In an isosceles trapezoid, a height 5 is known, a smaller base 7 and an angle at the base of 45, find a larger base.

Let’s designate our trapezoid ABCD. By condition we have: ВС = 7, ВН – height, ВН = 5, angle А = 45 °.
Consider a right-angled triangle ABН, in it one of the acute angles, the angle A = 45 °, which means that the angle B = 90 ° – 45 ° = 45 °. We get that the considered triangle ABН is isosceles (two angles are equal). This implies the equality of the sides BH = AH = 5.
Now we can find a larger base:
AD = BC + 2 * AH = 7 + 2 * 5 = 17.
Answer: The larger base of the trapezoid is 10.