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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 measures of the angles is always 180 degrees.
Let's set up an equation to solve for x:
2x + 3x + 10 + 4x - 19 = 180
Combine like terms:
9x - 9 = 180
Add 9 to both sides:
9x = 189
Divide by 9:
x = 21
Now we can find the measures of the angles:
Angle 1: 2x = 2(21) = 42 degrees
Angle 2: 3x + 10 = 3(21) + 10 = 63 + 10 = 73 degrees
Angle 3: 4x - 19 = 4(21) - 19 = 84 - 19 = 65 degrees
So, the value of x is 21, and the measures of the three angles are 42 degrees, 73 degrees, and 65 degrees.