Lesson Video: Parallel Lines in a Triangle Mathematics

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 let’s start with this triangle. Let’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 𝐴𝐵𝐶. And we’ll let the parallel segment be segment 𝐷𝐸.

We started with the larger triangle, triangle 𝐴𝐵𝐶. But now because of this line in the middle, we have a second triangle, the smaller triangle, triangle 𝐴𝐷𝐸. And if we want to compare triangle 𝐴𝐵𝐶 to triangle 𝐴𝐷𝐸, we need to be able to say something about its side lengths or about its angles.

To do this, let’s extend our parallel lines and the line segment 𝐴𝐵. This should remind us that when two parallel lines are crossed by a transversal, corresponding angles are equal. Line segment 𝐷𝐸 and line segment 𝐵𝐶 are parallel, which means line segment 𝐴𝐵 could be considered a transversal of these two parallel lines, which means that angle 𝐷 is a corresponding angle with angle 𝐵. And these angles will be equal.

For the same reasons, angle 𝐸 is a corresponding angle to angle 𝐶. And therefore, these two angles will be equal. Both of these triangles share angle 𝐴. So we can say that angle 𝐴 is equal to angle 𝐴, angle 𝐵 is equal to angle 𝐷, and angle 𝐶 is equal to angle 𝐸.

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.

Let’s look at the corresponding side lengths for these two triangles. In our larger triangle, we have side length 𝐴𝐵, which corresponds to the side length 𝐴𝐷 in the smaller triangle. This ratio will be equal to side length 𝐴𝐶 in our larger triangle over side length 𝐴𝐸 in our smaller triangle. And for our final sides, side length 𝐵𝐶 in the larger triangle corresponds to side length 𝐷𝐸 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 𝐴𝐵 and the other larger side length 𝐵𝐶, we can still set up a ratio. This time, in our numerator on the other side, we’ll need the corresponding side length to 𝐴𝐵, which is 𝐴𝐷. And then the denominator, we’ll need the corresponding side to side length 𝐵𝐶, which in our case is 𝐷𝐸. 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 we’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 it’s parallel to the third side. At this point, we’re ready to consider some examples.

Using the diagram, which of the following is equal to 𝐴𝐵 over 𝐴𝐷? (A) 𝐴𝐵 over 𝐷𝐵, (B) 𝐴𝐶 over 𝐴𝐸, (C) 𝐴𝐶 over 𝐸𝐶, (D) 𝐴𝐷 over 𝐷𝐵, or (E) 𝐴𝐸 over 𝐸𝐶.

In our diagram, line segment 𝐸𝐷 is parallel to line segment 𝐶𝐵. Because of that, we have two similar triangles. We can say that triangle 𝐴𝐸𝐷 is similar to triangle 𝐴𝐶𝐵. And in similar triangles, corresponding side lengths are proportional. We’re interested in the ratio 𝐴𝐵 over 𝐴𝐷. 𝐴𝐵 represents a larger side length, and 𝐴𝐷 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 𝐴𝐵 corresponding to 𝐷𝐵. But 𝐷𝐵 is not part of the smaller triangle, which means option (A) cannot work. Option (B) has side length 𝐴𝐶, which is part of our larger triangle, and then the distance from 𝐴 to 𝐸. 𝐴 to 𝐸 is the corresponding smaller side length from 𝐴𝐶. This is an equal ratio. But let’s check the others just in case.

Again, we have the side length 𝐴𝐶, but the denominator is 𝐸𝐶. And 𝐸𝐶 is not part of the smaller triangle, which makes this an invalid ratio. What about option (D)? 𝐴𝐷 is a smaller side length, and then 𝐷𝐵, which is not part of our similar triangles. And we see that again 𝐴𝐸 is a smaller triangle, but 𝐸𝐶 is not part of any of our similar triangles. The only ratio that is equal to 𝐴𝐵 over 𝐴𝐷 in this list is 𝐴𝐶 over 𝐴𝐸.

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

Find the value of 𝑥.

When we look at our diagram, we see a larger triangle that is cut by the segment 𝐷𝐸. And this line segment 𝐷𝐸 is parallel to one of the side lengths 𝐵𝐶. And when a triangle is cut by a line segment that’s parallel to one of its side lengths, two similar triangles are created. So we can say that if we’re given line segment 𝐷𝐸 is parallel to 𝐵𝐶, the smaller triangle, triangle 𝐴𝐷𝐸, is similar to the larger triangle, triangle 𝐴𝐵𝐶. So we can say that 𝐴𝐷 over 𝐴𝐵 is equal to 𝐷𝐸 over 𝐵𝐶. This is because in similar triangles corresponding side lengths are proportional.

