Question

select the correct answer.
given the degree measures, which statement is true?
image of angle diagram with angles 135° and 45° and lines t, ab, cd, and a transversal
a. t is parallel to \\(overleftrightarrow{cd}\\)

b. \\(overleftrightarrow{ab}\\) is parallel to \\(overleftrightarrow{cd}\\)

c. \\(overleftrightarrow{ab}\\) will intersect \\(overleftrightarrow{cd}\\)

d. t is perpendicular to \\(overleftrightarrow{ab}\\)

Explanation:

Step1: Analyze angle relationships for \( \overleftrightarrow{AB} \) and \( \overleftrightarrow{CD} \)

First, find the measure of \( \angle 4 \) and its corresponding angle with \( \angle 6 \). The angle adjacent to \( 135^\circ \) ( \( \angle 4 \)) and \( \angle 2 \) are supplementary? Wait, no, let's check the transversal \( t \) and the lines \( AB \) and \( CD \). The angle given for \( CD \) is \( 45^\circ \) ( \( \angle 6 \) adjacent angle). Let's find the consecutive interior angles or corresponding angles.

For \( \overleftrightarrow{AB} \) and \( \overleftrightarrow{CD} \) with transversal \( t \): The angle \( \angle 4 \) is \( 180^\circ - 135^\circ = 45^\circ \)? Wait, no, the angle marked \( 135^\circ \) and \( \angle 4 \) are adjacent, so \( \angle 4 = 180^\circ - 135^\circ = 45^\circ \)? Wait, no, actually, the angle next to \( 135^\circ \) (let's see, the angle between \( AB \) and \( t \) at the intersection: \( \angle 4 \) and the \( 135^\circ \) angle—wait, maybe it's a same - side interior angle or corresponding angle. Wait, the angle at \( CD \) with the transversal is \( 45^\circ \) (the angle marked \( 45^\circ \) near \( \angle 6 \)). Let's check the corresponding angles.

The angle \( \angle 4 \): Let's see, the angle adjacent to \( 135^\circ \) (if \( 135^\circ \) is \( \angle \) between \( AB \) and a line, wait, maybe \( \angle 4 \) and the \( 45^\circ \) angle are corresponding angles. Wait, \( \angle 4 \) and the angle at \( CD \) (the \( 45^\circ \) angle) — if we calculate \( \angle 4 \): the angle marked \( 135^\circ \) and \( \angle 4 \) are supplementary? Wait, no, the angle between \( AB \) and \( t \): \( \angle 2 \) and \( \angle 4 \) are vertical angles? Wait, maybe a better approach: For two lines to be parallel, the corresponding angles should be equal, alternate interior angles equal, or same - side interior angles supplementary.

Let's look at \( \overleftrightarrow{AB} \) and \( \overleftrightarrow{CD} \). The angle at \( CD \) (the \( 45^\circ \) angle) and the angle at \( AB \): Let's find the angle between \( AB \) and \( t \). The angle marked \( 135^\circ \) and the angle adjacent to it (let's say \( \angle 4 \)): \( \angle 4 = 180^\circ - 135^\circ = 45^\circ \). Now, the angle at \( CD \) (the \( 45^\circ \) angle) is equal to \( \angle 4 \) (corresponding angles). So by the corresponding angles postulate, if corresponding angles are equal, the lines are parallel. So \( \overleftrightarrow{AB} \parallel \overleftrightarrow{CD} \).

Now let's check other options:

• Option A: For \( t \) and \( \overleftrightarrow{CD} \), we don't have enough info to say they are parallel. The angle at \( CD \) with \( t \) is \( 45^\circ \), and the angle at \( t \) with \( AB \) — let's see, the angle between \( t \) and \( AB \): \( \angle 2 \) and \( \angle 4 \) — if \( \angle 4 = 45^\circ \), then \( \angle 2 = 45^\circ \)? Wait, no, maybe not. So we can't say \( t \parallel \overleftrightarrow{CD} \).

• Option D: For \( t \) and \( \overleftrightarrow{AB} \), the angle between them: if \( \angle 2 \) is, say, \( 45^\circ \) (from earlier), then \( t \) and \( AB \) are not perpendicular (perpendicular would be \( 90^\circ \)).

• Option C: Since \( \overleftrightarrow{AB} \parallel \overleftrightarrow{CD} \), they won't intersect (parallel lines never intersect). So option C is wrong.

So the correct option is B.

Answer:

B. \( \overleftrightarrow{AB} \) is parallel to \( \overleftrightarrow{CD} \)