The proof presented is meant to show that an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles, which is a property true for every triangle.

Here's how the reasoning works step by step:

Proof Outline:

1. Step 1:
   $m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ$
   Reason: The sum of the interior angles of any triangle equals $180^\circ$.

2. Step 2:
   $m\angle 2 + m\angle 4 = 180^\circ$
   Reason: $\angle 2$ and $\angle 4$ form a linear pair, so their measures add up to $180^\circ$.

3. Step 3:
   $m\angle 1 + m\angle 2 + m\angle 3 = m\angle 2 + m\angle 4$
   Reason: This is a result of substitution. From Step 1, we know the sum of the interior angles of the triangle, and from Step 2, we know the sum of the linear pair.

4. Step 4:
   $m\angle 1 + m\angle 3 = m\angle 4$
   Reason: By subtracting $m\angle 2$ from both sides of the equation in Step 3, we isolate the exterior angle ($\angle 4$) on one side and the sum of the non-adjacent interior angles ($\angle 1$ and $\angle 3$) on the other side.

Conclusion:

The proof shows that for any triangle, the exterior angle equals the sum of the two non-adjacent interior angles. Thus, the answer to the question is:

Choice A: Every triangle

This property is universally true for all triangles, not just the specific triangle in the diagram.

Would you like more details on any step or have any questions?

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Here are 5 related questions for further exploration:

1. How can this exterior angle theorem be applied to isosceles triangles?
2. What happens to the exterior angle property in a right triangle?
3. How can this property help in solving for unknown angles in more complex geometric shapes?
4. How would this proof change if we were working with a quadrilateral instead of a triangle?
5. Can you prove the same property using alternate interior angles and parallel lines?

Tip: Always remember that the sum of the angles in any triangle is always $180^\circ$. This fundamental property is the basis for many geometric proofs!