Answer:
From the information given, we can draw the following diagram:
S ------- P
/ \ / \
/ \ QR / \
/ \ / / \
/ Q \
R-----------------P
MN
Here, SP || QR, so we have ∠QSP = ∠PQR and ∠SPQ = ∠QRP.
Since m∠QRS = 120, we have m∠QRP = 360 - m∠QRS = 360 - 120 = 240.
Since m∠QPS = 135 and ∠QSP = ∠PQR, we have m∠PQR = 360 - m∠QPS = 360 - 135 = 225.
Let x = MN. Since MN is the median of the trapezoid, we have MP = NR = x.
Now, consider the triangles SPQ and RPQ. We have:
tan(∠SPQ) = x/12 (using the tangent ratio in triangle SPQ)
tan(∠RPQ) = x/12 (using the tangent ratio in triangle RPQ)
Since ∠SPQ = ∠RPQ (as they are corresponding angles), we have:
x/12 = x/12
x = 12
Therefore, MN = x = 12.