Question

for items 10 - 14, lines a, b, c, and d intersect as shown.

1. which pair of lines are parallel?

a a and b
c c and d
b a and c
d b and d

1. what is the value of x?
2. what is the value of y?

a 42
c 88
b 85
d 95

1. what is the value of z?

a 88
c 95
b 92
d 124

1. two lines intersect to form ∠abc. one step in constructing a line parallel to (overrightarrow{bc}) through point a is to construct an angle with vertex a. how should this angle be related to ∠abc?

a the angles should be congruent.
b the angles should be complementary.
c the angles should be supplementary.
d the angles should have different measures.
for items 15 - 18, use the part of a city map shown.

1. which street is parallel to 1st ave?

a 2nd ave
c central ave
b main road
d d street

1. a city planner wants to build a road perpendicular to d street. what is the slope of the new road?
2. what is the equation of a line that is parallel to the line (y = 2x + 7) and passes through the point ((-2, 4))?

a (y = -\frac{1}{2}x + 3)
b (y = 2x + 4)
c (y = -\frac{1}{2}x - 2)
d (y = 2x + 8)

1. what is the equation of a line that is perpendicular to the line (y = -\frac{1}{4}x - 1) and passes through the point ((3, 7))?

a (y = -4x + 19)
b (y = 4x - 5)
c (y = \frac{1}{4}x + \frac{25}{4})
d (y = -4x + 31)

Explanation:

Step1: Find parallel lines

For question 10, use the property that parallel lines have equal corresponding angles. Lines \(a\) and \(c\) have equal corresponding angles (both have an angle of \(88^{\circ}\) with the transversal).

Step2: Solve for \(x\)

For question 11, \(x = 180 - 124=56\) (linear - pair angles).

Step3: Solve for \(y\)

For question 12, \(y = 180-(88 + 42)=50\) (sum of angles in a triangle is \(180^{\circ}\)). Wait, no. Wait, looking at the angle relationships: If we consider the angle adjacent to \(124^{\circ}\) is \(180 - 124=56^{\circ}\). Then using the triangle - like angle sum (assuming a transversal - related angle sum). Wait, another approach: Since \(a\parallel c\), the angle adjacent to \(x\) is \(124^{\circ}\) (corresponding to the \(124^{\circ}\) angle). Then \(x = 180 - 124 = 56\). For \(y\), using the fact that the sum of angles around a point (related to parallel lines \(a\) and \(c\)): \(y=180-(88 + 42)=50\) (error in options? Wait, no. Wait, if we consider the angle adjacent to \(y\) is \(85^{\circ}\) (using the property of parallel lines and transversals). Wait, no. Wait, looking at the first figure: The angle adjacent to \(y\) is \(85^{\circ}\) (because of the vertical - angle and parallel - line relationships). So \(y = 180 - 95=85\) (using linear - pair with an angle that is \(95^{\circ}\) (since \(180-(88 + 42)=50\) was wrong. Wait, another way: The angle adjacent to \(y\) is \(95^{\circ}\) (because \(180 - 85=95\) is wrong. Wait, using the property of parallel lines \(a\) and \(c\): The angle \(y\) and the \(85^{\circ}\) - related angle (through vertical angles and parallel - line angle relationships) gives \(y = 85\).

Step4: Solve for \(z\)

For question 13, \(z=180 - 88=92\) (linear - pair with the \(88^{\circ}\) angle).

Step5: Parallel - line construction (question 14)

When constructing a line parallel to \(\overrightarrow{BC}\) through \(A\) using the angle - construction method (corresponding - angles postulate), the angles should be congruent.

Step6: Parallel streets (question 15)

For question 15, by observing the map (assuming grid - like parallelism where 2nd Ave is parallel to 1st Ave).

Step7: Slope of perpendicular line (question 16)

If \(D\) Street is vertical (assuming from the map, if it's a vertical line, its slope is undefined. But if we assume a non - vertical case: Wait, if we assume \(D\) Street is horizontal (wrong). Wait, if we use the general slope formula. But if \(D\) Street is vertical (in a coordinate - grid - like map), the slope of a line perpendicular to a vertical line is \(0\). But if we assume \(D\) Street has a slope \(m\), and for a perpendicular line \(m_1\times m=- 1\). But if we assume \(D\) Street is vertical (in a typical map - grid, vertical lines have undefined slope, and horizontal lines (perpendicular) have slope \(0\)).

Step8: Equation of parallel line (question 17)

For question 17, the slope of \(y = 2x+7\) is \(m = 2\). Using the point - slope form \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(-2,4)\), \(y - 4=2(x + 2)\), \(y=2x+8\).

Step9: Equation of perpendicular line (question 18)

For question 18, the slope of \(y=-\frac{1}{4}x - 1\) is \(m_1=-\frac{1}{4}\). The slope of the perpendicular line \(m_2 = 4\). Using the point - slope form \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(3,7)\), \(y - 7=4(x - 3)\), \(y=4x-5\).

Answer:

1. B. \(a\) and \(c\)
2. \(x = 56\)
3. B. \(85\)
4. B. \(92\)
5. A. The angles should be congruent.
6. A. 2nd Ave
7. \(0\) (assuming \(D\) Street is vertical)
8. D. \(y = 2x+8\)
9. B. \(y = 4x - 5\)