### Video Transcript

In the diagram, lines π΄π΅ and π·πΆ are parallel. Weβve then been given a diagram whichβs not been drawn accurately. Show that triangle π΄π΅π· is congruent to triangle π΅πΆπ·.

Now the fact that the diagram has not been drawn accurately means that we canβt use our protractor to measure any angles or a ruler to measure the length of any lines on the diagram. Weβve got to use geometric reasoning to come up with an argument for why triangle π΄π΅π· is congruent to triangle π΅πΆπ·. And congruent just means the same shape and size. We should be able to take triangle π΄π΅π· and lay exactly on top of triangle π΅πΆπ·. We may have to twist it round or turn it over. But we must be able to align it so that all the corresponding sides are exactly the same length and the corresponding angles are exactly the same.

So hereβs triangle π΄π΅π· in pink and hereβs triangle π΅πΆπ· in orange. And just before we get going on our geometric reasoning, letβs take a moment to realize that lines π΄π΅ and π·πΆ are parallel, so this line here and this line here are parallel. And thatβs what these two little arrows indicate. So weβve got two parallel lines and three transversals. Now transversals are just lines that go across two parallel lines. Now, weβll come back to this in a minute. But transversals and parallel lines have special properties things, like corresponding angles or alternate angles being equal. But as I say, weβll come back to that in a moment.

Now, the first thing we notice is that angles on a line sum to 180 degrees. So angle π΅πΆπ· β this angle here β plus 150 degrees β this angle here β sum to 180 degrees. So weβve created an equation using those angles. And if we subtract 150 degrees from each side of the equation, we can see that angle π΅πΆπ· is 30 degrees. Next, we know that vertically opposite angles are equal.

Now, vertically opposite angles are what you get when two straight lines cross each other, the vertically opposite angles here are equal and the vertically opposite angles here are equal. Now, on our diagram, angle π΅π·πΆ β this angle here β is vertically opposite this 30-degree angle here. And since theyβre equal, that means that angle π΅π·πΆ must be 30 degrees. And we can write this on our diagram.

Next, we know that alternate angle are equal. When a transversal crosses two parallel lines, we know that this angle here is equal to this angle here. Itβs also the case that this angle here is equal to this angle here. So on our diagram, we can see that this angle here, angle π·π΅π΄, is alternate to this angle here, angle π΅π·πΆ. So we know that angle π·π΅π΄ is 30 degrees.

Now we also know that angles in a triangle sum to 180 degrees. So angle π·π΅πΆ β this angle here β plus 30 degrees plus 30 degrees is equal to 180 degrees. And angle π΄π·π΅ β this angle here β plus 30 degrees plus 30 degrees is also equal to 180 degrees. And if we subtract 60 degrees from each side of those equations, we can see that angle π·π΅πΆ is 120 degrees and so is angle π΄π·π΅. Letβs clear a little bit more space for our working out.

Now, so far, weβve got enough information to tell us that triangles π΄π΅π· and π΅πΆπ· are similar triangles. Their corresponding angles are equal. They both have 30-degree, 30-degree, and 120-degree angles. But are they congruent? Well, we know that side π΅π· is common to both of those triangles. And that side is the same length in triangle π΄π΅π· as it is in triangle π΅πΆπ·. In triangle π΄π΅π·, we have an angle of 30 degrees adjacent to side π΅π· adjacent to an angle of 120 degrees. But the same is true in triangle π΅πΆπ·, we have an angle of 30 degrees followed by side length π΅π· followed by an angle of 120 degrees. So not only is side π΅π· common to both of those triangles, itβs also adjacent to angles of 30 degrees and 120 degrees in each case.

So we can say that triangle π΄π΅π· and triangle π΅πΆπ· are congruent using the ASA rule. And ASA just means that an adjacent angle, then a side, then an angle are the same in both of those triangles.

Now to get your full marks in questions like these, you do need to explain your geometric reasoning very carefully. In each case, you need to state the geometric rule that youβre using and show how it applies to your particular diagram and then draw a conclusion in each case about the size of an angle or the length of a side. And you have to do this for each step of your argument.