# Video: Pack 3 • Paper 1 • Question 13

Pack 3 • Paper 1 • Question 13

04:01

### Video Transcript

The lines 𝐿 two and 𝐿 three are parallel. The lines 𝐿 four and 𝐿 five are parallel. Segment 𝐴𝐺 is congruent to segment 𝐸𝐷. Prove that triangle 𝐴𝐵𝐺 is congruent to triangle 𝐶𝐷𝐸.

In order to prove that two triangles are congruent, we need to prove that they are exactly the same. There are four conditions for congruency. SSS stands for side, side, side. When we have two triangles with all three sides equal, those triangles must be congruent. SAS stands for side, angle, side. If we have two triangles where two of the sides are equal and the included angle that’s the angle that falls between those two sides is equal, then those triangles must also be congruent.

ASA, SAA, and AAS always have saying that the two triangles have two angles that are equal and one side that is equal. This differs to SAS, where it was important that the order mattered. The angle had to be the included angle. In this case, order is less important.

Finally, RHS stands for right angle, hypotenuse, and side. If we have two right-angled triangles with the same-length hypotenuse and the same length for one of the other sides, then those two right-angled triangles are congruent. Remember it’s not enough to show that three angles are the same since a triangle that’s been enlarged will have the same angles, but different-length sides. Two triangles that have the same angles are called “similar triangles.”

Let’s make sure we’re clear which triangles we’re interested in. 𝐴𝐵𝐺 and 𝐶𝐷𝐸 are the triangles highlighted. I’ve drawn them a little bit bigger so we can follow clearly what’s happening. One of the pieces of information we have is that segment 𝐴𝐺 and 𝐸𝐷 are congruent. This means they’re exactly the same. So we can write 𝐴𝐺 is equal to 𝐸𝐷.

Next, we’ll use the fact that the lines 𝐿 two and 𝐿 three are parallel and the lines 𝐿 four and 𝐿 five are also parallel. Alternate angles are equal. Those are the angles that look like they’re enclosed in the letter 𝑍. That means that angle 𝐺𝐴𝐵 is equal to angle 𝐶𝐷𝐸. Remember it’s not enough just to say alternate angles or to use the letter 𝑍. We must say alternate angles are equal.

Next, let’s look at angle 𝐴𝐵𝐺. 𝐴𝐵𝐺 is equal to angle 𝐶𝐵𝐹 because vertically opposite angles are equal. On our diagram, that’s the angles marked by the two arcs. We also know that corresponding angles are equal. Those are the angles that look like they’re enclosed in the letter 𝐹. What that means is angle 𝐶𝐵𝐹 is equal to angle 𝐷𝐶𝐸.

Since both angle 𝐴𝐵𝐺 and 𝐷𝐶𝐸 were equal to 𝐶𝐵𝐹, that must mean that angle 𝐴𝐵𝐺 is equal to angle 𝐷𝐶𝐸. We’ve shown that both of the triangles share two angles and one side.

By the condition AAS then, the triangles 𝐴𝐵𝐺 and 𝐶𝐷𝐸 are congruent.