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. triangle lmn is an obtuse triangle. triangle lmn is a right triangle. the angle at vertex n is right. the angle at vertex l is right. the angle at vertex l is acute.

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# 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 $\angle N$ must be acute. This statement can be true. - **Triangle $LMN$ is an obtuse triangle**: By definition, a triangle with one obtuse angle is an obtuse triangle. Since $\angle M$ is obtuse, $\triangle LMN$ is an obtuse triangle. This statement is true. - **Triangle $LMN$ is a right triangle**: A right - triangle has one right angle ($90^{\circ}$). If $\angle M>90^{\circ}$, then $\angle L+\angle N<90^{\circ}$, so there cannot be a right angle. This statement is false. - **The angle at vertex $N$ is right**: If $\angle N = 90^{\circ}$, and $\angle M>90^{\circ}$, then $\angle L+\angle M+\angle N>180^{\circ}$, which violates the triangle angle sum property. This statement is false. - **The angle at vertex $L$ is right**: If $\angle L = 90^{\circ}$, and $\angle M>90^{\circ}$, then $\angle L+\angle M+\angle N>180^{\circ}$, which violates the triangle angle sum property. This statement is false. - **The angle at vertex $L$ 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 $\angle N$ must be acute. This statement can be true. # Answer: - The angle at vertex $N$ is acute. - Triangle $LMN$ is an obtuse triangle. - The angle at vertex $L$ is acute.

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QUESTION IMAGE

Question

Explanation:

Step1: Recall triangle angle sum property

Step2: Analyze each statement

Answer: