The bases of an isosceles trapezoid are 6 and 4, one of the angles is 45 degrees. Find the height of the trapezoid.

Let us lower the height from the top of the obtuse angle C to the larger base AD.

Since the trapezoid is isosceles, the angles at the base are equal, the angle BAD = CDA = 45.

By the property of an isosceles trapezoid, the height lowered from the apex of an obtuse angle to the larger base divides it into two segments, the smaller of which is equal to the half-difference of the bases, and the larger half-sum of the bases.

DН = (АD – ВС) / 2 = (6 – 4) / 2 = 2/2 = 1 cm.

Consider a right-angled triangle СDН, in which the angle НDС is 45, then the angle НСD = 180 – 90 – 45 = 45. Then the triangle СНD is isosceles, DH = СН = 1 cm.

Answer: The height of the trapezoid is 1 cm.