Question

bisects \\(\angle dbc\\). find the value of \\(x\\) and \\(m\angle abc\\).

press the hotspot and type your answer. do not include symbols, just provide your numerical answer.

\\(x = \\) ______
\\(m\angle abc = \\) ______

Explanation:

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<reasoning>

Set up the geometric equation

Using the Angle Bisector Definition and Algebraic Angle Solving knowledge points
\[

$$\begin{aligned} &m\angle DBA = m\angle ABC\\ &x + 40 = 5x \end{aligned}$$

\]

Solve for x

Using the Algebraic Angle Solving knowledge point
\[

$$\begin{aligned} &40 = 4x\\ &x = 10 \end{aligned}$$

\]

Calculate the measure of angle ABC

Using the Angle Measurement and Algebraic Angle Solving knowledge points
\[

$$\begin{aligned} &m\angle ABC = 5x\\ &m\angle ABC = 5(10) = 50 \end{aligned}$$

\]
</reasoning>

<answer>
If \(\vec{BA}\) bisects \(\angle DBC\), find the value of \(x\) and \(m\angle ABC\).

\(x =\) <blank>10</blank>

\(m\angle ABC =\) <blank>50</blank>
</answer>

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"Mathematics",
"Geometry",
"Angle Bisector Definition"
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Answer:

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<reasoning>

Set up the geometric equation

Using the Angle Bisector Definition and Algebraic Angle Solving knowledge points
\[

$$\begin{aligned} &m\angle DBA = m\angle ABC\\ &x + 40 = 5x \end{aligned}$$

\]

Solve for x

Using the Algebraic Angle Solving knowledge point
\[

$$\begin{aligned} &40 = 4x\\ &x = 10 \end{aligned}$$

\]

Calculate the measure of angle ABC

Using the Angle Measurement and Algebraic Angle Solving knowledge points
\[

$$\begin{aligned} &m\angle ABC = 5x\\ &m\angle ABC = 5(10) = 50 \end{aligned}$$

\]
</reasoning>

<answer>
If \(\vec{BA}\) bisects \(\angle DBC\), find the value of \(x\) and \(m\angle ABC\).

\(x =\) <blank>10</blank>

\(m\angle ABC =\) <blank>50</blank>
</answer>

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"question_type": "Fill-in-the-blank",
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"Geometry",
"Angle Bisector Definition"
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