Question

in the diagram, \\(\vec{fo}\\) bisects \\(\angle rfg\\). drag to correctly order the steps to solve for \\(x\\) and the measure of \\(\angle rfo\\).

note: angles are not drawn to scale.

\\(m\angle rfo = 36 + 24\\)

\\(24 = 11x - 9\\)

\\(12x + 24 = 23x - 9\\)

\\(33 = 11x\\)

\\(m\angle rfo = 12(3) + 24\\)

\\(x = 3\\)

\\(m\angle rfo = 60^{\circ}\\)

Explanation:

Identify the geometric relationship

Since ray \(FO\) bisects \(\angle RFG\), the two adjacent angles \(\angle RFO\) and \(\angle OFG\) are congruent.
Using the Congruent Angles knowledge point:

$$ m\angle RFO = m\angle OFG $$

Set up the algebraic equation

Using the Algebraic Angle Solving knowledge point, substitute the given algebraic expressions:

$$ 12x + 24 = 23x - 9 $$

Solve for x

Subtract \(12x\) from both sides:

$$ 24 = 11x - 9 $$

Add \(9\) to both sides:

$$ 33 = 11x $$

Divide by \(11\):

$$ x = 3 $$

Calculate the measure of angle RFO

Substitute \(x = 3\) back into the expression for \(m\angle RFO\):

$$ m\angle RFO = 12(3) + 24 $$

Multiply:

$$ m\angle RFO = 36 + 24 $$

Add:

$$ m\angle RFO = 60^\circ $$

Order the steps

By arranging the logical progression of equations, the correct order is:

1. \(12x + 24 = 23x - 9\)
2. \(24 = 11x - 9\)
3. \(33 = 11x\)
4. \(x = 3\)
5. \(m\angle RFO = 12(3) + 24\)
6. \(m\angle RFO = 36 + 24\)
7. \(m\angle RFO = 60^\circ\)

Answer:

The correct order of the steps to solve for \(x\) and the measure of \(\angle RFO\) is:

1. \(12x + 24 = 23x - 9\)
2. \(24 = 11x - 9\)
3. \(33 = 11x\)
4. \(x = 3\)
5. \(m\angle RFO = 12(3) + 24\)
6. \(m\angle RFO = 36 + 24\)
7. \(m\angle RFO = 60^\circ\)