Question

a) if one angle of a triangle is obtuse, can another also be obtuse? why or why not?
b) if one angle of a triangle is acute, can the other two angles also be acute? why or why not?
c) can a triangle have two right angles? why or why not?
d) if a triangle has one acute angle, is the triangle necessarily acute? why or why not?

a) if one angle of a triangle is obtuse, can another also be obtuse? why or why not? choose the correct answer below.

a. yes, because every obtuse triangle has at least two obtuse angles
b. no, because the sum of two obtuse angles is more than 180°
c. yes, because a triangle with two obtuse angles is isosceles
d. no, because the sum of two obtuse angles is less than 180°

Explanation:

To determine the answer, we use the fact that the sum of the interior angles of a triangle is \(180^\circ\). An obtuse angle is greater than \(90^\circ\) and less than \(180^\circ\). If we have two obtuse angles, say \(\alpha\) and \(\beta\), where \(\alpha> 90^\circ\) and \(\beta>90^\circ\), then \(\alpha+\beta>90^\circ + 90^\circ=180^\circ\). But the sum of all three angles of a triangle must be \(180^\circ\), so having two obtuse angles would make the sum of just those two angles exceed \(180^\circ\), which is impossible.

• Option A is wrong because an obtuse triangle can only have one obtuse angle.
• Option C is wrong because the property of being isosceles (having two equal sides/angles) has nothing to do with having two obtuse angles, and a triangle can't have two obtuse angles anyway.
• Option D is wrong because the sum of two obtuse angles is more than \(180^\circ\), not less.
• Option B is correct as it correctly states that the sum of two obtuse angles is more than \(180^\circ\), so a triangle can't have two obtuse angles.

Answer:

B. No, because the sum of two obtuse angles is more than \(180^\circ\)