In a triangle 𝑋𝑌𝑍, if 𝑋𝑌 is
equal to 𝑋𝑍, what type of angle is the exterior angle at vertex 𝑍? Is it (A) acute, (B) right, (C)
obtuse, or (D) reflex?
We begin by recalling the
properties of our four angles. An acute angle is less than 90
degrees. A right angle is equal to 90
degrees. An obtuse angle lies between 90
degrees and 180 degrees. Finally, a reflex angle lies
between 180 degrees and 360 degrees. In this question, we are told that
in the triangle 𝑋𝑌𝑍, the length of 𝑋𝑌 is equal to the length of 𝑋𝑍. This means that we have an
isosceles triangle. In any isosceles triangle, two
interior angles are also equal. In this case, angle 𝑋𝑌𝑍 is equal
to angle 𝑋𝑍𝑌.
We know that angles in a triangle
sum to 180 degrees. This means that the interior angles
at vertex 𝑌 and vertex 𝑍 must both be acute. They must both be less than 90
degrees. Otherwise the sum would be greater
than 180. The exterior angle at vertex 𝑍 is
shown on the diagram. It is the angle between the side
length 𝑋𝑍 and the extension of side length 𝑌𝑍. We know that angles on a straight
line sum to 180 degrees. This means that the interior and
exterior angles at vertex 𝑍 must sum to 180. As the interior angle is acute, it
is less than 90, the exterior angle must be greater than 90 and obtuse. In the isosceles triangle drawn,
the exterior angle at vertex 𝑍 is obtuse.