Question

find the measures of angles 1 through 12.

\\(m\angle 7 = 57^{\circ}\\)
\\(m\angle 1 = 57^{\circ}\\)
\\(m\angle 2 = 59^{\circ}\\)
\\(m\angle 3 = 64^{\circ}\\)
\\(m\angle 4 = 64^{\circ}\\)
\\(m\angle 5 = 59^{\circ}\\)

Explanation:

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"concepts_used": [
"Triangle Angle Sum Theorem",
"Vertical Angles",
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"Corresponding Angles"
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"Corresponding Angles"
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</pre_analysis>

<reasoning>

Find the measure of angle 7

Using the Triangle Angle Sum Theorem knowledge point
\[

$$\begin{aligned} &m\angle 7 + 59^\circ + 64^\circ = 180^\circ \\ &m\angle 7 + 123^\circ = 180^\circ \\ &m\angle 7 = 57^\circ \end{aligned}$$

\]

Find the measure of angle 1

Using the Parallel Lines Cut by a Transversal and Corresponding Angles knowledge points
\[

$$\begin{aligned} &m\angle 1 = m\angle 7 \\ &m\angle 1 = 57^\circ \end{aligned}$$

\]

Find the measure of angle 2

Using the Vertical Angles knowledge point
\[

$$\begin{aligned} &m\angle 2 = 59^\circ \end{aligned}$$

\]

Find the measure of angle 3

Using the Parallel Lines Cut by a Transversal and Corresponding Angles knowledge points
\[

$$\begin{aligned} &m\angle 3 = 64^\circ \end{aligned}$$

\]

Find the measure of angle 4

Using the Vertical Angles knowledge point
\[

$$\begin{aligned} &m\angle 4 = m\angle 3 \\ &m\angle 4 = 64^\circ \end{aligned}$$

\]

Find the measure of angle 5

Using the Vertical Angles knowledge point
\[

$$\begin{aligned} &m\angle 5 = m\angle 2 \\ &m\angle 5 = 59^\circ \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

\(m\angle 7 =\) <blank>\(57\)</blank>\(^\circ\)

Question 2

\(m\angle 1 =\) <blank>\(57\)</blank>\(^\circ\)

Question 3

\(m\angle 2 =\) <blank>\(59\)</blank>\(^\circ\)

Question 4

\(m\angle 3 =\) <blank>\(64\)</blank>\(^\circ\)

Question 5

\(m\angle 4 =\) <blank>\(64\)</blank>\(^\circ\)

Question 6

\(m\angle 5 =\) <blank>\(59\)</blank>\(^\circ\)
</answer>

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"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Geometry",
"Parallel Lines Cut by a Transversal"
]
}
</post_analysis>

Answer:

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"Triangle Angle Sum Theorem",
"Vertical Angles",
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"Corresponding Angles"
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</pre_analysis>

<reasoning>

Find the measure of angle 7

Using the Triangle Angle Sum Theorem knowledge point
\[

$$\begin{aligned} &m\angle 7 + 59^\circ + 64^\circ = 180^\circ \\ &m\angle 7 + 123^\circ = 180^\circ \\ &m\angle 7 = 57^\circ \end{aligned}$$

\]

Find the measure of angle 1

Using the Parallel Lines Cut by a Transversal and Corresponding Angles knowledge points
\[

$$\begin{aligned} &m\angle 1 = m\angle 7 \\ &m\angle 1 = 57^\circ \end{aligned}$$

\]

Find the measure of angle 2

Using the Vertical Angles knowledge point
\[

$$\begin{aligned} &m\angle 2 = 59^\circ \end{aligned}$$

\]

Find the measure of angle 3

Using the Parallel Lines Cut by a Transversal and Corresponding Angles knowledge points
\[

$$\begin{aligned} &m\angle 3 = 64^\circ \end{aligned}$$

\]

Find the measure of angle 4

Using the Vertical Angles knowledge point
\[

$$\begin{aligned} &m\angle 4 = m\angle 3 \\ &m\angle 4 = 64^\circ \end{aligned}$$

\]

Find the measure of angle 5

Using the Vertical Angles knowledge point
\[

$$\begin{aligned} &m\angle 5 = m\angle 2 \\ &m\angle 5 = 59^\circ \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

\(m\angle 7 =\) <blank>\(57\)</blank>\(^\circ\)

Question 2

\(m\angle 1 =\) <blank>\(57\)</blank>\(^\circ\)

Question 3

\(m\angle 2 =\) <blank>\(59\)</blank>\(^\circ\)

Question 4

\(m\angle 3 =\) <blank>\(64\)</blank>\(^\circ\)

Question 5

\(m\angle 4 =\) <blank>\(64\)</blank>\(^\circ\)

Question 6

\(m\angle 5 =\) <blank>\(59\)</blank>\(^\circ\)
</answer>

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"knowledge_point": [
"Mathematics",
"Geometry",
"Parallel Lines Cut by a Transversal"
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