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?

c) can a triangle have two right angles? why or why not? choose the correct answer below
a. yes, because the sum of two right angles is less than 180°.
b. no, because right angles are perpendicular.
c. no, because the sum of two right angles is 180°, thus the sum of the three angles of the triangle would be more than 180°
d. yes, because every right triangle has at least two right angles.

d) if a triangle has one acute angle, is the triangle necessarily acute? why or why not? choose the correct answer below
a. yes, because if one angle is less than 90°, they are all less than 90°
b. yes, because acute angles are supplementary
c. no, because you can have a right or an obtuse triangle with one acute angle
d. no, because acute triangles have three acute angles

Explanation:

Part (c)

To determine if a triangle can have two right angles, we use the triangle angle - sum property, which states that the sum of the interior angles of a triangle is \(180^{\circ}\). A right angle is \(90^{\circ}\). If a triangle had two right angles, the sum of these two angles would be \(90^{\circ}+90^{\circ} = 180^{\circ}\). But a triangle has three angles, so the sum of all three angles would be greater than \(180^{\circ}\) (since we still have a third angle), which violates the triangle angle - sum property.

• Option A is wrong because the sum of two right angles is \(180^{\circ}\), not less than \(180^{\circ}\).
• Option B is wrong because the reason about right angles being perpendicular is not relevant to the triangle angle - sum.
• Option D is wrong because a right triangle has only one right angle.

An acute triangle is defined as a triangle with all three angles being acute (less than \(90^{\circ}\)). If a triangle has one acute angle, the other two angles could be a right angle (in a right triangle) or an obtuse angle (in an obtuse triangle) and one other angle. For example, a right triangle has one right angle (\(90^{\circ}\)) and two acute angles, and an obtuse triangle has one obtuse angle (greater than \(90^{\circ}\)) and two acute angles.

• Option A is wrong because just because one angle is less than \(90^{\circ}\), the other angles can be \(90^{\circ}\) or greater than \(90^{\circ}\).
• Option B is wrong because acute angles are not supplementary (supplementary angles sum to \(180^{\circ}\), and acute angles are less than \(90^{\circ}\), so their sum is less than \(180^{\circ}\)).
• Option D is wrong because while acute triangles have three acute angles, the question is about a triangle with one acute angle, and the reason in D is not the correct explanation for why a triangle with one acute angle is not necessarily acute.

Answer:

C. No, because the sum of two right angles is \(180^{\circ}\), thus the sum of the three angles of the triangle would be more than \(180^{\circ}\)

Part (d)