Question

"On a separate piece of paper, try to draw a triangle with side lengths of 3 centimeters and 6 centimeters, and an included angle of 120 degrees. Determine whether the given segments and angle produce a unique triangle, more than one triangle, or no triangle."

Answer 1

The given segments and angle can produce a unique triangle.

Step-by-step explanation:

To determine whether the given segments and angle produce a unique triangle, more than one triangle, or no triangle, we can use the triangle inequality theorem and the concept of angles in a triangle.

Applying the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In this case, 3 + 6 = 9, which is greater than the third side of a triangle. Therefore, the given segments can form a triangle.

Now, let's analyze the angle. The given included angle is 120 degrees. The sum of the three angles of a triangle is always 180 degrees. If the two sides adjacent to the given included angle are 3 cm and 6 cm, then the third angle would have to be 180 - 120 = 60 degrees.

So in this case, the segments and angle produce a unique triangle.