Proof: parallel lines have the same slope | High School Math | Khan Academy

Summary

This video demonstrates that parallel lines share the same slope. The explanation involves drawing parallel lines and transversals, identifying congruent angles formed by these intersections, and using the properties of similar triangles to conclude that the slopes of the lines are equal.

Takeaways

• 📐 The video aims to prove that parallel lines have the same slope.
• 🖌️ The presenter draws two parallel lines and introduces them as the main subjects.
• ✍️ Transversals are introduced to help in the demonstration, with a horizontal and a vertical one.
• 🟢 The horizontal transversal is assumed to be green and the vertical one blue.
• 🔍 The assumption is made that the transversals intersect at right angles, implying perpendicularity.
• 🔢 The presenter labels points A, B, C, D, and E to facilitate the explanation.
• 🔄 Angles CED and AEB are identified as congruent right angles.
• 🔄 Corresponding angles and vertical angles are used to establish congruence between angles on either side of the transversals.
• 🔄 The concept of alternate interior angles is introduced to further establish angle congruence.
• 🔄 Triangles CEB and ABE are identified as similar based on their congruent angles.
• 📈 The similarity of triangles leads to the conclusion that the ratios of corresponding sides are equal.
• 📉 The ratio of BE to AE is equated to the ratio of CE to DE, linking the slopes of the lines.
• 📚 The slope of a line is defined as the change in y over the change in x, leading to the conclusion that the slopes of the parallel lines are the same.

Q & A

What is the main goal of the video?

The main goal of the video is to prove that parallel lines have the same slope.

How does the video begin?

The video begins by drawing some parallel lines and introducing the concept that they will be used to demonstrate the property of having the same slope.

What are transversals in the context of this video?

Transversals in this video are lines that intersect two or more other lines, specifically the parallel lines being discussed, to help demonstrate their properties.

Why are horizontal and vertical transversals drawn?

Horizontal and vertical transversals are drawn to assume they are perpendicular to each other, which helps in establishing the similarity of triangles formed by the intersection of these transversals with the parallel lines.

What is the assumption made about the green and blue transversals?

The assumption made is that the green transversal is horizontal and the blue transversal is vertical, intersecting at right angles.

How are the angles formed by the transversals and the parallel lines related?

The angles formed by the transversals and the parallel lines are related in that they are congruent due to the properties of parallel lines and transversals, such as corresponding angles and alternate interior angles being equal.

What property of triangles is used to establish that the slopes of the parallel lines are the same?

The property of triangle similarity is used, where corresponding angles of the triangles formed by the transversals and the parallel lines are congruent, leading to the conclusion that the triangles are similar.

How does the video use the concept of similar triangles to prove the slopes are the same?

By showing that the triangles formed by the transversals and the parallel lines have all corresponding angles congruent, the video concludes that the triangles are similar, and thus the ratios of corresponding sides are equal, which implies the slopes are the same.

What is the ratio of BE to AE in the context of the video?

In the context of the video, the ratio of BE to AE represents the slope of the line connecting points A and B, which is one of the parallel lines.

How does the video conclude that the slopes of the two parallel lines are the same?

The video concludes that the slopes of the two parallel lines are the same by establishing the similarity of the triangles formed by the transversals and the parallel lines, and showing that the ratios of corresponding sides (which represent the slopes) are equal.

Outlines

00:00

📚 Proving Parallel Lines Have the Same Slope

This paragraph introduces the concept of parallel lines and their properties, specifically focusing on their slopes. The speaker begins by drawing parallel lines and then introduces transversals to these lines. The goal is to use the properties of parallel lines and transversals to prove that the slopes of the parallel lines are equal. The speaker labels points on the lines and discusses the congruence of angles formed by the transversals, including right angles and corresponding angles. The concept of similar triangles is introduced, where triangles formed by the intersection of parallel lines and transversals share corresponding angles, leading to the conclusion that the slopes of the parallel lines are the same.

