The measure of one angle in a triangle is 2x degrees, the second angle is 3x + 10 degrees, and the third angle is 4x - 19 degrees. Find the value of x and the measures of all three angles.
In a triangle, the sum of the angles is always 180 degrees.
So, we have the equation:
2x + (3x + 10) + (4x - 19) = 180.
Simplifying the equation, we get:
9x - 9 = 180.
Adding 9 to both sides, we get:
9x = 189.
Dividing both sides by 9, we get:
x = 21.
Plugging x = 21 into 2x, 3x + 10, and 4x - 19, we find:
2x = 2(21) = 42 degrees,
3x + 10 = 3(21) + 10 = 73 degrees, and
4x - 19 = 4(21) - 19 = 65 degrees.
Therefore, the value of x is 21, and the measures of the three angles are 42 degrees, 73 degrees, and 65 degrees, respectively.