Question

use the diagram below to fill - in the blank given the information
choose which lines, if any, are parallel with the information given:
a) if \\( \angle 13 \cong \angle 3 \\) which two lines are parallel? reason:
b) if \\( \angle 11 \cong \angle 9 \\) which two lines are parallel? reason:
c) if \\( \angle 1+\angle 6 = 180 \\) which two lines are parallel? reason:
d) if \\( \angle 12 \cong \angle 6 \\) which two lines are parallel? reason:

Explanation:

Step1: Analyze part a)

When \(\angle13\cong\angle3\), these are alternate exterior angles. By the Alternate Exterior Angles Converse, if two lines are cut by a transversal and the alternate exterior angles are congruent, then the lines are parallel. The lines \(m\) and \(p\) are cut by transversal \(t\).

Step2: Analyze part b)

For \(\angle11\cong\angle9\), these are alternate interior angles. By the Alternate Interior Angles Converse, when two lines are cut by a transversal and the alternate interior angles are congruent, the lines are parallel. The lines \(t\) and \(w\) are cut by transversal \(v\).

Step3: Analyze part c)

Given \(\angle1+\angle6 = 180^{\circ}\), these are consecutive exterior angles. By the Consecutive Exterior Angles Converse, if two lines are cut by a transversal and the consecutive exterior angles are supplementary, then the lines are parallel. The lines \(m\) and \(p\) are cut by transversal \(t\).

Step4: Analyze part d)

When \(\angle12\cong\angle6\), these are corresponding angles. By the Corresponding Angles Converse, if two lines are cut by a transversal and the corresponding angles are congruent, then the lines are parallel. The lines \(t\) and \(w\) are cut by transversal \(v\).

Answer:

a) \(m\parallel p\), Reason: Alternate Exterior Angles Converse
b) \(t\parallel w\), Reason: Alternate Interior Angles Converse
c) \(m\parallel p\), Reason: Consecutive Exterior Angles Converse
d) \(t\parallel w\), Reason: Corresponding Angles Converse