The degree measures of two angles of a triangle are consecutive even integers. If the measure of the third angle is 50 degrees more than twice the measure of the least angle. Find the measure of each angle.
Let x be the smallest angle.
x + 2 is the next smallest angle
The largest angle is 2x + 50
The sum of all angles is 180 degrees, since it is a triangle.
x + x + 2 + 2x + 50 = 180
4x = 180 - 52 = 128
x = 32
The other angles are 34 and 114
Let's denote the measure of the least angle as x.
Since the degree measures of the two angles are consecutive even integers, we can express them as (x + 2) and (x + 4).
According to the given information, the measure of the third angle is 50 degrees more than twice the measure of the least angle, which gives us the equation:
(x + 4) = 2x + 50
Now, we can solve this equation to find the value of x:
x + 4 = 2x + 50
Subtract x from both sides:
4 = x + 50
Subtract 50 from both sides:
-46 = x
Therefore, the measure of the least angle is -46 degrees, but since angles cannot be negative, this is not a valid solution.
Hence, there is no valid solution for this problem.
To solve this problem, let's assign variables to represent the unknown angles.
Let's say that the smallest angle measures x degrees. Since the consecutive even integers will be x, x+2, and x+4 (since consecutive even integers increase by 2 each time), we can write the following equation based on the sum of angles in a triangle:
x + (x+2) + (x+4) = 180
Now let's simplify and solve for x:
3x + 6 = 180
Subtracting 6 from both sides:
3x = 174
Dividing both sides by 3:
x = 58
So, the smallest angle measures 58 degrees. The other two angles can be found by substituting x back into our expressions:
x+2 = 58 + 2 = 60
x+4 = 58 + 4 = 62
Therefore, the three angles of the triangle measure 58 degrees, 60 degrees, and 62 degrees, respectively.