Question

1. write the slope-intercept form of the equation of the line passing through the point (3, 3) and perpendicular to the line $y=\frac{1}{3}x+4$.

a $y=-3x + 12$
b $y=\frac{5}{3}x - 14$
c $y=-3x + 14$
d $y=3x - 12$
28

1. explain how you know whether two lines are parallel or perpendicular, using their slopes.

i know when 2 lines are parallel 29
becnuse they have the same slope and perpendicular
slopes are opposite reciprocal and the product equal -1

(omit) dont do
complete the proof:
given: $\angle1\cong\angle2$
prove: $\ell\parallel m$

statements reasons

1. 1.
2. 2.
3. 3.
4. 4.
5. 5.

Explanation:

Question 28

Step1: Find perpendicular slope

The slope of $y=\frac{1}{3}x+4$ is $m_1=\frac{1}{3}$. Perpendicular slope $m_2 = -3$ (negative reciprocal).

Step2: Substitute point into slope-intercept

Use point $(3,3)$ and $y=mx+b$:
$3 = -3(3) + b$

Step3: Solve for intercept $b$

$3 = -9 + b \implies b = 3 + 9 = 12$

Step4: Write final equation

Combine slope and intercept: $y=-3x+12$

Two lines are parallel if their slopes are exactly equal (they have the same steepness and direction). Two lines are perpendicular if the product of their slopes equals $-1$; this means one slope is the negative reciprocal of the other (e.g., if one slope is $\frac{a}{b}$, the perpendicular slope is $-\frac{b}{a}$).

Step1: State given information

Statement 1: $\angle 1 \cong \angle 2$
Reason 1: Given

Step2: Identify vertical angles

Statement 2: $\angle 1 \cong \angle 3$
Reason 2: Vertical Angles Theorem

Step3: Apply transitive property

Statement 3: $\angle 2 \cong \angle 3$
Reason 3: Transitive Property of Congruence

Step4: Use parallel lines criterion

Statement 4: $\ell \parallel m$
Reason 4: Corresponding Angles Converse (if corresponding angles are congruent, lines are parallel)

Answer:

[A] $y = -3x + 12$

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Question 29