Question

statements
△xyz and ( overleftrightarrow{ay} ) is an auxiliary line parallel to ( overleftrightarrow{xz} )
( mangle 1 + mangle 5 = mangle ayx )
( mangle ayx + mangle 4 = 180^circ )

reasons

substitution
alternate interior angles theorem
definition of congruence
substitution

Explanation:

Step1: Analyze the first statement

The first statement is " $\triangle XYZ$ and $\overleftrightarrow{AY}$ is an auxiliary line parallel to $\overleftrightarrow{XZ}$". The reason for drawing an auxiliary line parallel to a side in a triangle (probably for angle - sum or parallel - line related proofs, like triangle angle - sum or using parallel lines to get angle relationships) is a construction step. But looking at the reasons given, we need to match the statements to reasons.

Step2: Analyze the second statement

The second statement is "$m\angle1 + m\angle5=m\angle AYX$". This looks like an angle - addition postulate (the sum of two adjacent angles forming a larger angle). But among the given reasons, we have to check. Wait, maybe we are doing a triangle angle - sum proof using parallel lines. Let's recall that when we draw a line parallel to a side of a triangle, we can use alternate interior angles.

Step3: Analyze the third statement

The third statement is "$m\angle AYX + m\angle4 = 180^{\circ}$". This is the linear pair postulate (angles on a straight line sum to $180^{\circ}$), but among the reasons we have "Substitution", "Alternate Interior Angles Theorem", "Definition of Congruence", "Substitution". Wait, maybe the first reason (for the first statement) is "Construction" (but it's not in the list). Wait, maybe the first statement's reason: when we draw $\overleftrightarrow{AY}\parallel\overleftrightarrow{XZ}$, the reason for the existence of such a line is the parallel postulate (but not in the list). Wait, maybe the second statement: $m\angle1 + m\angle5=m\angle AYX$ is angle addition postulate, but the reasons given are about substitution, alternate interior angles, etc. Wait, maybe the first statement (drawing the parallel line) has no given reason in the list? No, the table has statements and reasons to match.

Wait, maybe this is a proof of the triangle angle - sum theorem. Let's re - structure:

1. Statement: $\triangle XYZ$ and $\overleftrightarrow{AY}$ is an auxiliary line parallel to $\overleftrightarrow{XZ}$; Reason: Construction (but not in the list, maybe a mis - label). Wait, the reasons given are "Substitution", "Alternate Interior Angles Theorem", "Definition of Congruence", "Substitution" (and some dashed boxes). Wait, maybe the second statement: $m\angle1 + m\angle5=m\angle AYX$; Reason: Angle Addition Postulate (not in the list). Wait, maybe I misread. Let's look at the alternate interior angles: if $\overleftrightarrow{AY}\parallel\overleftrightarrow{XZ}$, then $\angle2\cong\angle4$ (alternate interior angles) and $\angle3\cong\angle5$ (alternate interior angles) by Alternate Interior Angles Theorem. Then, $m\angle1 + m\angle2+m\angle3 = 180^{\circ}$ (triangle angle - sum). If we substitute $\angle2$ with $\angle4$ and $\angle3$ with $\angle5$, and $\angle1+\angle5=\angle AYX$, then $\angle AYX+\angle4 = 180^{\circ}$, then substitute to get $m\angle1 + m\angle2+m\angle3 = 180^{\circ}$.

So for the first statement (drawing the parallel line), the reason is "Construction" (but not in the given reasons, maybe a mistake). For the second statement ($m\angle1 + m\angle5=m\angle AYX$), the reason could be "Angle Addition Postulate" (not in the list). For the third statement ($m\angle AYX + m\angle4 = 180^{\circ}$), the reason is "Linear Pair Postulate" (not in the list). But among the given reasons (Substitution, Alternate Interior Angles Theorem, Definition of Congruence, Substitution), let's match:

- The statement "$\triangle XYZ$ and $\overleftrightarrow{AY}$ is an auxiliary line parallel to $\overleftrightarrow{XZ}$" has no matching reason in the given list (maybe a construction step).
- The statement "$m\angle1 + m\angle5=m\angle AYX$" could be related to angle addition, but no. Wait, maybe the second statement's reason is "Substitution"? No. Wait, the third statement: $m\angle AYX + m\angle4 = 180^{\circ}$: if $\overleftrightarrow{AY}$ is a straight line (since it's an auxiliary line), then $\angle AYX$ and $\angle4$ are supplementary (linear pair), so the reason is "Linear Pair Postulate" (not in the list).

Wait, maybe the problem is to match the statements to the reasons. Let's list the statements and reasons:

Statements:
1. $\triangle XYZ$ and $\overleftrightarrow{AY}$ is an auxiliary line parallel to $\overleftrightarrow{XZ}$
2. $m\angle1 + m\angle5=m\angle AYX$
3. $m\angle AYX + m\angle4 = 180^{\circ}$

Reasons (from bottom, up? Wait the reasons are: Substitution, Alternate Interior Angles Theorem, Definition of Congruence, Substitution (and dashed boxes). Wait, maybe the first statement (drawing the parallel line) has reason "Construction" (not in the list), the second statement: $m\angle1 + m\angle5=m\angle AYX$ has reason "Angle Addition Postulate" (not in the list), the third statement: $m\angle AYX + m\angle4 = 180^{\circ}$ has reason "Linear Pair Postulate" (not in the list). But this is confusing. Wait, maybe the correct matching is:

- First statement (drawing $\overleftrightarrow{AY}\parallel\overleftrightarrow{XZ}$): no given reason (maybe a mistake in the problem).
- Second statement ($m\angle1 + m\angle5=m\angle AYX$): Angle Addition Postulate (not in the list).
- Third statement ($m\angle AYX + m\angle4 = 180^{\circ}$): Linear Pair Postulate (not in the list).

But since the problem is about triangle angle - sum using parallel lines, the key reason when using the parallel line is the Alternate Interior Angles Theorem. For example, if $\overleftrightarrow{AY}\parallel\overleftrightarrow{XZ}$, then $\angle2\cong\angle4$ (alternate interior angles) and $\angle3\cong\angle5$ (alternate interior angles) by Alternate Interior Angles Theorem. Then, $m\angle1 + m\angle2+m\angle3 = 180^{\circ}$ (triangle angle - sum). If we substitute $\angle2$ with $\angle4$ and $\angle3$ with $\angle5$, and $\angle1+\angle5=\angle AYX$, then $\angle AYX+\angle4 = 180^{\circ}$, then substitute to get $m\angle1 + m\angle2+m\angle3 = 180^{\circ}$.

So the first statement (drawing the parallel line) is a construction, the second statement is angle addition, the third is linear pair, and then substitution. But among the given reasons, the Alternate Interior Angles Theorem is for the angles formed by the parallel lines and the transversal.

Answer:

The subfield is Geometry (Mathematics). The key reason in the proof (when using the parallel auxiliary line) is the Alternate Interior Angles Theorem, which is a concept in Geometry. The steps involve using parallel lines, angle - sum properties, and substitution, all of which are part of Geometry.