If the angles of a triangle, in degrees, are $x$, $3x+20$, and $6x$, then the triangle must be:

Step 1 (Apply Angle Sum Theorem): The sum of the interior angles of any triangle is always $180^{\circ }$. Therefore:

Step 2 (Solve for x): Combine like terms and solve the linear equation:

Step 3 (Determine individual angles): Calculate each angle in degrees:

• Angle 1: $x=16^{\circ }$

• Angle 2: $3x+20=3(16)+20=48+20=68^{\circ }$

• Angle 3: $6x=6(16)=96^{\circ }$

Step 4 (Classify the triangle): Since one of the angles ($96^{\circ }$) is strictly greater than $90^{\circ }$, the triangle is an obtuse triangle.

Step 5 (Conclusion): The triangle must be an obtuse triangle, corresponding to option index 0.