Question

which of the following statements would guarantee that lines a and b are parallel? image of intersecting lines a, b and transversal t with angles 1-8 labeled options: - ( mangle 5 = 115^circ ) and ( mangle 6 = 65^circ ) - ( mangle 4 = 110^circ ) and ( mangle 6 = 110^circ ) - ( mangle 1 = 105^circ ) and ( mangle 4 = 105^circ ) - ( mangle 1 = 110^circ ) and ( mangle 5 = 110^circ ) - ( mangle 3 = 115^circ ) and ( mangle 4 = 65^circ )

Explanation:

Step1: Recall Parallel Line Theorems

To determine if lines \(a\) and \(b\) are parallel, we use angle - related theorems (like corresponding angles, alternate interior angles, consecutive interior angles, or supplementary angles on the same - side of the transversal). If corresponding angles are equal, alternate interior angles are equal, or consecutive interior angles are supplementary, then the lines are parallel.

Step2: Analyze Each Option

• Option 1: \(m\angle5 = 115^{\circ}\) and \(m\angle6 = 65^{\circ}\)

\(\angle5\) and \(\angle6\) are adjacent angles forming a linear pair. Their sum is \(115^{\circ}+65^{\circ}=180^{\circ}\), which is always true for a linear pair, but this does not give information about the relationship between lines \(a\) and \(b\) with respect to the transversal \(t\) to prove parallelism.

• Option 2: \(m\angle4 = 110^{\circ}\) and \(m\angle6 = 110^{\circ}\)

\(\angle4\) and \(\angle6\) are alternate interior angles. If alternate interior angles are equal, then the lines cut by the transversal are parallel. Since \(m\angle4=m\angle6 = 110^{\circ}\), by the Alternate Interior Angles Theorem, lines \(a\) and \(b\) are parallel.

• Option 3: \(m\angle1 = 105^{\circ}\) and \(m\angle4 = 105^{\circ}\)

\(\angle1\) and \(\angle4\) are vertical angles (for \(\angle1\) and \(\angle3\) are vertical, \(\angle3\) and \(\angle4\) are adjacent, so \(\angle1\) and \(\angle4\) are not alternate interior, corresponding or consecutive interior angles in a way that proves parallelism). Vertical angles are always equal, but this does not imply \(a\parallel b\).

• Option 4: \(m\angle1 = 110^{\circ}\) and \(m\angle5 = 110^{\circ}\)

\(\angle1\) and \(\angle5\) are corresponding angles? No, \(\angle1\) and \(\angle5\) are not in the correct position for corresponding angles. Corresponding angles would be \(\angle1\) and \(\angle5\) only if the lines are parallel, but we are trying to prove parallelism. Also, just having equal measure does not fit the corresponding angles theorem structure here (since their position is not correct for corresponding angles when considering transversal \(t\) cutting \(a\) and \(b\)).

• Option 5: \(m\angle3 = 115^{\circ}\) and \(m\angle4 = 65^{\circ}\)

\(\angle3\) and \(\angle4\) are adjacent angles forming a linear pair. Their sum is \(115^{\circ}+65^{\circ}=180^{\circ}\), which is always true for a linear pair, and this does not give information about the relationship between lines \(a\) and \(b\) to prove parallelism.

Answer:

\(m\angle4 = 110^{\circ}\) and \(m\angle6 = 110^{\circ}\) (the second option)