Question

can a quadrilateral have at most three obtuse angles?

Answer 1

yes

Answer 2

well, i guess you if are talking about non-Euclidean space, then yes a quadrilateral could have 3 obtuse angles.

however, assuming we're probably talking about only Euclidean space, no, it cannot have 3 obtuse angles.

think about it - draw a square, then attempt to increase 3 of the angles beyond 90 degrees while still keeping only 4 sides.

Answer 3

For a quadrilateral to have 3 obtuse angles, 3 of its angles must be greater than 90. Let X denote the measure of each of the obtuse angle, then 3X > 90*3=270. But the sum of all angles in an quadrilateral is 360. Let Y be the measure of the remaining angle. So 3X+Y=360 or 3X=360-Y. Therefore, 360-Y > 270 or Y < 360 - 270 = 90. So, if all the X are greater than 90 (obtuse), then if Y < 90, we satisfy these conditions.

For example. Let X = 91. Then Y = 360 - 3*91= 87: