### Video Transcript

From the figure below, determine
the correct inequality from the following: ๐ด๐ต is greater than ๐ถ๐ต, ๐ด๐ต is less
than ๐ถ๐ต, ๐ด๐ต is greater than ๐ด๐ถ, or ๐ด๐ถ is less than ๐ถ๐ต.

Looking at the diagram, we can see
that ๐ด๐ต, ๐ถ๐ต, and ๐ด๐ถ all represent the lengths of sides of a triangle. Weโve been given four possibilities
for relationships that could exist between the lengths of different pairs of
sides. We havenโt been given any lengths
in the diagram. Instead, weโve been given some
information about some of the angles. This suggests that we need to
consider the relationship between the lengths of sides and the size of angles in a
triangle. And therefore, weโre going to
approach this question using the angleโside triangle inequality.

Hereโs what the angleโside triangle
inequality tells us. If one angle of a triangle has a
greater measure than another angle, then the side opposite the greater angle is
longer than the side opposite the lesser angle. Basically, what this means is that
the longest side of a triangle is opposite the largest angle. The shortest side is opposite the
smallest angle. And the middle side is opposite the
middle angle. In the diagram, however, weโve only
currently got the size of one of the angles in the triangle. So we need to consider how we can
find the other angles.

First of all, letโs consider angle
๐ด๐ต๐ถ. We can see that the lines ๐ด๐ท and
๐ถ๐ต are parallel as theyโve been marked with blue arrows on their lengths. The line ๐ด๐ต is a transversal
through these parallel lines. And therefore, we can see that the
angle ๐ด๐ต๐ถ and the angle of 66 degrees are alternate interior angles. Which means that theyโre
congruent. So angle ๐ด๐ต๐ถ is also 66
degrees.

Now that we know the measures of
two of the angles in the triangle, we can calculate the third because the angle sum
in a triangle is always 180 degrees. So angle ๐ด๐ถ๐ต can be found by
subtracting 52 degrees and 66 degrees from 180 degrees. Itโs 62 degrees.

So now that we know the sizes of
all three angles in the triangle, we can deduce something about the lengths of the
three sides. The largest angle in the triangle
is 66 degrees. And the angleโside triangle
inequality tells us that the longest side of the triangle will be opposite this
angle. So the longest side of the triangle
is the side ๐ด๐ถ. The second biggest angle in the
triangle is the angle of 62 degrees which is opposite the side ๐ด๐ต. This means then that ๐ด๐ต is the
second longest side of the triangle. The smallest angle of 52 degrees is
opposite the shortest side of the triangle. So ๐ถ๐ต is the shortest side.

Now that we have the three sides of
the triangle ordered from longest to shortest, we can turn our attention to the four
inequalities and determining which are true. Firstly, is ๐ด๐ต greater than
๐ถ๐ต? Yes, ๐ด๐ต appears above ๐ถ๐ต in the
list. Which means this first inequality
is true. Is ๐ด๐ต less than ๐ถ๐ต? Well, this is the reverse of the
inequality that weโve just shown to be true. Therefore, this one must be
false. Thirdly, is ๐ด๐ต greater than
๐ด๐ถ? No, ๐ด๐ถ is the longest side of the
triangle. So this inequality is also
false. And finally, is ๐ด๐ถ less than
๐ถ๐ต? Again, this is false. ๐ด๐ถ is the longest side of the
triangle.

So we can conclude that of the four
inequalities, only one is true. ๐ด๐ต is greater than ๐ถ๐ต.