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?

b) if one angle of a triangle is acute, can the other two angles also be acute? why or why not? choose the correct answer below.
a. no, because the other two angles must be obtuse.
b. no, because you cant have more than two acute angles in a triangle.
c. yes, because three angles that are less than 90° can sum to 180°
d. yes, because acute angles are supplementary

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

Explanation:

Part b)

To determine if a triangle with one acute angle can have the other two acute, we use the triangle angle - sum property (the sum of the interior angles of a triangle is \(180^{\circ}\)). An acute angle is less than \(90^{\circ}\). If we have three acute angles (each less than \(90^{\circ}\)), say \(a<90^{\circ}\), \(b < 90^{\circ}\), and \(c<90^{\circ}\), then \(a + b + c<90^{\circ}+90^{\circ}+90^{\circ}=270^{\circ}\), and more importantly, it is possible for \(a + b + c = 180^{\circ}\) (for example, an equilateral triangle with all angles \(60^{\circ}\)). Option A is wrong because the other two angles don't have to be obtuse. Option B is wrong as a triangle can have three acute angles. Option D is wrong because supplementary angles sum to \(180^{\circ}\), and acute angles (less than \(90^{\circ}\)) are not supplementary in this context. The correct option is the one that says yes, because three angles less than \(90^{\circ}\) can sum to \(180^{\circ}\).

We use the triangle angle - sum property (sum of interior angles of a triangle is \(180^{\circ}\)). A right angle is \(90^{\circ}\). If a triangle had two right angles, their sum would be \(90^{\circ}+90^{\circ} = 180^{\circ}\). But a triangle has three angles, so the sum of the three angles would be at least \(180^{\circ}\) (and actually more, since the third angle would be greater than \(0^{\circ}\)), which violates the triangle angle - sum property. Option A is wrong because the sum of two right angles is \(180^{\circ}\), not less than \(100^{\circ}\). Option B is wrong because the perpendicularity of right angles is not relevant here. Option D is wrong because a right triangle has only one right angle. The correct option is the one that says no, because the sum of two right angles is \(180^{\circ}\), so the sum of the three angles of the triangle would be more than \(180^{\circ}\).

Answer:

C. Yes, because three angles that are less than \(90^{\circ}\) can sum to \(180^{\circ}\)

Part c)