(sat prep) in the figure, if pn = ln,
np ∥
mq
, and
ql
bisects ∠pqm, what is the value of x? do this i desperate

Hello from MrBillDoesMath!

x = 67

Discussion:

This is a tough one and it's late. But it looks good to me at 8:30pm );

Study attachment where I added angle labels "a", "b", and c"". I also drew  auxiliary line LS parallel to QM (and hence parallel to PN)

1. As PN =  NL, the bases angles ("b") of the isosceles triangle PNL are equal

2. As LQ bisects angles PQM,  let the equal angles be labeled "a".

3. Angle PLS = b as alternate interior angles of parallel lines (PN, SL) are equal.

4. Angle SLQ= a as  alternate interior angles of parallel lines (QM, SL) are equal.

5. In triangle QLM, a + angle QLM + 54 = 180, so angle QLM =  180 - 54 - a = 126- a

6. Line NLM has 180 degrees so

    180  = (126-a) + a + b + b =>    180 = 126 + 2b => b =27

7. Angle PLQ = 70 = a + b. Using 6) this gives 70 =  a + 27 or

   a = 70 - 27 = 43.

8. From triangle QPL,  

             a + x + 70 = 180    =>

             43 + x + 70 = 180   =>

              113 + x = 180         =>

              x = 180 - 113 = 67

Good luck. Please tell me if you see any errors.

Thank you,

MrB