Question

use the value of \\(x\\) from the previous question to determine the angle measurements.

press a hotspot, then type your response, and press submit. repeat for each response.

\\(m\angle abd = \underline{\quad}^{\circ}\\)
\\(m\angle abc = \underline{\quad}^{\circ}\\)

\\((3x + 5)^{\circ}\\)
\\((x + 15)^{\circ}\\)

Explanation:

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<reasoning>

Set up the linear pair equation

The angles \(\angle ABD\) and \(\angle ABC\) form a linear pair along the straight line \(DC\).
By the Linear Pair Postulate, their measures sum to \(180^\circ\):
\[(3x + 5) + (x + 15) = 180\]

Solve for x

Combine like terms and solve the linear equation:
\[4x + 20 = 180\]
\[4x = 160\]
\[x = 40\]

Calculate the measure of angle ABD

Substitute \(x = 40\) into the expression for \(m\angle ABD\):
\[m\angle ABD = 3(40) + 5 = 120 + 5 = 125^\circ\]

Calculate the measure of angle ABC

Substitute \(x = 40\) into the expression for \(m\angle ABC\):
\[m\angle ABC = 40 + 15 = 55^\circ\]

Verify the sum of the angles

Check that the sum of the calculated angles equals \(180^\circ\):
\[125^\circ + 55^\circ = 180^\circ\]
The calculations are correct.
</reasoning>

<answer>
Use the value of x from the previous question to determine the angle measurements.

\(m\angle ABD =\) <blank>125</blank>\(^\circ\)

\(m\angle ABC =\) <blank>55</blank>\(^\circ\)
</answer>

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"question_type": "Fill-in-the-blank",
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"Linear Pair Postulate"
]
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Answer:

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</pre_analysis>

<reasoning>

Set up the linear pair equation

The angles \(\angle ABD\) and \(\angle ABC\) form a linear pair along the straight line \(DC\).
By the Linear Pair Postulate, their measures sum to \(180^\circ\):
\[(3x + 5) + (x + 15) = 180\]

Solve for x

Combine like terms and solve the linear equation:
\[4x + 20 = 180\]
\[4x = 160\]
\[x = 40\]

Calculate the measure of angle ABD

Substitute \(x = 40\) into the expression for \(m\angle ABD\):
\[m\angle ABD = 3(40) + 5 = 120 + 5 = 125^\circ\]

Calculate the measure of angle ABC

Substitute \(x = 40\) into the expression for \(m\angle ABC\):
\[m\angle ABC = 40 + 15 = 55^\circ\]

Verify the sum of the angles

Check that the sum of the calculated angles equals \(180^\circ\):
\[125^\circ + 55^\circ = 180^\circ\]
The calculations are correct.
</reasoning>

<answer>
Use the value of x from the previous question to determine the angle measurements.

\(m\angle ABD =\) <blank>125</blank>\(^\circ\)

\(m\angle ABC =\) <blank>55</blank>\(^\circ\)
</answer>

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"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Geometry",
"Linear Pair Postulate"
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