This problem asks us to determine how many triangles, if any, can be formed given the following measurements:

• One 4-centimeter side
• One 6-centimeter side
• One 50° angle

Let's break it down:

Step 1: Understanding the problem

We are given:

• Two sides (4 cm and 6 cm)
• One angle (50°)

The goal is to determine if these measurements can form:

1. Zero triangles
2. Exactly one triangle
3. More than one triangle (i.e., two or more distinct triangles)

Step 2: Analyzing the case

The problem resembles the Ambiguous Case in triangle geometry, also known as the SSA (Side-Side-Angle) condition. This condition can lead to three possibilities:

1. Zero triangles: If the given measurements are invalid, no triangle can be formed.
2. Exactly one triangle: A valid triangle can be formed with the given measurements.
3. Two triangles (more than one): In some cases, the given measurements could form two distinct triangles, depending on the angle and sides.

Step 3: Use of the Law of Sines

To investigate, we can apply the Law of Sines, which states:

$\frac{a}{\sin(A)} = \frac{b}{\sin(B)}$

Where:

• $a$ and $b$ are sides of the triangle,
• $A$ and $B$ are the angles opposite those sides.

We know:

• $a = 4 \, \text{cm}$,
• $b = 6 \, \text{cm}$,
• One angle $A = 50^\circ$.

We need to determine whether there is a possibility of forming more than one triangle.

Step 4: Conclusion

Because this is an SSA case, it’s possible that more than one triangle can be formed depending on how the other angle and sides come together. Given that two sides and an angle are provided, and the ambiguity of SSA cases, more than one triangle can indeed be formed.

Thus, the answer is "More than one" triangle can be formed.

Let me know if you'd like further details or explanations!

Follow-up questions:

1. How does the Law of Sines help solve ambiguous triangle cases?
2. Can a triangle always be formed with two sides and one angle?
3. What conditions lead to zero triangles being formed in an SSA case?
4. How can you determine if one or two triangles are possible when given two sides and an angle?
5. What would change if the angle provided was an obtuse angle instead of 50°?

Tip: Whenever given two sides and a non-included angle, think about the ambiguous case and apply the Law of Sines to see how many triangles can be formed!