Question

if \\(\angle a\\) and \\(\angle b\\) are supplementary angles and the measure of \\(\angle a\\) is fourteen times as large as the measure of \\(\angle b\\), find the measures of \\(\angle a\\) and \\(\angle b\\).

the measure of \\(\angle b\\) is \\(\square^{\circ}\\).

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"tutor"
],
"needs_drawing": false,
"concepts_used": [
"Supplementary Angles",
"Systems of Linear Equations"
],
"new_concepts": [],
"current_concepts": [
"Supplementary Angles",
"Systems of Linear Equations"
]
}
</pre_analysis>

<reasoning>

Define variables for the angles

Let \(a\) represent the measure of \(\angle A\) in degrees, and let \(b\) represent the measure of \(\angle B\) in degrees.

Set up the system of equations

Using the Supplementary Angles concept, the sum of the measures of two supplementary angles is \(180^\circ\):
\[a + b = 180\]
Using the problem statement, the measure of \(\angle A\) is fourteen times the measure of \(\angle B\):
\[a = 14b\]

Solve the system of equations

Using the Systems of Linear Equations concept, substitute the expression for \(a\) from the second equation into the first equation:
\[14b + b = 180\]
\[15b = 180\]
\[b = \frac{180}{15}\]
\[b = 12\]

Calculate the measure of angle A

Substitute \(b = 12\) back into the equation for \(a\):
\[a = 14 \times 12\]
\[a = 168\]

Verify the solution

Check if the sum of the angles is \(180^\circ\):
\[168 + 12 = 180\]
The solution is correct.
</reasoning>

<answer>
If \(\angle A\) and \(\angle B\) are supplementary angles and the measure of \(\angle A\) is fourteen times as large as the measure of \(\angle B\), find the measures of \(\angle A\) and \(\angle B\).

The measure of \(\angle B\) is <blank>12</blank>\(^\circ\).
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Geometry",
"Supplementary Angles"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"tutor"
],
"needs_drawing": false,
"concepts_used": [
"Supplementary Angles",
"Systems of Linear Equations"
],
"new_concepts": [],
"current_concepts": [
"Supplementary Angles",
"Systems of Linear Equations"
]
}
</pre_analysis>

<reasoning>

Define variables for the angles

Let \(a\) represent the measure of \(\angle A\) in degrees, and let \(b\) represent the measure of \(\angle B\) in degrees.

Set up the system of equations

Using the Supplementary Angles concept, the sum of the measures of two supplementary angles is \(180^\circ\):
\[a + b = 180\]
Using the problem statement, the measure of \(\angle A\) is fourteen times the measure of \(\angle B\):
\[a = 14b\]

Solve the system of equations

Using the Systems of Linear Equations concept, substitute the expression for \(a\) from the second equation into the first equation:
\[14b + b = 180\]
\[15b = 180\]
\[b = \frac{180}{15}\]
\[b = 12\]

Calculate the measure of angle A

Substitute \(b = 12\) back into the equation for \(a\):
\[a = 14 \times 12\]
\[a = 168\]

Verify the solution

Check if the sum of the angles is \(180^\circ\):
\[168 + 12 = 180\]
The solution is correct.
</reasoning>

<answer>
If \(\angle A\) and \(\angle B\) are supplementary angles and the measure of \(\angle A\) is fourteen times as large as the measure of \(\angle B\), find the measures of \(\angle A\) and \(\angle B\).

The measure of \(\angle B\) is <blank>12</blank>\(^\circ\).
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Geometry",
"Supplementary Angles"
]
}
</post_analysis>