. Since we are asked for the average of these two angles, we divide 90 degrees by 2 to get the arithmetic mean.
90 degrees ÷ 2 = 45 degrees
Hence, the arithmetic mean of the measures of the two smallest angles in a right triangle ABC is 45 degrees.
A triangle has a total of 180 degrees.
Since the right angle = 90 degrees.
There is a total of 90 degrees left for the other two angles.
Their sum divided by 2 gives the arithmetic mean of 45 degrees