Question

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

Explanation:

Step1: Recall triangle angle sum property

The sum of angles in a triangle is \(180^{\circ}\). An obtuse angle is greater than \(90^{\circ}\) and less than \(180^{\circ}\).

Step2: Analyze each statement

• The angle at vertex \(N\) is acute:

Since \(\angle M>90^{\circ}\), and \(\angle L+\angle M+\angle N = 180^{\circ}\), then \(\angle L+\angle N=180^{\circ}-\angle M<90^{\circ}\). So both \(\angle L\) and \(\angle N\) can be acute. This statement can be true.

• The angle at vertex \(L\) is right (\(=90^{\circ}\)):

If \(\angle L = 90^{\circ}\) and \(\angle M>90^{\circ}\), then \(\angle L+\angle M>180^{\circ}\), which violates the angle - sum property of a triangle (\(\angle L+\angle M+\angle N=180^{\circ}\)). This statement is false.

• The angle at vertex \(N\) is right (\(=90^{\circ}\)):

If \(\angle N = 90^{\circ}\) and \(\angle M>90^{\circ}\), then \(\angle N+\angle M>180^{\circ}\), which violates the angle - sum property of a triangle (\(\angle L+\angle M+\angle N=180^{\circ}\)). This statement is false.

• Triangle \(LMN\) is an obtuse triangle:

By definition, a triangle with one obtuse angle is an obtuse triangle. Since \(\angle M\) is obtuse, this statement is true.

• The angle at vertex \(L\) is acute:

As \(\angle L+\angle N=180^{\circ}-\angle M<90^{\circ}\), \(\angle L\) can be acute. This statement can be true.

• Triangle \(LMN\) is a right triangle:

A right - triangle has one right angle (\(90^{\circ}\)). But if \(\angle M>90^{\circ}\), we can't have a right angle in the triangle (because of the angle - sum property \(\angle L+\angle M+\angle N = 180^{\circ}\)). This statement is false.

Answer:

The angle at vertex \(N\) is acute, Triangle \(LMN\) is an obtuse triangle, The angle at vertex \(L\) is acute.