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Vertical lines are lines that go straight up and down as you can see in the picture below :

Vertical Line Picture

Horizontal Line Picture

Other traits of Vertical lines

The equation of a vertical line does NOT have a y-intercept since a vertical line never crosses the y-axis.

The slope of a vertical line is undefined because the denominator of the slope (the change in X) is zero

Vertical lines help determine if a relation is a function in math.

The equation of a vertical line always takes the form x = k
where k is any number and k is also the x-intercept. For instance in the graph below, the vertical line has the equation x = 2 As you can see in the picture below, the line goes straight up and down at x = 2

Equation : x = 1

Below are several examples of the equation and graph of different vertical lines

Equation : x = 1

Equation : x = -1

Equation : x = 1.5

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Vertical line: traits, examples and usage in mathematics

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Vertical lines are lines that go straight up and down as you can see in the picture below : Vertical Line Picture Horizontal Line Picture The equation of a vertical line does NOT have a y-intercept since a vertical line never crosses the y-axis.

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parallel and  perpendicular lines (slope)

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parallel and  perpendicular lines (slope)

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The first line's equation was

This relationship will become very important when you start working with straight-line equations.

Now let's consider those two equations and their graphs.

For the first equation,

Notice how the line, as we move from left to right along the -axis, is edging upward toward the top of the drawing; technically, the line is an "increasing" line. And... the slope was positive.

This relationship always holds true: If a line is increasing, then its slope will be positive; and if a line's slope is positive, then its graph will be increasing.

For the second line, , the slope was , a negative number. The graph looked like this:

Notice how the line, as we move from left to right along the -axis, is edging downward toward the bottom of the drawing; technically, the line is a "decreasing" line. And... the slope was negative.

This relationship is always true: If a line is decreasing, then its slope will be negative; and if a line's slope is negative, then its graph will be decreasing.

This relationship between the sign on the slope and the direction of the line's graph can help you check your calculations: if you calculate a slope as being negative, but you can see from the graph of the equation that the line is actually increasing (so the slope must be positive), then you know you need to re-do your calculations. Being aware of this connection can save you points on a test because it will enable you to check your work before you hand it in.

So now we know: Increasing lines have positive slopes, and decreasing lines have negative slopes. With this in mind, let's consider the following horizontal line:

Its graph is shown below:

Is the horizontal line edging upward; that is, is it an increasing line? No, so its slope can't be positive. Is the horizontal line edging downward; that is, is it a decreasing line? No, so its slope can't be negative. What number is neither positive nor negative?

So the slope of this (and any other) horizontal line should, logically, be zero. Let's do the calculations to confirm this. Using the (arbitrary) points from the line, (-3, 4) and (5, 4), the slope computes as:

This relationship always holds: a slope of zero means that the line is horizontal, and a horizontal line means you'll get a slope of zero.

(By the way, all horizontal lines are of the form " = some number", and the equation " = some number" always graphs as a horizontal line.)

Now consider the following vertical line:

Its graph is below.

Is the vertical line going up on one end? Well, yes, kind of. So maybe the slope will be positive...? Is the vertical line going down on the other end? Well, again, kind of. So maybe the slope will be negative...?

But is there any number that is both positive and negative? Nope.

Verdict: vertical lines have NO SLOPE. The concept of slope simply does not work for vertical lines. The slope of a vertical line does not exist!

Let's do the calculations to confirm the logic. From the line's graph, I'll use the (arbitrary) points (4, 5) and (4, -3). Then the slope is:

We can't divide by zero, which is of course why this slope value is "undefined".

This relationship is always true: a vertical line will have no slope, and "the slope is undefined" or "the line has no slope" means that the line is vertical.

(By the way, all vertical lines are of the form " = some number", and " = some number" means the line is vertical. Any time your line involves an undefined slope, the line is vertical; and any time the line is vertical, you'll end up dividing by zero if you try to compute the slope.)

Warning: It is very common to confuse these two types of lines and their slopes, but they are very different.

Just as "horizontal" is not at all the same as "vertical", so also "zero slope" is not at all the same as "no slope".

Just as a "Z" (with its two horizontal lines) is not the same as an "N" (with its two vertical lines), so also "Zero" slope (for a horizontal line) is not the same as "No" slope (for a vertical line).

The number "zero" exists, so horizontal lines do indeed have a slope. But vertical lines don't have any slope; "slope" simply doesn't have any meaning for vertical lines.

It is very common for tests to contain questions regarding horizontals and verticals. Don't mix them up!

