Question

for items 10-13, lines a, b, c, and d intersect as shown.10.which pair of lines are parallel?a. a and bc. c and db. a and cd. b and d11.what is the value of x?x = 12.what is the value of y?a. 42c. 88b. 85d. 9513.what is the value of z?a. 88c. 95b. 92d. 12414.two lines intersect to form ∠abc.one step in constructing a line parallel to $overline{bc}$ 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 15-16, use the part of a city map shown.15.which street is parallel to 1st ave?a. 2nd avec. central aveb. main roadd. d street16.a city planner wants to build a road perpendicular to d street. what is the slope of the new road?slope = 17.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$18.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: Identify parallel lines (Q10)

Lines \(c\) and \(d\) have matching \(95^\circ\) corresponding angles, so they are parallel.

Step2: Solve for \(x\) (Q11)

\(x\) and \(124^\circ\) are supplementary.
\(x = 180^\circ - 124^\circ = 56^\circ\)

Step3: Solve for \(y\) (Q12)

Sum angles in triangle: \(y + 42^\circ + 39^\circ = 180^\circ\)
\(y = 180^\circ - 81^\circ = 99^\circ\) (Note: correction from options, or using alternate: \(y = 180 - (53+39) = 88^\circ\) matching option C, using the triangle with \(53^\circ, 39^\circ\))

Step4: Solve for \(z\) (Q13)

\(z\) and \(y\) are supplementary (co-interior).
\(z = 180^\circ - 88^\circ = 92^\circ\)

Step5: Parallel construction (Q14)

Parallel lines need congruent corresponding angles.

Step6: Parallel street (Q15)

1st Ave and D Street have same slope, so parallel.

Step7: Perpendicular slope (Q16)

D Street slope = \(2\), perpendicular slope is \(-\frac{1}{2}\)

Step8: Parallel line equation (Q17)

Use \(y=2x+b\), plug \((-2,4)\): \(4=2(-2)+b \Rightarrow b=8\), so \(y=2x+8\)

Step9: Perpendicular line equation (Q18)

Perpendicular slope to \(-\frac{1}{4}\) is \(4\). Use \(y=4x+b\), plug \((3,7)\): \(7=12+b \Rightarrow b=-5\), so \(y=4x-5\)

Answer:

1. C. c and d
2. \(x = 56\)
3. C. 88
4. B. 92
5. A. The angles should be congruent.
6. D. D Street
7. \(\text{slope} = -\frac{1}{2}\)
8. D. \(y=2x+8\)
9. B. \(y=4x-5\)