Video: Parallel Lines in a Triangle | Nagwa Video: Parallel Lines in a Triangle | Nagwa

# Video: Parallel Lines in a Triangle

In this video, we will learn how to find missing lengths in a triangle containing two or three parallel lines using proportionality.

16:08

### Video Transcript

In this video, we will learn how to find missing lengths in a triangle containing two or three parallel lines using proportionality. So letβs start with this triangle. Letβs say this triangle has three different angle measures. And then this triangle is cut by a line that is parallel to one of the side lengths. We can label our triangle π΄π΅πΆ. And weβll let the parallel segment be segment π·πΈ.

We started with the larger triangle, triangle π΄π΅πΆ. But now because of this line in the middle, we have a second triangle, the smaller triangle, triangle π΄π·πΈ. And if we want to compare triangle π΄π΅πΆ to triangle π΄π·πΈ, we need to be able to say something about its side lengths or about its angles.

To do this, letβs extend our parallel lines and the line segment π΄π΅. This should remind us that when two parallel lines are crossed by a transversal, corresponding angles are equal. Line segment π·πΈ and line segment π΅πΆ are parallel, which means line segment π΄π΅ could be considered a transversal of these two parallel lines, which means that angle π· is a corresponding angle with angle π΅. And these angles will be equal.

For the same reasons, angle πΈ is a corresponding angle to angle πΆ. And therefore, these two angles will be equal. Both of these triangles share angle π΄. So we can say that angle π΄ is equal to angle π΄, angle π΅ is equal to angle π·, and angle πΆ is equal to angle πΈ.

When two triangles have three congruent angles, we can say that the two triangles are similar. This means they are the same shape, but not the same size. And in similar triangles, corresponding sides are always proportional. That is, they always occur in the same ratio.

Letβs look at the corresponding side lengths for these two triangles. In our larger triangle, we have side length π΄π΅, which corresponds to the side length π΄π· in the smaller triangle. This ratio will be equal to side length π΄πΆ in our larger triangle over side length π΄πΈ in our smaller triangle. And for our final sides, side length π΅πΆ in the larger triangle corresponds to side length π·πΈ in the smaller triangle.

For these ratios, the numerator is a side length from the larger triangle and the denominator is the corresponding side length from the small triangle. And in order for our proportion to hold true, we have to maintain this pattern for the rest of the ratios, the side length from the larger triangle in the numerator and the corresponding smaller side length in the denominator.

However, there is another way we can write these proportions. If we take the larger side length π΄π΅ and the other larger side length π΅πΆ, we can still set up a ratio. This time, in our numerator on the other side, weβll need the corresponding side length to π΄π΅, which is π΄π·. And then the denominator, weβll need the corresponding side to side length π΅πΆ, which in our case is π·πΈ. In this case, we had the ratio of two of the larger side lengths set equal to the ratio of corresponding side lengths for the smaller triangle. This means we can line up corresponding side lengths this way or this way, as long as we are consistent for each of the fractions.

What weβve shown here can be summarized in a theorem about triangles. And that is the side splitter theorem. It says if a line is parallel to a side of a triangle and the line intersects the other two sides, then the line divides those sides proportionally. This is true because those parallel lines create two similar triangles. We should also note that that parallel line that intersects the other two sides can happen anywhere along the triangle and between any of the two sides, as long as itβs parallel to the third side. At this point, weβre ready to consider some examples.

Using the diagram, which of the following is equal to π΄π΅ over π΄π·? (A) π΄π΅ over π·π΅, (B) π΄πΆ over π΄πΈ, (C) π΄πΆ over πΈπΆ, (D) π΄π· over π·π΅, or (E) π΄πΈ over πΈπΆ.

In our diagram, line segment πΈπ· is parallel to line segment πΆπ΅. Because of that, we have two similar triangles. We can say that triangle π΄πΈπ· is similar to triangle π΄πΆπ΅. And in similar triangles, corresponding side lengths are proportional. Weβre interested in the ratio π΄π΅ over π΄π·. π΄π΅ represents a larger side length, and π΄π· is the corresponding smaller side length. This means that we are looking for the ratio that has a larger side length and the corresponding smaller side length.

Option (A) has π΄π΅ corresponding to π·π΅. But π·π΅ is not part of the smaller triangle, which means option (A) cannot work. Option (B) has side length π΄πΆ, which is part of our larger triangle, and then the distance from π΄ to πΈ. π΄ to πΈ is the corresponding smaller side length from π΄πΆ. This is an equal ratio. But letβs check the others just in case.

Again, we have the side length π΄πΆ, but the denominator is πΈπΆ. And πΈπΆ is not part of the smaller triangle, which makes this an invalid ratio. What about option (D)? π΄π· is a smaller side length, and then π·π΅, which is not part of our similar triangles. And we see that again π΄πΈ is a smaller triangle, but πΈπΆ is not part of any of our similar triangles. The only ratio that is equal to π΄π΅ over π΄π· in this list is π΄πΆ over π΄πΈ.

In our next example, weβll do something very similar, only this time weβll need to find the value of a missing side length.

Find the value of π₯.

