eitijan wrote:OA C

Tricky.

(1) is not sufficient because it simply tells us that Angle RSQ and Angle RQS are equivalent angles. We still have no way to calculate Angle QSU.

(2) likewise is not sufficient because it simply tells us Angle SUT and Angle TUS are equivalent angles.

However, combining (1) and (2), we can set up a convenient system of two equations in which two of the variables will cancel, allowing us to solve for x.

First, set Angles RSQ and QSU equal to y & set Angles SUT and UST equal to z.

Then, x + y + z = 180 (since they lie on line segment RT).

Also the angles of Quadrilateral PQSU sum to 360. Angle PQS must be 180-y and Angle PUS must be 180-z since they lie on line segments adjacent to angles of measure y and z, respectively.

So, 90 + 180 - y + 180 - z + x = 360.

This simplifies to x - y - z = -90.

Combining our two equations, we see x = 45.

Therefore, (1) and (2) combined are sufficient.

An alternate way to combine the two statements.

1. Plug in values for the angle measurements, satisfying the constraints in the problem and the rules of geometry.
2. Calculate the value of x.
3. Plug in a different set of values, again satisfying the constraints in the problem and the rules of geometry.
4. Calculate the value of x.

If the value of x is the SAME in each case, the two statements combined are SUFFICIENT.
If the value of x is NOT the same in each case, the two statements combined are INSUFFICIENT.

Below are two sets of angle measurements that satisfy the rules of geometry and the constraints in the two statements:

∠PRT + ∠PTR = 90 because triangle PRT is a right triangle.
Since QR=RS, ∠RQS = ∠RSQ.
Since ST=TU, ∠UST = ∠SUT.
Since the sum of angles that form a straight line is 180, x = 180 - ∠RQS - ∠UST.



In each case, x=45.
Thus, the two statements combined are SUFFICIENT to determine that x=45.

The correct answer is C.