In this image, the problem is asking to match the possible measures of angles $A$ and $C$ in a triangle where angle $B = 50^\circ$, based on the type of triangle given (Acute Triangle, Obtuse Triangle, or Right Triangle). You are given three possible angle pair options: $(60^\circ, 70^\circ)$, $(100^\circ, 30^\circ)$, and $(90^\circ, 40^\circ)$.

Analysis:

1. Acute Triangle: In an acute triangle, all angles are less than $90^\circ$. So both $A$ and $C$ must also be less than $90^\circ$.

   • $B = 50^\circ$, so $A + C = 130^\circ$.
   • The only valid pair of $A$ and $C$ that keeps both angles acute is $(60^\circ, 70^\circ)$.

2. Obtuse Triangle: In an obtuse triangle, one angle must be greater than $90^\circ$. Since $B = 50^\circ$, one of the other angles, either $A$ or $C$, must be greater than $90^\circ$.

   • The valid pair is $(100^\circ, 30^\circ)$ where $A = 100^\circ$, making it an obtuse triangle.

3. Right Triangle: In a right triangle, one of the angles must be $90^\circ$.

   • The valid pair is $(90^\circ, 40^\circ)$, where $A = 90^\circ$, making it a right triangle.

Correct matches:

• Acute Triangle: $60^\circ$ and $70^\circ$
• Obtuse Triangle: $100^\circ$ and $30^\circ$
• Right Triangle: $90^\circ$ and $40^\circ$

Let me know if you would like further explanation on any part of the reasoning.

Follow-up questions:

1. Can you explain how the sum of angles in a triangle is always $180^\circ$?
2. Why can't a triangle have more than one obtuse angle?
3. How do we classify triangles based on their angles?
4. How can we confirm if a given set of angles can form a triangle?
5. What is the relationship between angle measures and side lengths in triangles?

Tip:

When solving problems involving triangles, always remember that the sum of the interior angles of a triangle is $180^\circ$. This can help you quickly verify if angle measures are valid.