### Video Transcript

In the diagram below, π΄π΅ equals 10, π΅πΆ equals π₯ plus one, πΆπ· equals 20, πΈπΉ equals 10, and πΉπΊ equals 10. Find the value of π₯ and the length of line segment πΊπ».

Letβs begin this question by filling on the given length measurements. We arenβt given any units of measurement, but these would all be length units. We are asked firstly to find the value of π₯ which appears as part of the length of line segment π΅πΆ. In order to work out the value of π₯ and indeed the length of the line segment πΊπ», weβll need to use these four parallel lines. Lines π΄πΈ, π΅πΉ, πΆπΊ, and π·π» weβre all given are parallel. We also note that the other two lines π΄π· and πΈπ» are both transversals. A transversal is a line that intersects two or more lines in the same plane at distinct points.

Knowing that we have these four parallel lines and two transversals means that we could apply Thaleβs theorem, or the basic proportionality theorem. This theorem tells us that if three or more parallel lines intersect two transversals, then they cut off the transversals proportionally. And so we can say that the ratio of the line segment π΄π΅ to that of π΅πΆ must be equal to the ratio of the length of the line segment πΈπΉ to that of the line segment πΉπΊ. And so we could write the proportion that π΄π΅ over π΅πΆ is equal to πΈπΉ over πΉπΊ.

We can then substitute in the given length measurements and see if we can solve it for π₯. This gives us 10 over π₯ plus one is equal to 10 over 10. We can then simplify the right-hand side, since 10 over 10 is simply equal to one. Next, we can multiply both sides by π₯ plus one. So we have 10 is equal to one times π₯ plus one. And, of course, one times π₯ plus one is just π₯ plus one. Subtracting one from both sides, we have nine is equal to π₯. And so weβve answered the first part of the question. Weβve worked out that π₯ is equal to nine.

For the second part of the question, we need to work out the length of the line segment πΊπ», which is here at the top part of the diagram. To work this out, we will apply the basic proportionality theorem once more. Now we know that the line segment of πΆπ· will be proportional to this line segment of πΊπ». We can write a proportionality statement involving these two line segments and either π΅πΆ and πΉπΊ or π΄π΅ and πΈπΉ. Letβs use the line segments π΄π΅ and πΈπΉ. We could therefore write the proportionality statement that π΄π΅ over πΆπ· is equal to πΈπΉ over πΊπ». When we substitute in the given lengths, we have 10 over 20 is equal to 10 over πΊπ».

We could then cross multiply and work out the length of πΊπ». However, we might also notice that the numerators on both sides of the equation are the same. Theyβre both 10. And so in order to be equal, then the denominators must be the same. We could therefore immediately work out that πΊπ» must be equal to 20.

Itβs worth pointing out that even though in this question we have a number of pairs of congruent line segments, but this wonβt always be the case. The proportionality theorem just tells us that these line segments have to be proportional. In this question, they were congruent as well. But now we can give the two answers: π₯ is equal to nine and πΊπ» is equal to 20 length units.