05:02

🔍 Establishing Similarity of Triangles to Determine Slopes

In this paragraph, the speaker continues the discussion from the previous one, focusing on the similarity of triangles formed by the intersection of parallel lines and transversals. The speaker uses the congruence of angles to establish that triangles CEB and DEC are similar. This similarity is then used to equate the ratios of corresponding sides, which in turn equates the slopes of the lines. The speaker explains that the ratio of BE to AE (the slope of line AB) is equal to the ratio of CE to DE (the slope of line CD). This demonstrates that the slopes of the two lines are the same, thereby proving the initial claim that parallel lines have the same slope.

Keywords

💡Parallel lines

Parallel lines are two lines in a plane that do not intersect, no matter how far they are extended. In the video, the speaker draws two lines and claims they are parallel. This is a fundamental concept in geometry and is central to the video's theme of proving that parallel lines have the same slope.

💡Transversals

Transversals are lines that intersect two or more other lines in a plane. In the video, the speaker draws both horizontal and vertical transversals to intersect the parallel lines. This is crucial for demonstrating the properties of angles formed by the intersection of parallel lines and transversals.

💡Right angles

A right angle is an angle of exactly 90 degrees. The video assumes that the horizontal and vertical transversals intersect at right angles, forming right angles with the parallel lines. This assumption is key to establishing the congruence of certain angles in the subsequent geometric proofs.

💡Corresponding angles

Corresponding angles are pairs of angles that are in similar positions at the intersection of two lines cut by a transversal. In the video, the speaker identifies that certain angles formed by the transversals and the parallel lines are corresponding and therefore congruent. This is an important step in proving the similarity of triangles formed by the intersection.

💡Vertical angles

Vertical angles are the angles opposite each other when two lines intersect. They are always congruent. The video mentions that angles on one side of point B are congruent to each other because they are vertical angles. This is used to establish congruence in the geometric proof.

💡Alternate interior angles

Alternate interior angles are pairs of angles on opposite sides of a transversal and inside the two lines it intersects. In the video, the speaker uses the concept of alternate interior angles to show that certain angles formed by the transversals and the parallel lines are congruent, which is essential for proving the similarity of triangles.

💡Similar triangles

Similar triangles are triangles that have the same shape but may differ in size, with all corresponding angles being congruent and all corresponding sides being in proportion. The video uses the similarity of triangles formed by the intersection of the parallel lines and transversals to establish that the ratios of corresponding sides are equal.

💡Slope

Slope is a measure of the steepness of a line, calculated as the ratio of the change in the y-coordinate to the change in the x-coordinate between two points on the line. The video aims to prove that the slopes of parallel lines are the same, which is demonstrated by showing that the ratios of corresponding sides in similar triangles are equal.

💡Change in y over change in x

This phrase describes the method of calculating slope, where the change in y (vertical distance) is divided by the change in x (horizontal distance) between two points. The video uses this concept to define the slope of the lines and to show that the slopes of the parallel lines are equal.

💡Ratio of corresponding sides

In the context of similar triangles, the ratio of corresponding sides refers to the proportionality of the lengths of corresponding sides in the triangles. The video uses this concept to show that the slopes of the parallel lines are equal by demonstrating that the ratios of BE to AE and CE to DE are the same.

Highlights

The video aims to prove that parallel lines have the same slope.

Parallel lines are drawn to demonstrate the concept.

Transversals are introduced to intersect the parallel lines.

A horizontal transversal is drawn to intersect the lines.

A vertical transversal is also drawn, assumed to be perpendicular to the horizontal one.

The assumption is made that the green line is horizontal and the blue line is vertical.

The use of parallel line angle properties is mentioned to establish similarity.

Points A, B, C, D, and E are labeled on the diagram.

Angle CED is congruent to angle AEB because they are both right angles.

Corresponding angles formed by the transversal intersecting the parallel lines are congruent.

Vertical angles are congruent, as seen at point B.

Angle ABE is congruent to angle ECD, referred to as alternate interior angles.

Triangles CED and ABE are shown to have two angles in common.

The third angles of the triangles are determined to be congruent.

Triangles AEB and DEC are identified as similar by angle-angle-angle similarity.

