In each of the following, relationships among marked angles are given below the figure. Find the measures of the marked angles.

b. m(∠AOB ) is 30° less than 2m (∠BOC )

c. m(∠AOB ) – m(∠BOC ) = 50

hard to say, without knowing the relationship of the angles. In a triangle? Between parallel lines?

To find the measures of the marked angles, we need to first determine the value of a variable, let's say x, and then use that value to calculate the measures of the marked angles.

b. m(∠AOB) is 30° less than 2m(∠BOC)

Let's assume the measure of ∠BOC is x. According to the given relationship, m(∠AOB) is 30° less than 2m(∠BOC), which means m(∠AOB) = 2x - 30.

Now, since ∠AOB is a straight angle (180°), the sum of ∠AOB and ∠BOC should be 180°. Therefore, we have the equation:

m(∠AOB) + m(∠BOC) = 180

Replace m(∠AOB) with 2x - 30 and m(∠BOC) with x:

2x - 30 + x = 180

Combine like terms:

3x - 30 = 180

Now, solve for x by isolating 3x:

3x = 180 + 30
3x = 210
x = 210 / 3
x = 70

Once we have the value of x, we can calculate the measures of the marked angles:

m(∠AOB) = 2x - 30
m(∠AOB) = 2(70) - 30
m(∠AOB) = 140 - 30
m(∠AOB) = 110°

m(∠BOC) = x
m(∠BOC) = 70°

c. m(∠AOB) – m(∠BOC) = 50

In this case, we are given the difference between the measure of ∠AOB and ∠BOC, which is 50.

Let's assume the measure of ∠BOC is x. According to the given relationship, m(∠AOB) – m(∠BOC) = 50, which means m(∠AOB) = x + 50.

Again, since ∠AOB is a straight angle (180°), the sum of ∠AOB and ∠BOC should be 180°. Therefore, we have the equation:

m(∠AOB) + m(∠BOC) = 180

Replace m(∠AOB) with x + 50 and m(∠BOC) with x:

(x + 50) + x = 180

Combine like terms:

2x + 50 = 180

Now, solve for x by isolating 2x:

2x = 180 - 50
2x = 130
x = 130 / 2
x = 65

Once we have the value of x, we can calculate the measures of the marked angles:

m(∠AOB) = x + 50
m(∠AOB) = 65 + 50
m(∠AOB) = 115°

m(∠BOC) = x
m(∠BOC) = 65°

So, the measures of the marked angles for b are 110° and 70°, and for c are 115° and 65°.

b. To find the measure of ∠AOB, we can start by assigning a variable to ∠BOC. Let's say ∠BOC = x.

According to the given information, m(∠AOB) is 30° less than 2m(∠BOC).

Therefore, m(∠AOB) = 2x - 30°.

Now we can set up an equation using this information:

m(∠AOB) = 2m(∠BOC) - 30°.
2x - 30° = 2(x) - 30°.
2x - 30° = 2x - 30°.
-30° = -30°.

As we can see, both sides of the equation are equal. This means that the measure of ∠AOB can be any value as long as ∠BOC = x is consistent with the given information. Thus, the measure of ∠AOB cannot be determined with the provided information.

c. To find the measure of ∠AOB, we can start by assigning a variable to ∠BOC. Let's say ∠BOC = x.

According to the given information, m(∠AOB) - m(∠BOC) = 50°.

Therefore, m(∠AOB) = m(∠BOC) + 50°.

Now we can substitute m(∠BOC) with x and set up an equation:

m(∠AOB) = x + 50°.

As we can see, we still don't have enough information to determine the actual measure of ∠AOB. The measure of ∠AOB can be any value as long as ∠BOC = x is consistent with the given information. Thus, the measure of ∠AOB cannot be determined with only the provided information.