### Video Transcript

The lines ๐ฟ two and ๐ฟ three are
parallel. The lines ๐ฟ four and ๐ฟ five are
parallel. Segment ๐ด๐บ is congruent to
segment ๐ธ๐ท. Prove that triangle ๐ด๐ต๐บ is
congruent to triangle ๐ถ๐ท๐ธ.

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 thatโs 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 itโs not enough to show
that three angles are the same since a triangle thatโs been enlarged will have the
same angles, but different-length sides. Two triangles that have the same
angles are called โsimilar triangles.โ

Letโs make sure weโre clear which
triangles weโre interested in. ๐ด๐ต๐บ and ๐ถ๐ท๐ธ are the triangles
highlighted. Iโve drawn them a little bit bigger
so we can follow clearly whatโs happening. One of the pieces of information we
have is that segment ๐ด๐บ and ๐ธ๐ท are congruent. This means theyโre exactly the
same. So we can write ๐ด๐บ is equal to
๐ธ๐ท.

Next, weโll use the fact that the
lines ๐ฟ two and ๐ฟ three are parallel and the lines ๐ฟ four and ๐ฟ five are also
parallel. Alternate angles are equal. Those are the angles that look like
theyโre enclosed in the letter ๐. That means that angle ๐บ๐ด๐ต is
equal to angle ๐ถ๐ท๐ธ. Remember itโs not enough just to
say alternate angles or to use the letter ๐. We must say alternate angles are
equal.

Next, letโs look at angle
๐ด๐ต๐บ. ๐ด๐ต๐บ is equal to angle ๐ถ๐ต๐น
because vertically opposite angles are equal. On our diagram, thatโs the angles
marked by the two arcs. We also know that corresponding
angles are equal. Those are the angles that look like
theyโre enclosed in the letter ๐น. What that means is angle ๐ถ๐ต๐น is
equal to angle ๐ท๐ถ๐ธ.

Since both angle ๐ด๐ต๐บ and ๐ท๐ถ๐ธ
were equal to ๐ถ๐ต๐น, that must mean that angle ๐ด๐ต๐บ is equal to angle ๐ท๐ถ๐ธ. Weโve shown that both of the
triangles share two angles and one side.

By the condition AAS then, the
triangles ๐ด๐ต๐บ and ๐ถ๐ท๐ธ are congruent.