Question

multiple choice
identify the choice that best completes the statement or answers the question.

1. what is the equation of a line perpendicular to the line $y = -\frac{1}{2}x + 3$

a. $y = -\frac{1}{2}x + \frac{1}{3}$
c. $y = \frac{1}{2}x - 8$
b. $y = 2x + 12$
d. $y = -2x + 4$
write an equation for the line that is parallel to the given line and passes through the given point.

1. $y = 5x + 10; (2, 14)$

a. $y = \frac{1}{5}x + 4$
c. $y = 5x - 68$
b. $y = -\frac{1}{5}x - 4$
d. $y = 5x + 4$

1. what is the relationship between $\angle3$ and $\angle5$

a. corresponding angles
b. alternate exterior angles
c. alternate interior angles
d. same-side interior angles

1. find $m\angle g$. the diagram is not to scale.

a. $34^\circ$
b. $110^\circ$
c. $104^\circ$
d. $146^\circ$

Explanation:

Step1: Find perpendicular slope

The slope of $y=-\frac{1}{2}x+3$ is $m_1=-\frac{1}{2}$. Perpendicular slope $m_2 = 2$ (since $m_1 \times m_2 = -1$).

Step2: Match to option

Only option b has slope 2.

Step3: Parallel line slope rule

Parallel lines have equal slopes. Slope of $y=5x+10$ is $m=5$.

Step4: Solve for y-intercept

Use point $(2,14)$: $14 = 5(2) + b$ → $14=10+b$ → $b=4$. Equation: $y=5x+4$, which is option d.

Step5: Identify angle relationship

∠3 and ∠5 lie between lines $m,n$, on opposite sides of transversal: alternate interior angles (option c).

Step6: Find vertical angle of 34°

The $34^\circ$ angle has a vertical angle on line $r$, which is also $34^\circ$.

Step7: Find supplementary angle to ∠G

∠G and this $34^\circ$ angle are same-side interior angles (supplementary): $m\angle G = 180^\circ - 34^\circ = 146^\circ$, which is option d.

Answer:

1. b. $y=2x + 12$
2. d. $y=5x + 4$
3. c. alternate interior angles
4. d. $146^\circ$