Understanding Slope of Parallel and Perpendicular Lines in Geometry

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

The theorems seen in this lesson can be used to identify quadrilaterals and some of their properties.

Determine whether quadrilateral $A B C D$ is a parallelogram. Explain the reasoning.

Answer

Yes, see solution.

Hint

Recall that a parallelogram is a quadrilateral with two pairs of parallel sides.

Solution

From the graph, it appears that

\overline{A B}

and

\overline{C D}

are parallel and that

\overline{B C}

and

\overline{D A}

are parallel. To prove this claim, start by finding the slope of each side.

As it can be seen, the slopes of the sides

\overline{A B}

and

\overline{D C}

are the same, as well as the slopes of

\overline{A B}

and

\overline{D C} .

Therefore, by the Slopes of Parallel Lines Theorem,

\overline{A B}

and

\overline{D C}

are parallel and

\overline{B C}

and

\overline{A D}

are parallel.

\overline{A B} ∥ \overline{D C} and \overline{B C} ∥ \overline{A D}

Since the given quadrilateral has two pairs of parallel sides, it is a parallelogram.

Determine whether the diagonals of rhombus

K L M N

are perpendicular. Explain the reasoning.

Answer

Yes, see solution.

Hint

Start by drawing the diagonals of the rhombus. Then, find the slopes of the diagonals.

Solution

Start by drawing the diagonals of the rhombus. Then, find the slopes of the diagonals.

The slope

m_{1}

\overline{K L}

1

and the slope

m_{2}

\overline{N L}

- 1 .

Notice that their product is

- 1 .

m_{1} \cdot m_{2} = 1 (- 1) = - 1

By the Slopes of Perpendicular Lines Theorem, it can be concluded that the diagonals are perpendicular to each other.

\overline{K M} ⊥ \overline{L N}

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Explore

Investigating the Slopes of Parallel Lines

Discussion

Properties of Different Systems of Equations

Discussion

Properties of Perpendicular Lines

Closure

Using Properties of Parallel Lines To Classify Parallelograms

Answer

Hint

Solution

Answer

Hint

Solution

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