Question

A certain kite has exactly one acute angle, and it measures 16°. What is the maximum whole-number measure of the angle opposite that angle?
74º
90º
106º
164º

Answer 1

First lets start with the definition of a kite. :
Kite Definition: A quadrilateral with two distinct pairs of equal adjacent sides.
Wikipedia definition: In Euclidean geometry, a kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other.

Answer 2

Now there is a theorem we can prove in geometry, angles between unequal sides of a kite are equal . Let's label those two angles y and y in the diagram.
And the angle opposite to 16 lets label x.

Answer 3

Now we know the interior sum of a quadrilateral is 2*180 = 360 degrees.
Therefore we have
x + y + y + 16 = 360
x + y + y = 344

Now we are given that none of the other three angles are acute. That implies angle y is not an acute angle, so the minimum angle y can be is 90 degrees. The reason we are minimizing angle y is in order to maximize angle x.

x + 90 + 90 = 344
x + 180 = 344
x = 344 - 180 = 164