Question

In the figure, CA and CE are opposite rays, CH bisects ∠GCD, and GC bisects ∠BGD. If m∠GCF is 12° less than m∠BCG, what is m∠FCD? m∠FCD =

Answer 1

To find the measure of ∠FCD, we can start by noting that ∠GCF is 12° less than ∠BCG. Let's denote the measure of ∠BCG as "x" degrees.

So, m∠GCF = x - 12°.

Since GC bisects ∠BGD, we can also note that ∠BCG = ∠GCD (because they are vertically opposite angles).

Now, let's consider the angle ∠FCD. Since CH bisects ∠GCD, we can say that:

m∠FCD = (1/2) * m∠GCD.

Substituting the value of m∠GCD with m∠BCG, we get:

m∠FCD = (1/2) * x.

But we also know that m∠GCF = x - 12°. Therefore:

x - 12° = (1/2) * x.

Now, solve for x:

2(x - 12°) = x

2x - 24° = x

x = 24°.

Now that we have found the measure of ∠BCG (x), we can find the measure of ∠FCD:

m∠FCD = (1/2) * x = (1/2) * 24° = 12°.

So, the measure of ∠FCD is 12 degrees.

Answer 2

The measure of ∠FCD is 12 degrees.

To find the measure of ∠FCD, we can start by noting that ∠GCF is 12° less than ∠BCG. Let's denote the measure of ∠BCG as "x" degrees.

So, m∠GCF = x - 12°.

Since GC bisects ∠BGD, we can also note that ∠BCG = ∠GCD (because they are vertically opposite angles).

Now, let's consider the angle ∠FCD. Since CH bisects ∠GCD, we can say that:

m∠FCD = (1/2) * m∠GCD.

Substituting the value of m∠GCD with m∠BCG, we get:

m∠FCD = (1/2) * x.

But we also know that m∠GCF = x - 12°. Therefore:

x - 12° = (1/2) * x.

Now, solve for x:

2(x - 12°) = x

2x - 24° = x

x = 24°.

Now that we have found the measure of ∠BCG (x), we can find the measure of ∠FCD:

m∠FCD = (1/2) * x = (1/2) * 24° = 12°.