Question

use the given information and the diagram to determine which lines, if any, must be parallel. if no lines are parallel, write none.
a. \\( \angle 1 \cong \angle 7 \\) ______
b. \\( \angle 6 \cong \angle 7 \\) ______
c. \\( \angle 2 \\) and \\( \angle 3 \\) are supplementary ______
d. \\( \angle 1 \cong \angle 6 \\) ______
e. \\( \angle 4 \cong \angle 7 \\), and \\( \angle 4 \\) and \\( \angle 7 \\) are not right angles ______
word bank
\\( m\parallel n \\) \\( m\parallel n \\) \\( t\parallel u \\) \\( m\parallel n \\) \\( t\parallel u \\) \\( t\parallel u \\) \\( t\parallel u \\) none none \\( m\parallel n \\) \\( t\parallel u \\) none \\( m\parallel n \\) none none
blank 1:
blank 2:
blank 3:
blank 4:
blank 5:

Explanation:

Step1: Analyze \(\angle1\cong\angle7\)

\(\angle1\) and \(\angle7\) are not corresponding, alternate - interior or alternate - exterior angles for any pair of lines. So, no lines are parallel.

Step2: Analyze \(\angle6\cong\angle7\)

\(\angle6\) and \(\angle7\) are corresponding angles. If two corresponding angles are congruent, then the lines \(t\) and \(u\) (the transversals for which \(\angle6\) and \(\angle7\) are formed with respect to lines \(m\) and \(n\)) are parallel.

Step3: Analyze \(\angle2\) and \(\angle3\) are supplementary

\(\angle2\) and \(\angle3\) are same - side interior angles. If same - side interior angles are supplementary, then the lines \(t\) and \(u\) are parallel.

Step4: Analyze \(\angle1\cong\angle6\)

\(\angle1\) and \(\angle6\) are not corresponding, alternate - interior or alternate - exterior angles for any pair of lines. So, no lines are parallel.

Step5: Analyze \(\angle4\cong\angle7\), and \(\angle4\) and \(\angle7\) are not right angles

\(\angle4\) and \(\angle7\) are corresponding angles. If two corresponding angles are congruent, then the lines \(m\) and \(n\) are parallel.

Answer:

Blank 1: none
Blank 2: \(t\parallel u\)
Blank 3: \(t\parallel u\)
Blank 4: none
Blank 5: \(m\parallel n\)