Question

decide if each statement is always true, sometimes true, or never true.
scalene triangles are acute triangles. \t\talways \t\tsometimes \t\tnever
obtuse triangles are right triangles. \t\talways \t\tsometimes \t\tnever

Explanation:

Step1: Analyze "Scalene triangles are acute triangles"

A scalene triangle has all sides of different lengths. An acute triangle has all angles less than \(90^\circ\). A scalene triangle can be acute (e.g., sides 2, 3, 4, angles all acute), obtuse (e.g., sides 2, 3, 5? No, wait, 2, 3, 4: actually, a scalene triangle with sides 3, 4, 6: using the law of cosines, the largest angle (opposite 6) is \(\cos C=\frac{3^2 + 4^2-6^2}{2\times3\times4}=\frac{9 + 16 - 36}{24}=\frac{-11}{24}<0\), so it's obtuse. So scalene triangles can be acute, obtuse, or right (e.g., 3, 4, 5 is scalene and right). So "Scalene triangles are acute triangles" is sometimes true.

Step2: Analyze "Obtuse triangles are right triangles"

An obtuse triangle has one angle greater than \(90^\circ\). A right triangle has one angle equal to \(90^\circ\). A triangle can't have both an angle \(>90^\circ\) and an angle \(=90^\circ\) (since the sum of angles in a triangle is \(180^\circ\), so if one is \(>90^\circ\), the other two sum to \(<90^\circ\), so no right angle). So "Obtuse triangles are right triangles" is never true.

Answer:

For "Scalene triangles are acute triangles": Sometimes
For "Obtuse triangles are right triangles": Never