Question

can a right triangle also be obtuse? explain your reasoning.
o yes: by the exterior angle theorem, if one of the exterior angles is obtuse, then it is possible for the two nonadjacent interior angles to be obtuse.
o yes: a right angle measures 90°. by the corollary to the triangle sum theorem, the other two angles must be complementary. so, one of those angles can be obtuse as long as the sum of the two angles is 90°.
o no: by the corollary to the triangle sum theorem, the angles in a right triangle that are not right angles must be complementary.
o no: by the exterior angle theorem, if one of the exterior angles is obtuse, the two nonadjacent interior angles each have half the measure of the exterior angle, so they will both be acute.

Explanation:

A right - triangle has one angle that measures \(90^{\circ}\). The sum of the interior angles of a triangle is \(180^{\circ}\). So, if one angle is \(90^{\circ}\), the sum of the other two angles is \(180 - 90=90^{\circ}\). An obtuse angle is an angle that measures more than \(90^{\circ}\). If one of the non - right angles were obtuse (greater than \(90^{\circ}\)), then the sum of the two non - right angles would be greater than \(90^{\circ}\), which contradicts the fact that the sum of the two non - right angles in a right - triangle is \(90^{\circ}\).

Answer:

No. By the Corollary to the Triangle Sum Theorem, the angles in a right triangle that are not right angles are complementary. So, one of those angles cannot be obtuse as long as the sum of the two angles is \(90^{\circ}\).