# Video: Pack 1 • Paper 3 • Question 14

Pack 1 • Paper 3 • Question 14

03:24

### Video Transcript

𝐴𝐵𝐶𝐷 is a parallelogram. 𝐸 is the point where the diagonals 𝐴𝐶 and 𝐵𝐷 meet. Prove that triangles 𝐴𝐵𝐸 and 𝐶𝐷𝐸 are congruent.

Two shapes are congruent if they are exactly the same size and shape. This means that corresponding pairs of sides must always be equal in length and corresponding angles must always be the same size. There are number of different ways that we can prove that two triangles are congruent.

The first way is we can prove that all three pairs of corresponding sides are equal in length. This is referred to as SSS or Side Side Side congruency.

The second way is to prove that the two triangles have two sides and an included angle in common. An included angle is the angle between the two sides that we’re working with. So it must be Side Angle Side in this order.

The third way is to prove that the two triangles have two angles and one side in common. And it can be any two angles and any side. So this is referred to as ASA for Angle Side Angle or it can be AAS or even SAA.

There is a fourth method for proving that two triangles are congruent. But this is specifically for right-angled triangles. And we don’t know that the two triangles we’re working with here are right angled.

Now, I’ve shaded the two triangles that we’re interested in here: triangle 𝐴𝐵𝐸 in orange and triangle 𝐶𝐷𝐸 in pink. We need to have a closer look at their sides and angles. Remember 𝐴𝐵𝐶𝐷 is a parallelogram. And a key fact about parallelograms is that opposite sides are equal in length. This means that side 𝐴𝐵 is equal to side 𝐶𝐷. And so we have our first statement about the congruency of these two triangles. And it’s a statement about the sides.

Next, let’s consider the angles of this parallelogram. And we’ll start with the angle at the center at the point 𝐸. Angles 𝐴𝐸𝐵 and 𝐶𝐸𝐷 are vertically opposite one another. And a key fact about vertically opposite angles is that they are equal. This gives us our second statement about the congruency of these two triangles: angle 𝐴𝐸𝐵 is equal to angle 𝐶𝐸𝐷.

Finally, let’s consider another angle in these two triangles. And to do so, we need to recall that 𝐴𝐵𝐶𝐷 is a parallelogram, which means that the lines 𝐴𝐵 and 𝐶𝐷 are parallel. If you look at angles 𝐴𝐵𝐸 and 𝐶𝐷𝐸, we can see that they’re alternate angles in parallel lines. And a key fact about alternate angles is that they’re equal. This gives us our third statement about the congruency of the two triangles: angle 𝐴𝐵𝐸 is equal to angle 𝐶𝐷𝐸.

Now, if we look at the three statements that we’ve made, we made one about a side and two about angles, which means we’re using the third condition for triangle congruency — Side Angle Angle. We can conclude then that triangles 𝐴𝐵𝐸 and 𝐶𝐷𝐸 are congruent using the Side Angle Angle rule.

The statements we made about the equality of sides and angles are also an essential part of our answer to this question.