### Video Transcript

๐ด๐ต๐ถ๐ท is a parallelogram. ๐ธ is the point where the diagonals
๐ด๐ถ and ๐ต๐ท meet. Prove that triangles ๐ด๐ต๐ธ and
๐ถ๐ท๐ธ 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 weโre 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 donโt know that the two
triangles weโre working with here are right angled.

Now, Iโve shaded the two triangles
that weโre interested in here: triangle ๐ด๐ต๐ธ in orange and triangle ๐ถ๐ท๐ธ in
pink. We need to have a closer look at
their sides and angles. Remember ๐ด๐ต๐ถ๐ท is a
parallelogram. And a key fact about parallelograms
is that opposite sides are equal in length. This means that side ๐ด๐ต is equal
to side ๐ถ๐ท. And so we have our first statement
about the congruency of these two triangles. And itโs a statement about the
sides.

Next, letโs consider the angles of
this parallelogram. And weโll start with the angle at
the center at the point ๐ธ. Angles ๐ด๐ธ๐ต and ๐ถ๐ธ๐ท 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 ๐ด๐ธ๐ต is equal to angle
๐ถ๐ธ๐ท.

Finally, letโs consider another
angle in these two triangles. And to do so, we need to recall
that ๐ด๐ต๐ถ๐ท is a parallelogram, which means that the lines ๐ด๐ต and ๐ถ๐ท are
parallel. If you look at angles ๐ด๐ต๐ธ and
๐ถ๐ท๐ธ, we can see that theyโre alternate angles in parallel lines. And a key fact about alternate
angles is that theyโre equal. This gives us our third statement
about the congruency of the two triangles: angle ๐ด๐ต๐ธ is equal to angle
๐ถ๐ท๐ธ.

Now, if we look at the three
statements that weโve made, we made one about a side and two about angles, which
means weโre using the third condition for triangle congruency โ Side Angle
Angle. We can conclude then that triangles
๐ด๐ต๐ธ and ๐ถ๐ท๐ธ 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.