In triangle ABC, angle B = 135, find the height AH if side AB = 12.

AHB triangle is rectangular, because AH is the height perpendicular to HB. AB – hypotenuse, AH – leg opposite to the angle ABH.
Angles ABH and ABC are adjacent, their sum is 180 degrees, angle ABH = 180-angle ABC = 180-135 = 45 degrees.
The ratio of the opposite leg to the hypotenuse is equal to the sine of the angle, which means sinABH = AH / AB. From here we can find AН: AH = AB * sinABH = AB * sin45 = 12 * √2 / 2 = 6√2≈8.485.