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 6 \cong \angle 5 \\)
b. \\( \angle 5 \cong \angle 7 \\)
c. \\( \angle 2 \\) and \\( \angle 5 \\) are supplementary
d. \\( \angle 4 \cong \angle 6 \\), and \\( \angle 4 \\) and \\( \angle 6 \\) are not right angles
e. \\( \angle 3 \cong \angle 5 \\)
word bank:
none t\\( \\| \\)u t\\( \\| \\)u none none m\\( \\| \\)n m\\( \\| \\)n none m\\( \\| \\)n t\\( \\| \\)u t\\( \\| \\)u none m\\( \\| \\)n t\\( \\| \\)u m\\( \\| \\)n
blank 1:
blank 2:
blank 3:
blank 4:
blank 5:

Explanation:

Step1: Analyze ∠6 ≅ ∠5

∠6 and ∠5 are not corresponding, alternate - interior, or alternate - exterior angles for any pair of lines. So, no lines are parallel.

Step2: Analyze ∠5 ≅ ∠7

∠5 and ∠7 are alternate - exterior angles. When alternate - exterior angles are congruent, the lines cut by the transversal are parallel. Here, the transversal is \(u\) and the lines are \(m\) and \(n\). So, \(m\parallel n\).

Step3: Analyze ∠2 and ∠5 are supplementary

∠2 and ∠5 are adjacent angles. Supplementary adjacent angles do not give information about parallel lines. So, no lines are parallel.

Step4: Analyze ∠4 ≅ ∠6 (not right angles)

∠4 and ∠6 are alternate - interior angles. When alternate - interior angles are congruent, the lines cut by the transversal are parallel. Here, the transversal is \(t\) and the lines are \(m\) and \(n\). So, \(m\parallel n\).

Step5: Analyze ∠3 ≅ ∠5

∠3 and ∠5 are alternate - interior angles. When alternate - interior angles are congruent, the lines cut by the transversal are parallel. Here, the transversal is \(u\) and the lines are \(t\) and \(u\). So, \(t\parallel u\).

Answer:

a. none
b. \(m\parallel n\)
c. none
d. \(m\parallel n\)
e. \(t\parallel u\)