When we look at our diagram, we see a larger triangle that is cut by the segment π·πΈ. And this line segment π·πΈ is parallel to one of the side lengths π΅πΆ. And when a triangle is cut by a line segment thatβs parallel to one of its side lengths, two similar triangles are created. So we can say that if weβre given line segment π·πΈ is parallel to π΅πΆ, the smaller triangle, triangle π΄π·πΈ, is similar to the larger triangle, triangle π΄π΅πΆ. So we can say that π΄π· over π΄π΅ is equal to π·πΈ over π΅πΆ. This is because in similar triangles corresponding side lengths are proportional.

Side length π΄π· measures 10, but what about π΄π΅? And this is where we need to be very careful. Side length π΄π΅ is not equal to 11. Side length π΄π΅ is the full distance from vertex π΄ to vertex π΅, which is 21. So π΄π· over π΄π΅ will be equal to 10 over 21. We know that side length π·πΈ is equal to 10 as well. So we now have a proportion that says 10 over 21 is equal to 10 over π΅πΆ. And that means π΅πΆ must be equal to 21 so that these side lengths stay in proportion. Since side length π΅πΆ equals 21, our missing π₯-value is 21.

Our next example is another case where we need to find a missing length in a triangle.

In a figure, segments ππ and π΅πΆ are parallel. If π΄π equals 18, ππ΅ equals 24, and π΄π equals 27, what is the length of line segment ππΆ?

The first thing we can do is take the information in the question and add it to our figure. We know that π΄π equals 18, ππ΅ equals 24, and π΄π equals 27. Our missing length is from π to πΆ, here. Letβs call it π. Before we start solving for π, letβs see what we recognize in the figure. The line segment ππ cuts our triangle π΄π΅πΆ and is parallel to the line segment π΅πΆ. We know that if a line is parallel to a side of a triangle and the line intersects the other two sides, then the line divides those sides proportionally. That is, it creates two similar triangles. So we can start out by saying the smaller triangle, triangle π΄ππ, is similar to the larger triangle, triangle π΄π΅πΆ. And since this is true, corresponding side lengths will be proportional. If we set up a ratio for our smaller triangle side lengths, we can say that π΄π over π΄π will be equal to π΄π΅ over π΄πΆ.

In order to solve for our missing length, weβll need to carefully plug the information we know into this ratio. First of all, π΄π is equal to 18 and π΄π equals 27. But we need to be careful with π΄π΅. To find π΄π΅, we need to add 18 and 24, which gives us 42. But what about the length π΄πΆ? Well, π΄πΆ is made up of π΄π and ππΆ. π΄π measures 27, and ππΆ measures π, a missing value. So we plug in 27 plus π plus our missing value for the length of line segment π΄πΆ.

Once we get to this point, we can use cross multiplication to solve for our missing variable. 18 times 27 plus π is equal to 27 times 42. 27 times 42 is 1134. From here, we can divide both sides by 18, which gives us 27 plus π is equal to 63. And when we subtract 27 from both sides, our missing side length π is equal to 36. If we plug that back into our diagram, we show that ππΆ equals 36.

In our final example, weβll look at a figure that includes a parallelogram and a triangle.

πΉπ·πΈπΆ is a parallelogram, where πΉ and π· are the midpoints of line segment π΄π΅ and line segment π΄πΆ, respectively, and πΆπΈ equals six centimeters. Determine the length of π΅πΆ.

Letβs take the information weβre given and add that in to our diagram. If πΉπ·πΈπΆ is a parallelogram, the opposite sides are parallel, meaning πΉπ· is parallel to πΆπΈ and πΉπΆ is parallel to π·πΈ. We also know that πΆπΈ measures six centimeters. But based on our parallelogram properties, we also know that opposite side lengths will be equal. And that means that π·πΉ must also measure six centimeters.

But our missing side length is π΅πΆ. And so weβll need to be able to say something else here. If πΆπΈ is parallel to πΉπ·, we can also say that π΅πΈ is parallel to πΉπ·. And if we can say that line segment π΅πΆ is parallel to line segment πΉπ·, then we have a line that is parallel to a side of our triangle. And that line intersects the other two sides, which means the line segment πΉπ· creates two similar triangles. The smaller triangle, triangle π΄π·πΉ, is similar to the larger triangle, triangle π΄πΆπ΅. And in similar triangles, corresponding side lengths are proportional. The side length π΄π· on the smaller triangle corresponds to the side length π΄πΆ on the larger triangle. And that will have to be equal to the smaller triangleβs line segment πΉπ· over the larger triangleβs line segment π΅πΆ.

If we try to plug in the information we know, we only end up filling in the side length πΉπ·, which is six centimeters. But we can think carefully about this midpoint π·. The midpoint π· divides line segment π΄πΆ in half. And so we can say that π΄πΆ is equal to two times π΄π·. What weβre saying is the ratio of the smaller triangle to the larger triangle would be one-half since the larger triangle is always two times greater than the smaller triangle. And if the side lengths of the larger triangle is two times that of the smaller triangle and πΉπ· is equal to six centimeters, we know that π΅πΆ will have to be equal to 12 centimeters, as 12 is twice six.

Before we finish, letβs just review some key points from this video. If a line is parallel to a side of a triangle and the line intersects the other two sides, then that line creates two similar triangles. And in similar triangles, corresponding side lengths are proportional. For this example, triangle π΄π·πΉ would be similar to triangle π΄π΅πΆ. Therefore, the side length π΄π· over the side length π΄π΅ will be equal to the side length π΄πΉ over the side length π΄πΆ.

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