### Video Transcript

Is polygon ๐ด๐ต๐ถ๐ท similar to
polygon ๐บ๐น๐ธ๐?

From the figure, we can see that
the two polygons weโve been given are each parallelograms. So we can conclude that they are at
least the same type of shape to begin with. To determine whether theyโre
similar, we need to test two things. Firstly, we need to test whether
corresponding pairs of angles are congruent. And secondly, we need to test
whether corresponding pairs of sides are in proportion or in the same ratio.

Now, itโs important to remember
that when weโre working with similar polygons, the order of the letters is
important. So if these polygons are similar,
then the angle at ๐ด will correspond to the angle at ๐บ. The angle at ๐ต will correspond to
the angle at ๐น, and so on. We can therefore deduce that the
polygons have been drawn in the same orientation.

Letโs consider the angles first of
all then. In polygon ๐บ๐น๐ธ๐, weโve been
given a marked angle of 110 degrees. And in polygon ๐ด๐ต๐ถ๐ท, weโve been
given a marked angle of 70 degrees. One thing we do know about
parallelograms is that their opposite angles are equal. So in parallelogram ๐บ๐น๐ธ๐, the
angle at ๐ will be 110 degrees. And in parallelogram ๐ด๐ต๐ถ๐ท, the
angle at ๐ถ will be 70 degrees.

Letโs consider the angle at ๐บ in
the polygon ๐บ๐น๐ธ๐. By extending the line ๐น๐บ, we now
see that we have two parallel lines ๐๐บ and ๐ธ๐น and a transversal ๐บ๐น. We know that corresponding angles
in parallel lines are equal, which means that the angle above the line ๐๐บ will be
equal to the angle above the line ๐ธ๐น. Itโs 110 degrees. We also know that angles on a
straight line sum to 180 degrees, which means the angle below ๐๐บ will be 180 minus
110. Itโs 70 degrees. This shows us that the angle at ๐บ
in polygon ๐บ๐น๐ธ๐ is equal to the angle at ๐ด in polygon ๐ด๐ต๐ถ๐ท.

We already said that opposite
angles in parallelograms are equal. So the angle at ๐ธ is also 70
degrees, which is equal to the angle at ๐ถ in polygon ๐ด๐ต๐ถ๐ท. We could use the same logic in
parallelogram ๐ด๐ต๐ถ๐ท to show that the angles at ๐ต and ๐ท are each 110 degrees,
which are the same as the angles at ๐น and ๐ in the larger polygon. Weโve shown then that all pairs of
corresponding angles are indeed congruent. So our answer to the first check is
yes.

Letโs now consider whether
corresponding pairs of sides are in proportion. Firstly, from the figure, we can
see weโve been given side length ๐ด๐ต; itโs 13 centimeters. And if the two polygons are
similar, this will correspond to ๐บ๐น. We havenโt been given the length of
๐บ๐น. But we know that opposite sides in
a parallelogram are equal in length. So it will be the same as ๐๐ธ. Comparing the ratio of these two
sides then, we find that ๐ด๐ต over ๐บ๐น is equal to 13 over 26, which simplifies to
one-half.

The other pair of potentially
corresponding sides weโve been given are ๐ต๐ถ and ๐ธ๐น. The ratio here is 11.5 over 23,
which again simplifies to one-half. Now, Iโve written ๐ธ๐น here. But really, if weโre to be
consistent with the order of letters, then we should really write ๐น๐ธ as point ๐น
corresponds to point ๐ต and point ๐ธ corresponds to point ๐ถ. However, in calculating the ratio,
the length ๐ธ๐น is of course the same as the length ๐น๐ธ. So it makes no practical
difference.

The ratio of these pairs of
corresponding sides is therefore the same. As opposite sides in a
parallelogram are equal in length, the same will be true for the remaining two pairs
of corresponding sides. And so the answer to our second
check, โare corresponding pairs of sides in proportion?โ, is also yes. Hence, both of the criteria for
these two polygons to be similar are fulfilled. And so we can answer, yes, polygon
๐ด๐ต๐ถ๐ท is similar to polygon ๐บ๐น๐ธ๐.