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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.

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

1,2,5

Answer: Triangle LMN is an obtuse triangle.

The angle at vertex L is acute.

The angle at vertex N is acute.

Step-by-step 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.

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