To find the degree measures of angles BOC and AOB, we can use the fact that the sum of angles in a triangle is 180 degrees. Since AOC, BOC, and AOB form a triangle, we can set up an equation:
m AOC + m BOC + m AOB = 180°
Given:
m AOC = 67°
m BOC = 2x + 10
m AOB = 4x - 15
Now, substitute these values into the equation:
67° + (2x + 10) + (4x - 15) = 180°
Combine like terms:
67° + 2x + 10 + 4x - 15 = 180°
Combine constants:
(67 + 10 - 15) + (2x + 4x) = 180°
62 + 6x = 180°
Now, isolate the variable x:
6x = 180° - 62
6x = 118°
Divide both sides by 6 to solve for x:
x = 118° / 6
x = 19.67° (approximately)
Now that we have found the value of x, we can find the degree measures of angles BOC and AOB:
m BOC = 2x + 10
m BOC = 2(19.67) + 10
m BOC ≈ 49.34°
m AOB = 4x - 15
m AOB = 4(19.67) - 15
m AOB ≈ 68.68°
So, the degree measure of angle BOC is approximately 49.34°, and the degree measure of angle AOB is approximately 68.68°.