Question

if triangle def has a \\(90^{\circ}\\) 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:

Identify the given properties

We are given a triangle \(DEF\) with a \(90^\circ\) angle at vertex \(E\).

$$ m\angle E = 90^\circ $$

Classify the triangle

A triangle containing a \(90^\circ\) angle is classified as a right triangle.
Therefore, the statement "Triangle \(DEF\) is a right triangle" is true.

Analyze the remaining angles

Using the Triangle Angle Sum Theorem, the sum of all interior angles in triangle \(DEF\) is \(180^\circ\).

$$ m\angle D + m\angle E + m\angle F = 180^\circ $$

Substitute \(m\angle E = 90^\circ\):

$$ m\angle D + 90^\circ + m\angle F = 180^\circ \implies m\angle D + m\angle F = 90^\circ $$

Since angle measures in a triangle must be positive:

$$ m\angle D < 90^\circ \quad \text{and} \quad m\angle F < 90^\circ $$

This means both \(\angle D\) and \(\angle F\) must be acute angles.
Therefore, the statement "The angle at vertex \(D\) is acute" is true.

Answer:

• Triangle DEF is an obtuse triangle.
• The angle at vertex D is acute. (Correct answer)
• The angle at vertex F is obtuse.
• Triangle DEF is a right triangle. (Correct answer)
• The angle at vertex D is obtuse.