-> OB bisects <AOC

m<AOB=6x, m<AOC=2x+2y, m<BOC=y+5

1. If OB bisects <AOC, then <AOB + <BOC = <AOC.

6x + y + 5 = 2x + 2y
solve for y and you get 4x + 5 = y

2. Also, since OB bisect <AOC, then m<AOB = m<BOC
6x = y + 5
Now, use what y equals from step one and subsitute it in so that:
6x = (4x + 5) + 5
subtract 4x from both sides and add the fives, so that: 2x = 10, so x = 5
3. Now check it:
6x = 6(5) = 30 for m<AOB
y + 5 = 4x +5 +5 = 4(5) + 10 = 30 for m<BOC. This makes sense because <AOB = <BOC.
Next, check the m<AOC. It needs to equal 60 since <AOB and <BOC are 30 each.
m<AO = 2X + 2y
2X = 2(5) = 10
2y = 2[4(5) + 5] = 2[20 + 5] = 2[25] = 50
Now add what you got for 2X to 2Y: 50 + 1- = 60.
Everything checks.

To find the measures of angles <AOB, <AOC, and <BOC, we will use the property that the sum of the angles in a triangle is always 180 degrees.

Since OB bisects <AOC, we know that <AOC is divided into two equal angles. Let's call these angles x and x.

Therefore, we have:
m<AOB = 6x (given)
m<AOC = 2x + 2y (given)
m<BOC = y + 5 (given)

Now, we need to find the values of x and y.

Since OB bisects <AOC, the sum of the measures of <AOB and <BOC should equal the measure of <AOC. So, we can write the equation:

m<AOB + m<BOC = m<AOC

Substituting the given measures, we have:
6x + (y + 5) = 2x + 2y

Now, let's solve for x and y:

6x + y + 5 = 2x + 2y
4x - y = - 5

We have one equation with two variables, so we need another equation to solve for x and y.

Since the sum of the measures of the angles in a triangle is 180 degrees, we can write another equation:

m<AOB + m<AOC + m<BOC = 180

Substituting the given angles, we have:
6x + (2x + 2y) + (y + 5) = 180

Now, let's solve this equation for x and y:

6x + 2x + 2y + y + 5 = 180
8x + 3y + 5 = 180
8x + 3y = 175

We now have a system of equations:
4x - y = -5
8x + 3y = 175

Solving this system of equations will give us the values of x and y.

To find the values of x and y in this problem, we'll use the fact that OB bisects angle AOC. This means that the angles AOB and BOC are congruent.

Given:
m<AOB = 6x
m<AOC = 2x + 2y
m<BOC = y + 5

We can set up an equation using the fact that the angles AOB and BOC are congruent:

6x = y + 5

Now, let's solve for the values of x and y.

1. Distribute the 2 in 2x + 2y:
2x + 2y = y + 5

2. Simplify the equation:
2x + y + 5 = y + 5

3. Subtract y from both sides of the equation:
2x = 0

4. Divide both sides of the equation by 2:
x = 0

Now, we have the value of x, which is 0.

To solve for y, we'll substitute the value of x back into the original equation (6x = y + 5):

6(0) = y + 5

0 = y + 5

Subtract 5 from both sides of the equation:

-5 = y

So, the values of x and y are x = 0 and y = -5 respectively.