Question

if triangle def has a 90° angle at vertex e, which statements are true? choose two correct answers. triangle def is a right triangle. triangle def is an obtuse triangle. the angle at vertex f is obtuse. the angle at vertex d is acute. the angle at vertex d is obtuse.

Explanation:

• A triangle with a \(90^{\circ}\) angle is a right - triangle. So, "Triangle DEF is a right triangle" is correct.
• The sum of angles in a triangle is \(180^{\circ}\). If one angle (\(\angle E = 90^{\circ}\)), then the sum of the other two angles (\(\angle D+\angle F=180 - 90=90^{\circ}\)). Acute angles are angles between \(0^{\circ}\) and \(90^{\circ}\). Since \(\angle D+\angle F = 90^{\circ}\), both \(\angle D\) and \(\angle F\) must be acute. So, "The angle at vertex D is acute" is correct.
• An obtuse triangle has one angle greater than \(90^{\circ}\). Since \(\angle E = 90^{\circ}\), it is not an obtuse triangle. Also, \(\angle D\) and \(\angle F\) are acute (as shown above).

Answer:

Triangle DEF is a right triangle; The angle at vertex D is acute.