Question

1. in this figure, lines a and b are intersected by line t.

which of these statements proves that lines a and b are parallel?
a. ∠2 and ∠3 are complementary
b. ∠1 = ∠3
c. ∠1 = ∠2
d. ∠1 and ∠2 are supplementary

1. in the figure below, ∠1 ≅ ∠2.

what reason can be used to prove that m || n?
a. if a pair of vertical angles is congruent, then the lines forming them are parallel.
b. if a pair of corresponding angles is supplementary then the lines forming them are parallel.
c. if a pair of vertical angles is supplementary, then the lines forming them are parallel.
d. if a pair of corresponding angles is congruent, then the lines forming them are parallel.

Explanation:

Question 3

Step1: Recall parallel line theorems

When two lines are cut by a transversal, if corresponding angles are equal, the lines are parallel.

Step2: Analyze each option

• Option A: Complementary angles ($\angle2+\angle3 = 90^{\circ}$) do not prove parallel lines.
• Option B: $\angle1$ and $\angle3$ are corresponding angles. If $\angle1=\angle3$, by the corresponding - angles postulate, lines \(a\) and \(b\) are parallel.
• Option C: $\angle1$ and $\angle2$ are adjacent angles (linear pair: $\angle1+\angle2 = 180^{\circ}$), equality ($\angle1=\angle2$ implies each is \(90^{\circ}\)) does not prove \(a\parallel b\).
• Option D: $\angle1$ and $\angle2$ being supplementary ($\angle1+\angle2 = 180^{\circ}$) is a property of a linear pair, not a parallel - line proof.

Step1: Recall parallel line theorems

When two lines are cut by a transversal, if corresponding angles are congruent, the lines are parallel.

Step2: Analyze each option

• Option A: Vertical angles are formed by two intersecting lines. $\angle1$ and $\angle2$ are not vertical angles (vertical angles are formed by the intersection of the same two lines).
• Option B: Supplementary corresponding angles ($\angle1+\angle2 = 180^{\circ}$) is not a standard parallel - line proof (standard is congruent corresponding angles).
• Option C: $\angle1$ and $\angle2$ are not vertical angles, and supplementary vertical angles ($\angle1+\angle2 = 180^{\circ}$, vertical angles are equal, so this would imply each is \(90^{\circ}\)) is not relevant.
• Option D: $\angle1$ and $\angle2$ are corresponding angles. If $\angle1\cong\angle2$ (congruent), by the corresponding - angles postulate, \(m\parallel n\).

Answer:

B. $\angle1=\angle3$

Question 4