URL: https://www.purplemath.com/modules/slope2.htm

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The first line's equation was This relationship will become very important when you start working with straight-line equations. Now let's consider those two equations and their graphs. For the first equation, Notice how the line, as we move from left to right along the -axis, is edging upward toward the top of the drawing; technically, the line is an "increasing" line.

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Sal shows how to find the slope of a line.

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Corresponding Angles: Definition, Theorem & Examples - Video & Lesson Transcript | Study.com

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Corresponding angles occur when a transversal crosses two lines, creating angles that are formed in the same position. Explore the correlating…

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https://www.mathsisfun.com/geometry/consecutive-interior-angles.html

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proving lines are parallel with the converse

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proving lines are parallel with the converse

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Parallel Lines

To begin, we know that a pair of parallel lines is a pair that never intersect and are always the same distance apart. Think of the tracks on a roller coaster ride. Do you see how they never intersect each other and are always the same distance apart? This is what parallel lines are about.

Converse Statements

Along with parallel lines, we are also dealing with converse statements. Don't worry, it's nothing complicated. If we had a statement such as 'If a square is a rectangle, then a circle is an oval,' then its converse would just be the same statement but in reverse order, like this: 'If a circle is an oval, then a square is a rectangle.' We started with 'If this, then that,' and we ended up with 'If that, then this.' So just think of the converse as flipping the order of the statement.

Now, with parallel lines, we have our original statements that tell us when lines are parallel.

Parallel Lines Statements

We have four original statements we can make. But in order for the statements to work, for us to be able to prove the lines are parallel, we need a transversal, or a line that cuts across two lines. This line creates eight different angles that we can compare with each other. So if you're still picturing the tracks on a roller coaster ride, now add in a straight line that cuts across the tracks. You will see that it forms eight different angles. This is your transversal. You need this to prove parallel lines because you need the angles it forms because it's the properties of the angles that either make or break a pair of parallel lines.

What are the properties that the angles must have if the lines are parallel?

1. If the lines are parallel, then the corresponding angles are congruent. These are the angles that are on the same corner at each intersection. You will see that the transversal produces two intersections, one for each line. So, a corresponding pair of angles will both be at the same corner at their respective intersections. So if one angle was at the top left corner at one intersection, the corresponding angle at the other intersection will also be at the top left. For parallel lines, these angles must be equal to each other.

2. If the lines are parallel, then the alternate interior angles are congruent. The word 'alternate' means that you will have one angle on one side of the transversal and the other angle on the other side of the transversal. 'Interior' means that both angles are between the two lines that are parallel. These angles must be equal to each other for parallel lines.

3. If the lines are parallel, then the alternate exterior angles are congruent. This is similar to the one we just went over except now the angles are outside the pair of parallel lines. So these angles must likewise be equal to each for parallel lines.

4. Last but not least, if the lines are parallel, then the interior angles on the same side of the transversal are supplementary. Here, the angles are the ones between the two lines that are parallel, but both angles are not on the same side of the transversal. These must add up to 180 degrees.

Using Converse Statements

Now let's look at how our converse statements will look like and how we can use it with the angles that are formed by our transversal. All I need is for one of these to be satisfied in order to have a successful proof.

1. If the corresponding angles are congruent, then the lines are parallel. To use this statement to prove parallel lines, all we need is to find one pair of corresponding angles that are congruent. That is all we need. So we look at both intersections and we look for matching angles at each corner. For example, if we found that the top-right corner at each intersection is equal, then we can say that the lines are parallel using this statement.

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using converse statements to prove lines are parallel - video & lesson transcript | study.com

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Using Converse Statements to Prove Lines Are Parallel - Video & Lesson Transcript | Study.com

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To begin, we know that a pair of parallel lines is a pair that never intersect and are always the same distance apart. Think of the tracks on a roller coaster ride. Do you see how they never intersect each other and are always the same distance apart?

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When two lines are crossed by another line (called the Transversal):

Alternate Exterior Angles are a pair of angles on the outer side of each of those two lines but on opposite sides of the transversal.

In this example, these are two pairs of Alternate Exterior Angles:

And

To help you remember: the angle pairs are on Alternate sides of the Transversal, and they are on the Exterior of the two crossed lines.

Parallel Lines

When the two lines being crossed are Parallel Lines the Alternate Exterior Angles are equal.
(Click on "Alternate Exterior Angles" to have them highlighted for you.)

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Learn about Alternate Exterior Angles: When two lines are crossed by another line (called the Transversal), Alternate Exterior Angles are a pair of angles on the outer side of each of those two lines but on opposite sides of the transversal.

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http://www.mathnstuff.com/math/spoken/here/2class/260/trans.htm

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Coplanar lines that do not intersect