Question

1. in this figure, \\(m\angle 1 = 135^{\circ}\\).

which of these statements is enough to prove that line \\(m\\) and line \\(n\\) are parallel?
(lesson 4.3) (1 point)

a \\(m\angle 2 = 45^{\circ}\\)
b \\(m\angle 3 = 135^{\circ}\\)
c \\(m\angle 5 = 135^{\circ}\\)
d \\(m\angle 4 = 45^{\circ}\\)

Explanation:

Identify the given information

Using the Linear Pair Postulate knowledge point

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Analyze the conditions for parallel lines

To prove that line \(m\) and line \(n\) are parallel, we need a relationship between an angle on line \(m\) (angles 1, 2, 3, 4) and an angle on line \(n\) (angles 5, 6, 7, 8).

Evaluate the given options

• Option A: \(m\angle 2 = 45^\circ\). This only involves line \(m\) and does not connect to line \(n\).
• Option B: \(m\angle 3 = 135^\circ\). This only involves line \(m\) and does not connect to line \(n\).
• Option C: \(m\angle 5 = 135^\circ\). Since \(\angle 1\) and \(\angle 5\) are corresponding angles, if \(m\angle 1 = m\angle 5 = 135^\circ\), then line \(m \parallel n\).
• Option D: \(m\angle 4 = 45^\circ\). This only involves line \(m\) and does not connect to line \(n\).

Apply the parallel line converse theorem

Using the Converse of Corresponding Angles Postulate knowledge point

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Answer:

• A \(m\angle 2 = 45^\circ\)
• B \(m\angle 3 = 135^\circ\)
• C \(m\angle 5 = 135^\circ\) (Correct answer)
• D \(m\angle 4 = 45^\circ\)