An OC beam is drawn inside the right angle AOB. Find the degree measures of the AOC

An OC beam is drawn inside the right angle AOB. Find the degree measures of the AOC and AOC angles if: a) the AOC angle is 5 times greater than the COB angle b) the difference in the degree measures of the AOC and AO angles is 46 degrees c) the AOC angle is 4 times less than the COB angle

Decision:

Let’s denote the value of the AOC angle through X degrees, and the value of the COB angle through Y degrees.

Since the angle AOB is straight, its value is 90.

Then X + Y = 90.

a)

By condition, the AOC angle is 5 times greater than the COB angle, then:

X = 5 x Y.

Let’s solve the system of two equations.

X + Y = 90.

X = 5 x Y.

Let us express the value of X through Y in the first equation, and substitute it in the second.

X = 90 – U.

(90 – Y) = 5 x Y.

6 x Y = 90.

Y = 90/6.

Y = 15.

Then:

X = 90 – 15 = 75.

Answer: Angle AOC = 75, angle COB = 15.

B)

We solve in a similar way, solving the system of two equations by the substitution method.

By condition, the COB angle is 46 degrees greater than the AOB angle, then:

X + Y = 90.

Y – X = 46.

Y – 90 + Y = 46.

2 x Y = 136.

Y = 68.

X = 90 – 68 = 22.

Answer: Angle AOC = 22, angle COB = 68.

c)

We solve in a similar way, solving the system of two equations by the substitution method.

According to the condition of the problem, the AOC angle is 4 times less than the COI angle, then:

X + Y = 90.

4 x X = Y.

4 X (90 – Y) = Y.

360 – 4 x Y = Y.

5 x Y = 360.

Y = 360/5 = 72.

X = 90 – 72 = 18.

Answer: Angle AOC = 18, angle COB = 72.