Understanding Slope of Parallel and Perpendicular Lines in Geometry

Since the theorem consists of a biconditional statement, the proof consists of two parts.

If two distinct non-vertical lines are parallel, then their slopes are equal.
If the slopes of two distinct non-vertical lines are equal, then the lines are parallel.

Part $1$

Consider two distinct non-vertical parallel lines in a coordinate plane. Their equations can be written in slope-intercept form.

ℓ_{1} : y = m_{1} x + b_{1} ℓ_{2} : y = m_{2} x + b_{2}

Suppose that the slopes of the lines are not the same. The system of equations formed by the equations above can be solved by using the Substitution Method.

{y = m_{1} x + b_{1} y = m_{2} x + b_{2} (I) (II)

Substitute

$(I):$ $y = m_{2} x + b_{2}$

{m_{2} x + b_{2} = m_{1} x + b_{1} y = m_{2} x + b_{2}

(I):

Solve for

x

SubEqn

$(I):$ $LHS - b_{2} = RHS - b_{2}$

{m_{2} x = m_{1} x + b_{1} - b_{2} y = m_{2} x + b_{2}

SubEqn

$(I):$ $LHS - m_{1} x = RHS - m_{1} x$

{m_{2} x - m_{1} x = b_{1} - b_{2} y = m_{2} x + b_{2}

FactorOut

$(I):$ Factor out $x$

{(m_{2} - m_{1}) x = b_{1} - b_{2} y = m_{2} x + b_{2}

DivEqn

$(II):$ $LHS / (m_{2} - m_{1}) = RHS / (m_{2} - m_{1})$

{x = \frac{b _{1} - b _{2}}{m _{2} - m _{1}} y = m_{2} x + b_{2}

Since

m_{1} \neq = m_{2},

the expression

\frac{b _{1} - b _{2}}{m _{2} - m _{1}}

is not undefined because its denominator cannot be zero. To find the value of the

y -

variable,

\frac{b _{1} - b _{2}}{m _{2} - m _{1}}

can be substituted for

x

in Equation (II).

{x = \frac{b _{1} - b _{2}}{m _{2} - m _{1}} y = m_{2} x + b_{2}

Substitute

$(II):$ $x = \frac{b _{1} - b _{2}}{m _{2} - m _{1}}$

{x = \frac{b _{1} - b _{2}}{m _{2} - m _{1}} y = m_{2} (\frac{b _{1} - b _{2}}{m _{2} - m _{1}}) + b_{2}

The solution to the system formed by the equations was found. Since there is a solution for the system, the lines

ℓ_{1}

and

ℓ_{2}

intersect each other. However, this contradicts the fact that the lines are parallel. Therefore, the assumption that the slopes are different is false. Consequently, the slopes of the lines are equal.

$ℓ_{1} ∥ ℓ_{2} \Rightarrow m_{1} = m_{2}$

Part $2$

Now, consider two distinct non-vertical lines

ℓ_{1}

and

ℓ_{2}

that have the same slope

m .

Their equations can be written in slope-intercept form.

ℓ_{1} : y = m x + b_{1} ℓ_{2} : y = m x + b_{2}

Since these lies are distinct,

b_{1}

and

b_{2}

are not equal. With this information in mind, suppose that the lines intersect. Solving the system of equations will give the point of intersection. The Substitution Method will be used again.

{y = m x + b_{1} y = m x + b_{2} (I) (II)

Substitute

$(I):$ $y = m x + b_{2}$

{m x + b_{2} = m x + b_{1} y = m x + b_{2}

SubEqn

$(I):$ $LHS - m x = RHS - m x$

{b_{2} = b_{1} y = m x + b_{2}

The obtained result contradicts the fact that

b_{1}

and

b_{2}

are different. Therefore, there is no point of intersection between the lines

ℓ_{1}

and

ℓ_{2} .

This means that they are parallel lines.

$m_{1} = m_{2} \Rightarrow ℓ_{1} ∥ ℓ_{2}$

Both directions of the biconditional statement have been proved.

$ℓ_{1} ∥ ℓ_{2} \Leftrightarrow m_{1} = m_{2}$

Equation	Slope	$y -$ Intercept
$y = - 5 x + (- 3)$	$- 5$	$- 3$
$y = - 5 x + (- 1)$	$- 5$	$- 1$

Equation

Slope

y -

Intercept

y = - 5 x + (- 3)

- 5

- 3

y = - 5 x + (- 1)

- 5

- 1

Condition	Conclusion	Example
The lines of the system have the same slope and the same $y -$ intercept.	The lines are coincidental. This means that there are infinitely many points of intersection. Therefore, the system has infinitely many solutions.	${y = 3 x + 1 y = 3 x + 1$
The lines of the system have the same slope but different $y -$ intercept.	The lines are parallel. This means that there is not a point of intersection. Therefore, the system has no solution.	${y = 3 x + 1 y = 3 x + 5$
The lines of the system have different slopes.	The lines are neither parallel nor coincidental. This means that there is one point of intersection. Therefore, the system has one solution.	${y = 3 x + 1 y = - 2 x + 5$

Condition

Conclusion

Example

The lines of the system have the same slope and the same

y -

intercept.

The lines are coincidental. This means that there are infinitely many points of intersection. Therefore, the system has infinitely many solutions.

{y = 3 x + 1 y = 3 x + 1

The lines of the system have the same slope but different

y -

intercept.

The lines are parallel. This means that there is not a point of intersection. Therefore, the system has no solution.

{y = 3 x + 1 y = 3 x + 5

The lines of the system have different slopes.

The lines are neither parallel nor coincidental. This means that there is one point of intersection. Therefore, the system has one solution.

{y = 3 x + 1 y = - 2 x + 5

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Explore

Investigating the Slopes of Parallel Lines

Discussion

Slopes of Parallel Lines Theorem

Proof

Part $1$

Part $2$

Pop Quiz

Practice Finding the Slope of Parallel lines

Example

Identifying the Equation of a Parallel Line

Hint

Solution

Example

Systems of Equations With Parallel or Overlapping Lines

Answer

Hint

Solution

Discussion

Properties of Different Systems of Equations

Discussion

Properties of Perpendicular Lines

Example

Identifying the Equation of a Perpendicular Line

Hint

Solution

Example

Finding a Perpendicular Line When Given a Graph and a Point

Hint

Solution

Closure

Using Properties of Parallel Lines To Classify Parallelograms

Answer

Hint

Solution

Answer

Hint

Solution

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Proof

Part 1

Part 2

Hint

Solution

Answer

Hint

Solution

Hint

Solution

Hint

Solution

Answer

Hint

Solution

Answer

Hint

Solution

Part $1$

Part $2$