Question

1. in the figure shown, determine which lines, if any, are parallel. 13. given that a||b, solve for x. 15. in the map below, ny lane and va street are parallel and intersected by tx road and wi drive. what is the value of x?

Explanation:

Step1: Analyze parallel - line angle relationships for question 12

When two lines are parallel, corresponding angles are equal, alternate - interior angles are equal, and same - side interior angles are supplementary. For lines \(x\) and \(y\), and lines \(a\) and \(b\), we can see that the angle relationships satisfy the parallel - line criteria. For example, the pairs of \(75^{\circ}\) and \(105^{\circ}\) angles show supplementary same - side interior angles and equal corresponding and alternate - interior angles. So \(x\parallel y\) and \(b\parallel a\).

Step2: Solve for \(x\) in question 13

Since \(a\parallel b\), we can use the angle - sum property of a triangle formed by the transversals. The angle adjacent to the \(37^{\circ}\) angle on line \(a\) (vertically opposite) and the \(33^{\circ}\) angle and \(x\) form a triangle. The angle adjacent to \(37^{\circ}\) is also \(37^{\circ}\) (vertically opposite angles are equal). Using the fact that the sum of angles in a triangle is \(180^{\circ}\), we have \(x+37 + 33=180\). Then \(x=180-(37 + 33)=110\).

Step3: Solve for \(x\) in question 15

Since NY Lane and VA Street are parallel, we can use the angle - relationships formed by the transversals TX Road and WI Drive. The angle corresponding to the \(59^{\circ}\) angle on the other parallel line and the \(33^{\circ}\) angle and \(x\) are related. The angle corresponding to \(59^{\circ}\) is also \(59^{\circ}\) (corresponding angles). Then, using the fact that the sum of angles on a straight - line is \(180^{\circ}\), we consider the angles formed by the intersection of WI Drive and TX Road. We know that \(x + 59+33 = 180\), so \(x=180-(59 + 33)=88\).

Answer:

1. C. \(x\parallel y\) and \(b\parallel a\)
2. \(x = 110\)
3. \(x = 88\)