Video Transcript
Is polygon 𝐴𝐵𝐶𝐷 similar to
polygon 𝐺𝐹𝐸𝑋?
From the figure, we can see that
the two polygons we’ve 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 they’re
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, it’s important to remember
that when we’re working with similar polygons, the order of the letters is
important. So if these polygons are similar,
then the angle at 𝐴 will correspond to the angle at 𝐺. The angle at 𝐵 will correspond to
the angle at 𝐹, and so on. We can therefore deduce that the
polygons have been drawn in the same orientation.
Let’s consider the angles first of
all then. In polygon 𝐺𝐹𝐸𝑋, we’ve been
given a marked angle of 110 degrees. And in polygon 𝐴𝐵𝐶𝐷, we’ve 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 𝐺𝐹𝐸𝑋, the
angle at 𝑋 will be 110 degrees. And in parallelogram 𝐴𝐵𝐶𝐷, the
angle at 𝐶 will be 70 degrees.
Let’s consider the angle at 𝐺 in
the polygon 𝐺𝐹𝐸𝑋. By extending the line 𝐹𝐺, we now
see that we have two parallel lines 𝑋𝐺 and 𝐸𝐹 and a transversal 𝐺𝐹. We know that corresponding angles
in parallel lines are equal, which means that the angle above the line 𝑋𝐺 will be
equal to the angle above the line 𝐸𝐹. It’s 110 degrees. We also know that angles on a
straight line sum to 180 degrees, which means the angle below 𝑋𝐺 will be 180 minus
110. It’s 70 degrees. This shows us that the angle at 𝐺
in polygon 𝐺𝐹𝐸𝑋 is equal to the angle at 𝐴 in polygon 𝐴𝐵𝐶𝐷.
We already said that opposite
angles in parallelograms are equal. So the angle at 𝐸 is also 70
degrees, which is equal to the angle at 𝐶 in polygon 𝐴𝐵𝐶𝐷. We could use the same logic in
parallelogram 𝐴𝐵𝐶𝐷 to show that the angles at 𝐵 and 𝐷 are each 110 degrees,
which are the same as the angles at 𝐹 and 𝑋 in the larger polygon. We’ve shown then that all pairs of
corresponding angles are indeed congruent. So our answer to the first check is
yes.
Let’s now consider whether
corresponding pairs of sides are in proportion. Firstly, from the figure, we can
see we’ve been given side length 𝐴𝐵; it’s 13 centimeters. And if the two polygons are
similar, this will correspond to 𝐺𝐹. We haven’t been given the length of
𝐺𝐹. But we know that opposite sides in
a parallelogram are equal in length. So it will be the same as 𝑋𝐸. Comparing the ratio of these two
sides then, we find that 𝐴𝐵 over 𝐺𝐹 is equal to 13 over 26, which simplifies to
one-half.
The other pair of potentially
corresponding sides we’ve been given are 𝐵𝐶 and 𝐸𝐹. The ratio here is 11.5 over 23,
which again simplifies to one-half. Now, I’ve written 𝐸𝐹 here. But really, if we’re to be
consistent with the order of letters, then we should really write 𝐹𝐸 as point 𝐹
corresponds to point 𝐵 and point 𝐸 corresponds to point 𝐶. However, in calculating the ratio,
the length 𝐸𝐹 is of course the same as the length 𝐹𝐸. 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, “are corresponding pairs of sides in proportion?”, is also yes. Hence, both of the criteria for
these two polygons to be similar are fulfilled. And so we can answer, yes, polygon
𝐴𝐵𝐶𝐷 is similar to polygon 𝐺𝐹𝐸𝑋.
