One interior angle of a triangle has a measure that is equal to the sum of the measures of the other two angles of the triangle. What is the measure of the smallest exterior angle of the triangle in degrees?
If the interior angle is x, then the sum of the other two interior angles is also x. Since they add up to 180, x=90.
Also, since the sum of an interior angle and its adjacent exterior angle is a straight line, they sum to 180.
So, the largest interior angle has the smallest exterior angle. In this case, that's 90.
To find the measure of the smallest exterior angle of a triangle, we first need to know that the sum of the measures of the exterior angles of any polygon is always 360 degrees.
In a triangle, the sum of the three exterior angles is also 360 degrees. Since we're looking for the smallest exterior angle, let's call it "a", and the other two angles "b" and "c".
According to the given information, one interior angle (let's call it "b") is equal to the sum of the measures of the other two angles (let's call them "a" and "c").
So we have the equation: b = a + c
Since the sum of all three interior angles of a triangle is always 180 degrees, we also know that a + b + c = 180.
Substituting the first equation into the second equation, we get: a + (a + c) + c = 180
Simplifying this equation, we have: 2a + 2c = 180
Dividing both sides of the equation by 2, we get: a + c = 90
Now we know that the sum of the measures of the two angles is 90 degrees.
To find the smallest exterior angle, we subtract the measure of the corresponding interior angle from 180 degrees.
Therefore, the measure of the smallest exterior angle is 180 - a.
Substituting the value we found for a, we have: 180 - (a + c) = 180 - 90 = 90 degrees.
So, the measure of the smallest exterior angle of the triangle is 90 degrees.