Question

which of the following statements is true about triangles?
a. an obtuse triangle must have at least 2 obtuse angles.
b. all triangles contain at least 2 acute angles.
c. a right triangle also has an obtuse angle.
d. an acute triangle has only two acute angles.
○ a
○ b
○ c
○ d

Explanation:

Step1: Analyze option A

An obtuse triangle has only one obtuse angle (since the sum of angles in a triangle is \(180^{\circ}\)). If there were two obtuse angles (each \(> 90^{\circ}\)), their sum would exceed \(180^{\circ}\), which is not possible.

Step2: Analyze option B

Let's consider the three cases:

• Acute triangle: All three angles are acute (\(<90^{\circ}\)).
• Right triangle: One right angle (\( = 90^{\circ}\)) and two acute angles (since \(180 - 90=90\), and each of the remaining two angles must be \(<90^{\circ}\)).
• Obtuse triangle: One obtuse angle (\(>90^{\circ}\)) and two acute angles (because \(180-\text{obtuse angle}<90^{\circ}\) for each of the remaining two angles). So all triangles have at least two acute angles.

Step3: Analyze option C

A right triangle has one right angle (\(=90^{\circ}\)) and two acute angles. It does not have an obtuse angle.

Step4: Analyze option D

An acute triangle has all three angles acute (\(<90^{\circ}\)), not just two.

Answer:

B. All triangles contain at least 2 acute angles.