Question

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

Explanation:

1. A right triangle has one \(90^\circ\) angle (right angle) and the other two angles are acute (less than \(90^\circ\)) because the sum of angles in a triangle is \(180^\circ\).

• For "Triangle DEF is an obtuse triangle": Obtuse triangles have one angle greater than \(90^\circ\), but DEF has a \(90^\circ\) angle, so it's a right triangle, not obtuse. Eliminate this.
• For "The angle at vertex D is acute": Since \(\angle E = 90^\circ\), \(\angle D+\angle F=180 - 90=90^\circ\). So both \(\angle D\) and \(\angle F\) must be less than \(90^\circ\) (acute). This is true.
• For "The angle at vertex F is obtuse": As shown above, \(\angle F\) must be acute (since \(\angle D+\angle F = 90^\circ\)), so this is false.
• For "Triangle DEF is a right triangle": A triangle with a \(90^\circ\) angle is a right triangle. This is true.
• For "The angle at vertex D is obtuse": \(\angle D\) is part of \(\angle D+\angle F = 90^\circ\), so it must be acute, not obtuse. Eliminate this.

Answer:

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