Bunuel wrote:

If a = 25 in the figure above, which of the following statements must be true?

I. n is parallel to p

II. q is parallel to r

III. b = 65

(A) None

(B) I only

(C) I and II only

(D) II and III only

(E) I, II and III

Attachment:

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Since we don’t know the measure of the interior angle of the triangle that also contains angles a and b, there is no way we can determine the measure of angle b, even when we are given angle a = 25. Thus, Roman numeral III is not true. This eliminates answer choices D and E.

Now we are left with statements I and II. Since we need to determine the statement or statements that MUST BE true, let’s see if it’s possible for them not to be true. That is, let’s assume the contrary.

The contrary to statement I is: lines n and p are not parallel. If they are not parallel, then the angle formed by lines p and q that is the corresponding angle to angle a would not be 25 degrees.

The contrary to statement II is: lines q and r are not parallel. If they are not parallel, then the angle formed by lines p and q that is the alternate interior angle to angle a would not be 25 degrees.

We can see that the angle formed by lines p and q in the contrary to statement I is the same angle in the contrary to statement II. Since it is possible that it would not be 25 degrees, it is possible that lines n and p are not parallel and lines q and r are not parallel. Thus, statements I and II don’t have to be true.

Answer: A

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