Prove:

Let the midpoint of AC be r and connect QR, a1r.

Because q and r are midpoint,

So QR//AB and QR=( 1/2)AB.

Because A 1P//AB, and A 1P=( 1/2)AB.

So QR//=A 1P

So quadrilateral A 1PQR is a parallelogram.

So PQ//A 1R

Because m is the midpoint of AA 1 and n is the quarter point of AC,

So A 1R//MN

So PQ//MN

Because MN is on the BMN plane,

So PQ// plane BMN

2

Take the midpoint of B 1C 1 as s and connect MQ.

Firstly, it is proved that plane A 1AQS⊥ plane BMC.

Because BC⊥AQ, BC⊥A 1A, and AQ ∩ A1A = A.

So BC⊥ plane A 1AQS.

And BC is contained in a planar BMC,

So aircraft A 1AQS⊥ Aircraft BMC.

The intersection of two vertical planes is MQ, A⊥MQ is in H, connecting BH.

Then AH⊥ aircraft BMC

Therefore, ∠ABH is the included angle between AB and planar BMC.

AQ=2，AM=4，MQ=2√5

So AH=AQ*AM/MQ=4√5/5.

So sin ∠ abh = ah/ab = ∠ 5/5.