The angles in a triangle are such that the measure of one angle is 10° more than the measure of the smallest angle, while the measure of the third angle is three times the measure of the smallest angle. Find the measures of all three angles, in degrees.
smallest --- x
middle ---- x+10
largest ---- 3x
x + x+10 + 3x = 180
etc
To solve this problem, let's assume the measure of the smallest angle in the triangle is x degrees.
According to the problem, the measure of one angle is 10° more than the measure of the smallest angle. Therefore, the second angle can be expressed as (x + 10) degrees.
The problem also states that the measure of the third angle is three times the measure of the smallest angle. So, the third angle can be expressed as 3x degrees.
Since the sum of the angles in a triangle is always 180 degrees, we can set up an equation:
x + (x + 10) + 3x = 180
Now, let's solve this equation to find the value of x:
5x + 10 = 180
Subtracting 10 from both sides:
5x = 170
Dividing both sides by 5:
x = 34
Now, we can substitute the value of x back into our expressions to find the measures of all three angles:
The measure of the smallest angle = x = 34 degrees
The measure of the second angle = x + 10 = 34 + 10 = 44 degrees
The measure of the third angle = 3x = 3 * 34 = 102 degrees
Therefore, the measures of the three angles are 34 degrees, 44 degrees, and 102 degrees.