Question

analyzing obtuse triangles
triangle lmn has an obtuse angle at vertex m, which statements could be true? choose three correct answers.
the angle at vertex l is right.
the angle at vertex l is acute.
the angle at vertex n is acute.
the angle at vertex n is right.
triangle lmn is an obtuse triangle.
triangle lmn is a right triangle.

Explanation:

Step1: Recall triangle types

An obtuse triangle has one obtuse angle (greater than \(90^\circ\)) and two acute angles (less than \(90^\circ\)). A right triangle has one right angle (\(90^\circ\)), and a triangle can't be both obtuse and right.

Step2: Analyze each statement

• "The angle at vertex L is right." → Obtuse triangle can't have right angle (since it has one obtuse, sum of angles \(180^\circ\); obtuse + right + acute > \(180^\circ\)) → False.
• "The angle at vertex L is acute." → Obtuse triangle has two acute angles, so L can be acute → Possible.
• "The angle at vertex N is acute." → Obtuse triangle has two acute angles, so N can be acute → Possible.
• "The angle at vertex N is right." → Same as right at L, can't have right angle in obtuse triangle → False.
• "Triangle LMN is an obtuse triangle." → Given it has an obtuse angle at M, so this is true.
• "Triangle LMN is a right triangle." → Can't be both obtuse and right → False.

Answer:

• The angle at vertex L is acute.
• The angle at vertex N is acute.
• Triangle LMN is an obtuse triangle.