If mangle upper A upper O upper C= 85°, mangle upper B upper O upper C = 2x + 10, and mangle upper A upper O upper B = 4x – 15, find the degree measure of angle upper B upper O upper Cand angle upper A upper O upper B
To find the degree measure of angle BOC, we can use the fact that the sum of the interior angles of a triangle is 180°. Since angle AOC and angle AOB are known, we can find angle BOC.
Given:
angle AOC = 85°,
angle AOB = 4x - 15°
Since the sum of the interior angles of a triangle is 180°, we have:
angle BOC + angle AOC + angle AOB = 180°
Substituting the given values:
angle BOC + 85° + (4x - 15°) = 180°
Simplifying the equation:
angle BOC + 4x + 70° = 180°
angle BOC + 4x = 110°
Now, we are given that angle BOC = 2x + 10°, so:
2x + 10° + 4x = 110°
Combining like terms:
6x + 10° = 110°
Subtracting 10° from both sides:
6x = 100°
Dividing by 6:
x = 100° / 6
x = 16.67° (rounded to the nearest hundredth)
Now we can find the degree measure of angle BOC using the value of x we just obtained:
angle BOC = 2x + 10°
angle BOC = 2(16.67°) + 10°
angle BOC = 33.33° + 10°
angle BOC = 43.33°
Therefore, angle BOC measures approximately 43.33°.
To find the degree measure of angle AOB, we can substitute the value of x into the given expression:
angle AOB = 4x - 15°
angle AOB = 4(16.67°) - 15°
angle AOB = 66.67° - 15°
angle AOB = 51.67°
Therefore, angle AOB measures approximately 51.67°.
To find the degree measure of angle BOC, we can use the fact that angles in a triangle sum up to 180°.
We know that:
∠AOC = 85° (Given)
∠BOC = 2x + 10 (Given)
Using the fact that angles in a triangle sum up to 180°, we can write the equation:
∠AOC + ∠BOC + ∠AOB = 180°
Substituting the given values, we get:
85° + (2x + 10) + (4x - 15) = 180°
Combining like terms:
6x + 80 = 180
Subtracting 80 from both sides:
6x = 100
Dividing by 6:
x = 100/6 = 16.67
Now, we can find the degree measure of angle BOC:
∠BOC = 2x + 10
∠BOC = 2(16.67) + 10
∠BOC = 33.34 + 10
∠BOC = 43.34°
Therefore, the degree measure of angle BOC is 43.34°.
To find the degree measure of angle AOB, we can substitute the value of x into the equation:
∠AOB = 4x - 15
∠AOB = 4(16.67) - 15
∠AOB = 66.68 - 15
∠AOB = 51.68°
Therefore, the degree measure of angle AOB is 51.68°.