Question

If triangle LMN has an obtuse angle at vertex M, which statements could be true? Check all that apply.

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

Answer 1

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

1,2,5

Answer 2

Answer: Triangle LMN is an obtuse triangle.

The angle at vertex L is acute.

The angle at vertex N is acute.

Explanation:

Here, triangle LMN has an obtuse angle at vertex M,

Thus, by the definition of obtuse angle triangle LMN is an obtuse triangle,

Now, Angle M is obtuse,

⇒ 90° < m∠ M < 180°

Since, by the property of a triangle,

m∠ L + m∠ M + m∠ N = 180°

⇒ m∠ M = 180° - ( m∠ L+ m∠ N )

⇒ 90° < 180° - ( m∠ L+ m∠ N ) < 180°

⇒ 90° - 180° < - ( m∠ L+ m∠ N ) < 0

⇒ -90° < - ( m∠ L+ m∠ N ) < 0

⇒ 90° > ( m∠ L+ m∠ N ) > 0 ( Since, a < b ⇒ - a > - b)

⇒ 90° > m∠ L and 90° > m∠ N

⇒ Both angle L and angle N are acute.