Question

lines \\(l_1\\) and \\(l_2\\) are parallel lines. determine the measures of \\(\angle 1\\) through \\(\angle 12\\).

\\(m\angle 2 = 59^\circ\\)
\\(m\angle 3 = 64^\circ\\)
\\(m\angle 4 = 64^\circ\\)
\\(m\angle 5 = 57^\circ\\)
\\(m\angle 6 = \square^\circ\\)

Explanation:

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<reasoning>

Find angle 6 and its vertical counterpart

Using the Vertical Angles knowledge point
\[

$$\begin{aligned} &m\angle 6 = 180^\circ - 64^\circ = 116^\circ\\ &m\angle 4 = m\angle 64^\circ = 64^\circ \end{aligned}$$

\]

Find angle 3 and angle 1 using parallel lines

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

$$\begin{aligned} &m\angle 3 = m\angle 64^\circ = 64^\circ\\ &m\angle 1 = m\angle 3 = 64^\circ \end{aligned}$$

\]

Find angle 2 and angle 5 using the triangle sum

Using the Triangle Angle Sum Theorem and Supplementary Angles knowledge points
\[

$$\begin{aligned} &\text{Triangle interior angles: } 64^\circ, 59^\circ, m\angle 7\\ &m\angle 7 = 180^\circ - (64^\circ + 59^\circ) = 57^\circ\\ &m\angle 2 = 59^\circ\\ &m\angle 5 = m\angle 7 = 57^\circ \end{aligned}$$

\]

Find remaining angles on line 1 and line 2

Using the Supplementary Angles and Vertical Angles knowledge points
\[

$$\begin{aligned} &m\angle 9 = 180^\circ - m\angle 6 = 64^\circ\\ &m\angle 10 = m\angle 6 = 116^\circ\\ &m\angle 8 = 180^\circ - m\angle 7 = 123^\circ\\ &m\angle 11 = m\angle 8 = 123^\circ\\ &m\angle 12 = m\angle 7 = 57^\circ \end{aligned}$$

\]
</reasoning>

<answer>

No.	Answer
2	\(59^\circ\)
3	\(64^\circ\)
4	\(64^\circ\)
5	\(57^\circ\)
6	\(116^\circ\)
7	\(57^\circ\)
8	\(123^\circ\)
9	\(64^\circ\)
10	\(116^\circ\)
11	\(123^\circ\)
12	\(57^\circ\)

</answer>

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</post_analysis>

Answer:

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</pre_analysis>

<reasoning>

Find angle 6 and its vertical counterpart

Using the Vertical Angles knowledge point
\[

$$\begin{aligned} &m\angle 6 = 180^\circ - 64^\circ = 116^\circ\\ &m\angle 4 = m\angle 64^\circ = 64^\circ \end{aligned}$$

\]

Find angle 3 and angle 1 using parallel lines

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

$$\begin{aligned} &m\angle 3 = m\angle 64^\circ = 64^\circ\\ &m\angle 1 = m\angle 3 = 64^\circ \end{aligned}$$

\]

Find angle 2 and angle 5 using the triangle sum

Using the Triangle Angle Sum Theorem and Supplementary Angles knowledge points
\[

$$\begin{aligned} &\text{Triangle interior angles: } 64^\circ, 59^\circ, m\angle 7\\ &m\angle 7 = 180^\circ - (64^\circ + 59^\circ) = 57^\circ\\ &m\angle 2 = 59^\circ\\ &m\angle 5 = m\angle 7 = 57^\circ \end{aligned}$$

\]

Find remaining angles on line 1 and line 2

Using the Supplementary Angles and Vertical Angles knowledge points
\[

$$\begin{aligned} &m\angle 9 = 180^\circ - m\angle 6 = 64^\circ\\ &m\angle 10 = m\angle 6 = 116^\circ\\ &m\angle 8 = 180^\circ - m\angle 7 = 123^\circ\\ &m\angle 11 = m\angle 8 = 123^\circ\\ &m\angle 12 = m\angle 7 = 57^\circ \end{aligned}$$

\]
</reasoning>

<answer>

No.	Answer
2	\(59^\circ\)
3	\(64^\circ\)
4	\(64^\circ\)
5	\(57^\circ\)
6	\(116^\circ\)
7	\(57^\circ\)
8	\(123^\circ\)
9	\(64^\circ\)
10	\(116^\circ\)
11	\(123^\circ\)
12	\(57^\circ\)

</answer>

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