Find the value of X in the given quadrilateral.

This question tests equilateral triangles, the triangle inequality theorem,
and the relationship between sides and angles in a triangle.

Draw a line that connects vertex A (on the diagram) to vertex C.
The larger triangle, ABC, has two equal sides.
Having seen answer choices, ask: can x = 60° or 90°?

Can x = 60°?
By definition of an equilateral triangle, if AC = 10, then x = 60°
x can = 60° if ∆ ABC is equilateral*
∆ ABC is equilateral only if AC = 10

Can side AC = 10?

Now consider ∆ ACD
Triangle inequality theorem:
The third side of a triangle should be greater than the difference between the other two sides and
less than the sum of the other two sides. So:
(9-1) < AC < (9+1)
8 < AC < 10
AC cannot = 10
AC must be less than 10

If AC must be less than 10, then
x must be less than 60°
If the length of a side of a triangle decreases, the angle measure opposite that side decreases, too.

The sides that are opposite the y angles both = 10
(AB and BC = 10)
ONLY if AC = 10, then x = 60°

AC must be less than 10. Therefore
x must be less than 60°

Answer C


* Or plug in.
If x = 60, then the other two angles, y, must also each be 60°
x + 2y = 180°
60 + 2y = 180
2y = 120
y = 60