Side length 𝐴𝐷 measures 10, but what about 𝐴𝐵? And this is where we need to be very careful. Side length 𝐴𝐵 is not equal to 11. Side length 𝐴𝐵 is the full distance from vertex 𝐴 to vertex 𝐵, which is 21. So 𝐴𝐷 over 𝐴𝐵 will be equal to 10 over 21. We know that side length 𝐷𝐸 is equal to 10 as well. So we now have a proportion that says 10 over 21 is equal to 10 over 𝐵𝐶. And that means 𝐵𝐶 must be equal to 21 so that these side lengths stay in proportion. Since side length 𝐵𝐶 equals 21, our missing 𝑥-value is 21.

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

In a figure, segments 𝑋𝑌 and 𝐵𝐶 are parallel. If 𝐴𝑋 equals 18, 𝑋𝐵 equals 24, and 𝐴𝑌 equals 27, what is the length of line segment 𝑌𝐶?

The first thing we can do is take the information in the question and add it to our figure. We know that 𝐴𝑋 equals 18, 𝑋𝐵 equals 24, and 𝐴𝑌 equals 27. Our missing length is from 𝑌 to 𝐶, here. Let’s call it 𝑚. Before we start solving for 𝑚, let’s see what we recognize in the figure. The line segment 𝑋𝑌 cuts our triangle 𝐴𝐵𝐶 and is parallel to the line segment 𝐵𝐶. 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 𝐴𝑋𝑌, is similar to the larger triangle, triangle 𝐴𝐵𝐶. 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 𝐴𝑋 over 𝐴𝑌 will be equal to 𝐴𝐵 over 𝐴𝐶.

In order to solve for our missing length, we’ll need to carefully plug the information we know into this ratio. First of all, 𝐴𝑋 is equal to 18 and 𝐴𝑌 equals 27. But we need to be careful with 𝐴𝐵. To find 𝐴𝐵, we need to add 18 and 24, which gives us 42. But what about the length 𝐴𝐶? Well, 𝐴𝐶 is made up of 𝐴𝑌 and 𝑌𝐶. 𝐴𝑌 measures 27, and 𝑌𝐶 measures 𝑚, a missing value. So we plug in 27 plus 𝑚 plus our missing value for the length of line segment 𝐴𝐶.

Once we get to this point, we can use cross multiplication to solve for our missing variable. 18 times 27 plus 𝑚 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 𝑚 is equal to 63. And when we subtract 27 from both sides, our missing side length 𝑚 is equal to 36. If we plug that back into our diagram, we show that 𝑌𝐶 equals 36.

In our final example, we’ll look at a figure that includes a parallelogram and a triangle.

𝐹𝐷𝐸𝐶 is a parallelogram, where 𝐹 and 𝐷 are the midpoints of line segment 𝐴𝐵 and line segment 𝐴𝐶, respectively, and 𝐶𝐸 equals six centimeters. Determine the length of 𝐵𝐶.

Let’s take the information we’re given and add that in to our diagram. If 𝐹𝐷𝐸𝐶 is a parallelogram, the opposite sides are parallel, meaning 𝐹𝐷 is parallel to 𝐶𝐸 and 𝐹𝐶 is parallel to 𝐷𝐸. We also know that 𝐶𝐸 measures six centimeters. But based on our parallelogram properties, we also know that opposite side lengths will be equal. And that means that 𝐷𝐹 must also measure six centimeters.

But our missing side length is 𝐵𝐶. And so we’ll need to be able to say something else here. If 𝐶𝐸 is parallel to 𝐹𝐷, we can also say that 𝐵𝐸 is parallel to 𝐹𝐷. And if we can say that line segment 𝐵𝐶 is parallel to line segment 𝐹𝐷, 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 𝐹𝐷 creates two similar triangles. The smaller triangle, triangle 𝐴𝐷𝐹, is similar to the larger triangle, triangle 𝐴𝐶𝐵. And in similar triangles, corresponding side lengths are proportional. The side length 𝐴𝐷 on the smaller triangle corresponds to the side length 𝐴𝐶 on the larger triangle. And that will have to be equal to the smaller triangle’s line segment 𝐹𝐷 over the larger triangle’s line segment 𝐵𝐶.

If we try to plug in the information we know, we only end up filling in the side length 𝐹𝐷, which is six centimeters. But we can think carefully about this midpoint 𝐷. The midpoint 𝐷 divides line segment 𝐴𝐶 in half. And so we can say that 𝐴𝐶 is equal to two times 𝐴𝐷. What we’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 𝐹𝐷 is equal to six centimeters, we know that 𝐵𝐶 will have to be equal to 12 centimeters, as 12 is twice six.

Before we finish, let’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 𝐴𝐷𝐹 would be similar to triangle 𝐴𝐵𝐶. Therefore, the side length 𝐴𝐷 over the side length 𝐴𝐵 will be equal to the side length 𝐴𝐹 over the side length 𝐴𝐶.

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