Video Transcript
Which of the following is
correct? Is it option (A) the measure of
angle 𝑋𝑍𝑇 is greater than angle 𝑋 which is greater than angle 𝑋𝑍𝑌? Option (B) angle 𝑋𝑌𝑆 is greater
than angle 𝑋𝑍𝑌 which is greater than angle 𝑋𝑌𝑍. Option (C) angle 𝑋 is greater than
angle 𝑋𝑍𝑌 which is greater than angle 𝑋𝑌𝑍. Option (D) angle 𝑋𝑍𝑌 is greater
than angle 𝑋𝑌𝑍 which is greater than angle 𝑋. Or finally, option (E) angle 𝑋𝑌𝑍
is greater than angle 𝑋𝑌𝑆 which is greater than angle 𝑋𝑍𝑌.
As many of the angles in our
options are the same, it makes sense to fill in all the missing angles on the
diagram first. We can do this using our properties
of interior and exterior angles. We know that the angles on a
straight line sum to 180 degrees. We could use this to calculate the
interior angle at vertex 𝑌 and vertex 𝑍. 180 minus 137 is equal to 43. This means that the interior angle
at vertex 𝑍 is 43 degrees. 180 minus 115 is equal to 65. So the interior angle at vertex 𝑌
is 65 degrees.
To calculate the third interior
angle, the one at vertex 𝑋, we could use the fact that angles in a triangle sum to
180 degrees. We could add 43 and 65 and then
subtract this from 180. Alternatively, we could use the
fact that any exterior angle of a triangle is equal to the sum of the nonadjacent
interior angles. 137 is therefore equal to 65 plus
𝑥. Subtracting 65 from both sides of
this equation gives us 𝑥 is equal to 72. We could also have calculated this
by subtracting 43 degrees from 115 degrees. The third interior angle is 72
degrees.
We now have the five angles
required. Angle 𝑋 is equal to 72
degrees. Angle 𝑋𝑌𝑍 is equal to 65
degrees. Angle 𝑋𝑌𝑆 is equal to 115
degrees. Angle 𝑋𝑍𝑇 is equal to 137
degrees. And finally, angle 𝑋𝑍𝑌 is equal
to 43 degrees. We can now substitute these in to
our five options. All five of our options want the
numbers to be in descending order. This is because the first angle
needs to be greater than the second angle which needs to be greater than the third
angle.
In option (E), 65 is not greater
than 115. In option (D), 43 is not greater
than 65. In option (C), while 72 is greater
than 43, this is not greater than 65. The same problem occurs with option
(B). 115 is greater than 43, but 43 is
not greater than 65. In option (A), our numbers are in
descending order. 137 is greater than 72 which is
greater than 43. This means that the correct answer
is angle 𝑋𝑍𝑇 is greater than angle 𝑋 which is greater than angle 𝑋𝑍𝑌.
