write an indirect proof..if two angles are supplementary, than they both cannot be obtuse angles.

Accepted Solution

Let's assume that both angles are, in fact, obtuse supplements. We know that supplementary angles must add up to 180°. We also know by definition that obtuse angles are greater than 90°. If we were to take the two supplementary, obtuse angles, ∡A=90+x and ∡B=90+y, with x and y equaling positive real numbers, then we should be able to say that 180=m∡A+mB, or 180=90+x+90+y. By simplifying we get that 180=180+x+y. Simplify further and you get that 0=x+y. If we define x in terms of y, then x= -y. If we define y in terms of x, then y= -x. Because either x or y must be negative to make this statement true, one of the angle measures must be less than 90. If one of the angles must be less than 90 while the other is greater than 90, then one angle MUST be acute if the other is obtuse in order for them to be supplements of each other.