The inner angles of a convex quadrilateral are in the ratio 1: 2: 3: 4. Find the sum of the smallest and largest angles.

Let’s denote the value of the angle BAD through X0, then, by condition, the angle ABC = 2 * X0, the angle BCD = 3 * X0, the angle ADC = 4 * X0.

The sum of the interior angles of a quadrilateral is 360, then:

X + 2 * X + 3 * X + 4 * X = 360.

10 * X = 360.

X = BAD = 360/10 = 360.

Angle ABC = 2 * 36 = 72, angle BCD = 3 * 36 = 108, angle ADC = 4 * 36 = 144.

Smaller angle BAD = 360, larger angle ADC = 144.

Then BAD + ADC = 36 + 144 = 180.

Answer: The sum of the smallest and largest angles is 180.