Question

for exercises 2 and 3, use the figure below right. question 2 1 pts which statement proves that ( a parallel b )? ( angle 8 ) is supplementary to ( angle 12 ). ( angle 1 cong angle 6 ) ( angle 5 cong angle 13 ) ( angle 10 ) is supplementary to ( angle 11 ).

Explanation:

Step1: Recall the properties of parallel lines

When two lines are parallel, certain angle relationships hold. For example, if two lines are cut by a transversal, corresponding angles are congruent, alternate - interior angles are congruent, and consecutive interior angles are supplementary.

Step2: Analyze each option

• Option 1: \(\angle8\) is supplementary to \(\angle12\)

\(\angle8\) and \(\angle12\) are vertical angles. Vertical angles are congruent (\(\angle8\cong\angle12\)), not supplementary (unless they are right angles). This does not prove \(a\parallel b\).

• Option 2: \(\angle1\cong\angle6\)

\(\angle1\) and \(\angle6\) are not in a position (corresponding, alternate - interior, etc.) that would imply \(a\parallel b\) based on angle - line parallelism theorems.

• Option 3: \(\angle5\cong\angle13\)

\(\angle5\) and \(\angle13\) are corresponding angles. If two lines are cut by a transversal and the corresponding angles are congruent, then the two lines are parallel. Here, the transversal is line \(y\) (assuming the lines \(a\) and \(b\) are the ones we want to check for parallelism).

• Option 4: \(\angle10\) is supplementary to \(\angle11\)

\(\angle10\) and \(\angle11\) are adjacent angles (they form a linear pair). A linear pair of angles is always supplementary (\(\angle10+\angle11 = 180^{\circ}\) by the definition of a linear pair), and this has nothing to do with the parallelism of \(a\) and \(b\).

Answer:

\(\angle5\cong\angle13\) (the third option)