Parallel Lines

What Are Parallel Lines?

Two lines are parallel if they do not intersect, no matter how far they are extended. This can occur when:

The general equation of a line is $y = mx + c$ , where mm is the slope and $c$ is the $y$ -intercept.

Parallel lines have:

Equal slopes: If $l_1$ and $l_2$ are parallel, $m_1 = m_2$
Different intercepts: The $c$ values differ unless the lines are coincident (identical).

To determine if two lines are parallel, compare their slopes:

Lines represented as $y = m_1x + c_1$ and $y = m_2x + c_2$ are parallel if $m_1 = m_2$
Vertical lines are parallel if their equations are of the form $x = k_1$ and $x = k_2$ , with $k_1 \neq k_2$

If two parallel lines are cut by a transversal, the following angle pairs are equal:
1. Corresponding angles.
2. Alternate interior angles.
3. Alternate exterior angles. These angle properties are useful for proving lines are parallel using geometry.

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Problem: Determine if the lines $y = 2x + 3$ and $y = 2x - 5$ are parallel.

Solution:

Step 1: Identify the slopes:

$m_1 = 2$

$m_2 = 2$

Step 2: Since $m_1 = m_2$ , the lines are parallel.

Answer: Yes, the lines are parallel.

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Problem: Find the equation of a line parallel to $y = -3x + 7$ that passes through the point $(2, 1)$

Solution:

Step 1: Identify the slope of the given line: $m = -3$

Step 2: Use the point-slope formula:

y - y_1 = m(x - x_1)

Step 3: Substitute $m = -3$ , $x_1 = 2$ , and $y_1 = 1$ :

y - 1 = -3(x - 2)

Step 4: Simplify:

y = -3x + 6 + 1 = -3x + 7

Answer: The equation is $y = -3x + 7$

Definition: Parallel lines have the same slope and do not intersect.
Key Property: For parallel lines $y = m_1x + c_1$ and $y = m_2x + c_2$ , $m_1 = m_2$
Angle Properties: Corresponding, alternate interior, and alternate exterior angles are equal when parallel lines are intersected by a transversal.
Use the slope and point-slope formulas to solve parallel line problems.
Practice identifying and working with parallel lines in coordinate geometry to master this concept.