Statement 1):  Two angles are acute, and two angles are obtuse.

This statement is not necessarily true.  Two intersecting lines may also be perpendicular to each other, which means that all four angles are 90 degrees.

There is not enough information to justify this statement.

Statement 2:  Any two non-perpendicular intersecting lines with known equations. 

 This is a tricky statement. 

When two functions meet, they must have an intersecting point .  Both functions  can be set equal to each other to determine that intersecting point.  

Draw an imaginary line  where the line is perpendicular to the first function and passes through the second function at some known arbitrary point . Point  will need to be determined.

The equation of the third function can be determined since imaginary line  intersects equation  at , and  is also perpendicular to .  The slope of  can be determined since it's the negative reciprocal of the slope of .

After the equation  has been determined by using point  and the slope of , the point  can also be determined by setting the functions  equal to each other.

Once the points  have been determined, the distance formula may be used to determine the lengths from , , and .

The Law of Sines can then be used to determine the interior angles of the triangle bounded by .  Knowing one angle at the intersection of  is sufficient to solve for all four angles by supplementary and opposite angle rules.

Therefore: 

With just statement 1, there is no definitive relationship between angle y and angle x.

With just statement 2, there is a difinitive relationship between angle y and angle x, but we don't know the measure of angle y.

If you have the information from both statements 1 and 2, you can determine the measure of , so .

Fortunately, this is a data sufficiency question, so you don't have to actually do the math, you just have to know that you have all the information to do the math.

With just statement 1, we know  , so we could determine the measure of angle y, but there is no definitive relationship between z and x nor y and x, because we don't know if lines p and q are parallel.

With just statement 2,we know  , so we could determine the measure of angle z, but there is no definitive relationship between y and x nor z and x, because we don't know if lines p and q are parallel.

Even if we have the information from both statements 1 and 2, we still do not know if lines p and q are parallel, therefore there is no difinitive relationship between angle y and angle x nor angle z and angle x.

Although the question itself tells us that lines p and q are parallel, the iinformation in statement 1 is insufficient to determine a definitive value for either y or z.

With the information from the question that lines p and q are parallel, and the added information from statement 2 that , using the rules of supplementary angles and alternate interior angles, we can determine the value of x.

Statement 1 alone gives insufficient information. It only gives a relationship between  and , but no further clues about the measures of any angle are given.

Statement 2 alone gives insufficient information; , since the angles with those measures are vertical; since no measures are known,  cannot be calculated.

Now assume both statements are true. Again, from Statement 1, ; from Statement 2, . Again, from the diagram, . Three angles with measures  together form a straight angle, so 

Therefore, both statements together are sufficient to answer the question.

Assume Statement 1 alone. From the diagram, the three angles of measure  together form a straight angle, so

From Statement 1,

,

so by the subtraction property of equality,

Assume Statement 2 alone. , but there is no clue about the value of  or any other angle measure, so the value of  cannot be computed.

Assume Statement 1 alone.  and  are vertical angles, so they must have the same measure; .

Assume Statement 2 alone.  and  are alternating exterior angles, which are congruent if and only if ; however, we do not know whether , so no conclusions can be made about .

Assume Statement 1 alone. , since the three angles of these measures together form a straight angle. Also, from Statement 1, . Therefore:

Assume Statement 2 alone. Again, , and from Statement 2, . Therefore,

Since the angles of measures  and  form a linear pair, they are supplementary, and  .

We show that both statements together give insufficient information. Assume both statements together. Consider the following two cases:

Case 1: 

Since the angles of measures  and  form linear pairs with the angle of measure , each is supplementary to that angle and, subsequently,  .

The angle of measure  is vertical to the angle of measure , so the two must be congruent; .

From Statement 1, 

Since  and , then .

These values are therefore consistent with the diagram and with both statements.

Case 2: 

We can find the values of the other variables as before:

, so .

Again, all values are consistent with the diagram and both statements. 

Since at least two different values of  satify the conditions, the two statements are insufficient.

Statement 1 alone gives us that , but reveals no clues about any of the eight angle measures. From Statement 2 alone, that , we can assume that , and  all have measure , but no clues are given about any of the other four angles—in particular, .

Assume both statements are true. From Statement 1, , and by way of the Parallel Postulate, corresponding angles have the same measure—in particular, . From Statement 2, we know that . From these two statements, .