The ratio of corresponding sides in similar triangles is the same.

The ratio of BE to AE is equated to the ratio of CE to DE.

The slope of line AB is determined by the ratio of BE to AE.

The slope of line CD is determined by the ratio of CE to DE.

The conclusion is reached that the slopes of the two lines are the same, proving the initial claim.

Transcripts

- [Voiceover] What I wanna do in this video is prove

that parallel lines have the same slope.

So let's draw some parallel lines here.

So, that's one line and then let me draw another line

that is parallel to that.

I'm claiming that these are parallel lines.

And now, I'm gonna draw some transversals here.

So first let me draw a horizontal transversal.

So, just like that.

And then let me do a vertical transversal.

So,

just like that.

And I'm assuming that the green one is horizontal

and the blue one is vertical.

So we assume that they are perpendicular to each other,

that these intersect at right angles.

And from this, I'm gonna figure out,

I'm gonna use some parallel line angle properties

to establish that this triangle

and this triangle are similar

and then use that to establish that both of these lines,

both of these yellow lines have the same slope.

So actually let me label some points here.

So let's call that point A, point B, point C,

point D, and point E.

So, let's see.

First of all we know that angle CED

is going to be congruent to angle AEB,

because they're both right angles.

So that's a right angle and then that is a right angle

right over there.

We also know some things about corresponding angles

for where our transversal intersects parallel lines.

This angle corresponds to this angle if we look

at the blue transversal as it intersects those two lines.

And so they're going to be, they're going to have

the same measure, they're going to be congruent.

Now this angle on one side of this point B

is going to also be congruent to that,

because they are vertical angles.

We've seen that multiple times before.

And so we know that this angle, angle ABE

is congruent to angle ECD.

Sometimes this is called alternate interior angles

of a transversal and parallel lines.

Well, if you look at triangle CED and triangle ABE,

we see they already have two angles in common,

so if they have two angles in common,

well, then their third angle has to be in common.

So, because this third angle's just gonna be

180 minus these other two, and so this third angle

is just gonna be 180 minus this, the other two.

And so just like that, we notice we have all three angles

are the same in both of these triangles,

well, they're not all the same,

but all of the corresponding angles,

I should say, are the same.

This blue angle has the same measure as this blue angle,

this magenta angle has the same measure

as this magenta angle, and then the other angles

are right angles, these are right triangles here.

So we could say triangle AEB,

triangle AEB

is similar, similar

similar to triangle DEC,

triangle DEC

by, and we could say by angle, angle, angle,

all the corresponding angles are congruent,

so we are dealing with similar triangles.

And so we know similar triangles are a ratio

of corresponding sides are going to be the same.

So we could say that the ratio of let's say

the ratio of BE, the ratio of BE, let me write this down,

this is this side right over here, the ratio of BE

to AE, to AE, to AE,

is going to be equal to, so that side over that side,

well what is the corresponding side?

The corresponding side to BE is side CE.

So that's going to be the same

as the ratio between CE and DE, and DE.

And this just comes out of similar,

the similarity of the triangles, CE to DE.

So once again, once we established

these triangles are similar, we can say the ratio

of corresponding sides are going to be the same.

Now what is the ratio between BE and AE?

The ratio between BE and AE.

Well that is the slope of this top line right over here.

We could say that's the slope of line AB,

slope of line connecting,

connecting

A to B.

All right, let me just use, I could write it like this,

that is slope of, slope of A,

slope of line AB.

Remember slope is, when you're going from A to B,

it's change in y over change in x.

So when you're going from A to B, your change in x is AE,

and your change in y is BE, or EB,

however you want to refer to it.

So this right over here is change in y,

and this over here is change in x.

Well, now let's look at this second expression

right over here, CE over DE, CE over DE.

Well, now, this is going to be change in y

over change in x between point C and D.

So this is, this right over here, this is the slope of

line, of line CD.

And so just like that, by establishing similarity,

we were able see the ratio

of corresponding sides are congruent,

which shows us that the slopes of these two lines

are going to be the same.

And we are done.