Question

items 11 - 15. lines a, b, c, and d intersect as shown. 11. which pairs of lines are parallel? a a and b b a and c c c and d d b and d 12. what is x? 13. what is y? a 37 b 84 c 88 d 92 14. what is z? a 84 b 92 c 96 d 109 15. if the slope of line d is given, the slope of which other line is known? 16. what is the equation of a line that is perpendicular to the line y = 2x + 1 and passes through the point (4, 6)? a y=-1/2x + 6 b y = 2x + 6 c y=-1/2x + 8 d y = 2x + 8 17. what is the slope of a line parallel to the line y = 3x - 4? items 18 - 20. part of a city map is shown. 18. which street is perpendicular to 1st ave? a 2nd ave b main road c central ave d d street 19. a city planner wants to build a road parallel to 2nd ave. what is the slope of the new road? 20. if m∠5 = x, which angles have a measure of 180 - x? select all that apply. a ∠2 b ∠3 c ∠7 d ∠11

Explanation:

Step1: Recall parallel - line property

Parallel lines have equal slopes. For perpendicular lines, the product of their slopes is - 1. The equation of a line is in the form \(y = mx + b\), where \(m\) is the slope.

Step2: Solve question 11

Lines \(b\) and \(d\) are parallel because the corresponding angles formed by a transversal are equal (both have an angle of \(92^{\circ}\)). So the answer to 11 is D.

Step3: Solve question 12

The angle adjacent to \(109^{\circ}\) and \(x\) are vertical - angles. The angle adjacent to \(109^{\circ}\) is \(180 - 109=71^{\circ}\). So \(x = 71^{\circ}\).

Step4: Solve question 13

The sum of angles in a triangle formed by the intersection of the lines is \(180^{\circ}\). One angle of the triangle is \(37^{\circ}\) and another is \(55^{\circ}\). So \(y=180-(37 + 55)=88^{\circ}\), and the answer is C.

Step5: Solve question 14

\(z\) and the angle adjacent to \(92^{\circ}\) (which is \(88^{\circ}\)) are supplementary. So \(z = 92^{\circ}\), and the answer is B.

Step6: Solve question 15

Since \(b\) and \(d\) are parallel, if the slope of line \(d\) is given, the slope of line \(b\) is known.

Step7: Solve question 16

The slope of the line \(y = 2x+1\) is \(m_1 = 2\). The slope of a line perpendicular to it, \(m_2\), satisfies \(m_1m_2=-1\), so \(m_2=-\frac{1}{2}\). Using the point - slope form \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(4,6)\) and \(m =-\frac{1}{2}\), we have \(y - 6=-\frac{1}{2}(x - 4)\), which simplifies to \(y=-\frac{1}{2}x + 8\). The answer is C.

Step8: Solve question 17

Parallel lines have the same slope. The slope of the line \(y = 3x-4\) is \(3\), so the slope of a line parallel to it is also \(3\).

Step9: Solve question 18

2nd Ave is perpendicular to 1st Ave. The answer is A.

Step10: Solve question 19

The slope of 2nd Ave is \(0\) (it is a horizontal line). The slope of a line parallel to it is also \(0\).

Step11: Solve question 20

\(\angle5\) and \(\angle7\) are supplementary (linear - pair), and \(\angle5\) and \(\angle11\) are also supplementary. So the angles with measure \(180 - x\) are \(\angle7\) and \(\angle11\), and the answers are C and D.

Answer:

1. D. \(b\) and \(d\)
2. \(71^{\circ}\)
3. C. \(88\)
4. B. \(92\)
5. Line \(b\)
6. C. \(y =-\frac{1}{2}x + 8\)
7. \(3\)
8. A. 2nd Ave
9. \(0\)
10. C. \(\angle7\), D. \(\angle11\)