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:

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<reasoning>

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. \(m\angle RFO = 60^\circ\)

</reasoning>

<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. \(m\angle RFO = 60^\circ\)

</answer>

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"knowledge_point": [
"Mathematics",
"Geometry",
"Angle Bisector Definition"
]
}
</post_analysis>

Answer:

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<reasoning>

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. \(m\angle RFO = 60^\circ\)

</reasoning>

<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. \(m\angle RFO = 60^\circ\)

</answer>

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"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Geometry",
"Angle Bisector Definition"
]
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</post_analysis>