The problem asks to prove the Angle Sum Theorem for a triangle, which states that the sum of the interior angles of a triangle is 180 degrees. Given triangle ABC, we need to prove that $m\angle C + m\angle 2 + m\angle B = 180$. In the provided diagram, there's also a line XY passing through vertex A. Angles 1, 2, and 3 are formed around point A on line XY.

1. Problem Description

The problem asks to prove the Angle Sum Theorem for a triangle, which states that the sum of the interior angles of a triangle is 180 degrees. Given triangle ABC, we need to prove that $m\angle C + m\angle 2 + m\angle B = 180$. In the provided diagram, there's also a line XY passing through vertex A. Angles 1, 2, and 3 are formed around point A on line XY.

2. Solution Steps

We will use the following facts:

* A straight angle is 180 degrees.

* Alternate interior angles formed by parallel lines are congruent.

We need to construct a proof with statements and reasons. Here is a possible proof:

Statement 1: Triangle ABC

Reason 1: Given

Statement 2: Draw line XY through A parallel to BC

Reason 2: Parallel Postulate

Statement 3: $m\angle 1 + m\angle 2 + m\angle 3 = 180$

Reason 3: Definition of Straight Angle / Angles on a line add up to 180

Statement 4: $m\angle 1 = m\angle C$

Reason 4: Alternate Interior Angles (Since XY || BC)

Statement 5: $m\angle 3 = m\angle B$

Reason 5: Alternate Interior Angles (Since XY || BC)

Statement 6: $m\angle C + m\angle 2 + m\angle B = 180$

Reason 6: Substitution (Substitute $m\angle C$ for $m\angle 1$ and $m\angle B$ for $m\angle 3$ in Statement 3)

3. Final Answer

$m\angle C + m\angle 2 + m\angle B = 180$