Question

1. which line is parallel to the line $2x - 5y = -14$?

a. $y = -\frac{5}{2}x + 3$ b. $y = -\frac{2}{5}x + 6$ c. $y = \frac{2}{5}x - 8$ d. $y = \frac{5}{2}x - 4$

1. which line is perpendicular to the line $2x + 5y = -14$?

a. $y = -\frac{5}{2}x + 3$ b. $y = -\frac{2}{5}x + 6$ c. $y = \frac{2}{5}x - 8$ d. $y = \frac{5}{2}x - 4$

1. write an equation of a line parallel to the given line through $(8, 2)$ in slope intercept form. $y = 3x + 2$
2. write an equation of a line perpendicular to the given line through $(6, 10)$ in slope intercept form. $y = 3x + 2$
3. use the diagram below, determine which lines, if any, can be proved parallel given the angle relationship. give the converse to justify your answer.

a. corresponding angles converse
b. alternate interior angles converse
c. alternate exterior angles converse
d. consecutive interior angles converse
e. consecutive exterior angles converse
f. no lines are parallel.

angle relationship	parallel lines	converse
b. $\angle 5 \cong \angle 7$	___		___
c. $m\angle 4 + m\angle 16 = 180$	___		___
d. $\angle 1 \cong \angle 22$	___		___
e. $m\angle 7 + m\angle 21 = 180$	___		___
f. $m\angle 14 + m\angle 18 = 180$	___		___

1. given: $\angle 1$ and $\angle 2$ are supplementary

given: $\angle 4 \cong \angle 5$
prove: $q \parallel r$

statement	reason
2.	2.
3.	3.
4.	4.
5.	5.
6.	6.
7.	7.

Explanation:

Step1: Find the slope of the line \(2x - 5y=-14\)

Rewrite the equation in slope - intercept form \(y = mx + b\) (where \(m\) is the slope).

$$
LATEXBLOCK0
$$

The slope \(m_1=\frac{2}{5}\).

Step2: Determine the parallel line

Parallel lines have the same slope.
For option C: \(y=\frac{2}{5}x - 8\), the slope \(m_2=\frac{2}{5}\). Since \(m_1 = m_2\), the line \(y=\frac{2}{5}x - 8\) is parallel to \(2x - 5y=-14\).

Step3: Find the slope of the line \(2x + 5y=-14\)

Rewrite in slope - intercept form:

$$
LATEXBLOCK1
$$

The slope \(m_3=-\frac{2}{5}\).

Step4: Determine the perpendicular line

If two lines with slopes \(m\) and \(m'\) are perpendicular, then \(m\times m'=- 1\). Let the slope of the perpendicular line be \(m_4\).

$$-\frac{2}{5}\times m_4=-1\Rightarrow m_4=\frac{5}{2}$$

For option D: \(y = \frac{5}{2}x-4\), the slope \(m_4=\frac{5}{2}\). So the line \(y=\frac{5}{2}x - 4\) is perpendicular to \(2x + 5y=-14\).

Answer:

1. C. \(y=\frac{2}{5}x - 8\)
2. D. \(y=\frac{5}{2}x - 4\)