In the diagram, lines 𝐴𝐵 and 𝐷𝐶 are parallel. We’ve then been given a diagram which’s not been drawn accurately. Show that triangle 𝐴𝐵𝐷 is congruent to triangle 𝐵𝐶𝐷.
Now the fact that the diagram has not been drawn accurately means that we can’t use our protractor to measure any angles or a ruler to measure the length of any lines on the diagram. We’ve got to use geometric reasoning to come up with an argument for why triangle 𝐴𝐵𝐷 is congruent to triangle 𝐵𝐶𝐷. And congruent just means the same shape and size. We should be able to take triangle 𝐴𝐵𝐷 and lay exactly on top of triangle 𝐵𝐶𝐷. 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 here’s triangle 𝐴𝐵𝐷 in pink and here’s triangle 𝐵𝐶𝐷 in orange. And just before we get going on our geometric reasoning, let’s take a moment to realize that lines 𝐴𝐵 and 𝐷𝐶 are parallel, so this line here and this line here are parallel. And that’s what these two little arrows indicate. So we’ve got two parallel lines and three transversals. Now transversals are just lines that go across two parallel lines. Now, we’ll 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, we’ll 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 𝐵𝐶𝐷 — this angle here — plus 150 degrees — this angle here — sum to 180 degrees. So we’ve created an equation using those angles. And if we subtract 150 degrees from each side of the equation, we can see that angle 𝐵𝐶𝐷 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 𝐵𝐷𝐶 — this angle here — is vertically opposite this 30-degree angle here. And since they’re equal, that means that angle 𝐵𝐷𝐶 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. It’s 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 𝐷𝐵𝐴, is alternate to this angle here, angle 𝐵𝐷𝐶. So we know that angle 𝐷𝐵𝐴 is 30 degrees.
Now we also know that angles in a triangle sum to 180 degrees. So angle 𝐷𝐵𝐶 — this angle here — plus 30 degrees plus 30 degrees is equal to 180 degrees. And angle 𝐴𝐷𝐵 — this angle here — 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 𝐷𝐵𝐶 is 120 degrees and so is angle 𝐴𝐷𝐵. Let’s clear a little bit more space for our working out.
Now, so far, we’ve got enough information to tell us that triangles 𝐴𝐵𝐷 and 𝐵𝐶𝐷 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 𝐵𝐷 is common to both of those triangles. And that side is the same length in triangle 𝐴𝐵𝐷 as it is in triangle 𝐵𝐶𝐷. In triangle 𝐴𝐵𝐷, we have an angle of 30 degrees adjacent to side 𝐵𝐷 adjacent to an angle of 120 degrees. But the same is true in triangle 𝐵𝐶𝐷, we have an angle of 30 degrees followed by side length 𝐵𝐷 followed by an angle of 120 degrees. So not only is side 𝐵𝐷 common to both of those triangles, it’s also adjacent to angles of 30 degrees and 120 degrees in each case.
So we can say that triangle 𝐴𝐵𝐷 and triangle 𝐵𝐶𝐷 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 you’re 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.