To prove that the sum of the interior angle measures of a triangle is [tex]\(180^\circ\)[/tex], we need to carefully follow the geometric principles and theorems specified.
### Given
- Lines [tex]\(y\)[/tex] and [tex]\(z\)[/tex] are parallel ([tex]\(y \| z\)[/tex]).
### To Prove
- The sum of the measures of angles in the triangle, specifically [tex]\( m \angle 5 + m \angle 2 + m \angle 6 = 180^\circ \)[/tex].
### Proof
1. Identify the given angles in the triangle and their relations:
- [tex]\( \angle 5 \)[/tex]
- [tex]\( \angle 2 \)[/tex]
- [tex]\( \angle 6 \)[/tex]
2. Label the angles formed by the intersection of the transversal with parallel lines:
##### Statements and Reasons:
- Statement 1: [tex]\( \angle 5 = \angle 1 \)[/tex]
Reason 1: Alternate interior angles theorem (Lines [tex]\( y \| z \)[/tex] and transversal creates alternate interior angles).
- Statement 2: [tex]\( \angle 6 = \angle 3 \)[/tex]
Reason 2: Alternate interior angles theorem.
- Statement 3: [tex]\( m \angle 1 + m \angle 2 + m \angle 3 = 180^\circ \)[/tex]
Reason 3: Angle addition postulate for a straight line. Since [tex]\( \angle 1 \)[/tex], [tex]\( \angle 2 \)[/tex], and [tex]\( \angle 3 \)[/tex] form a straight line and thus form [tex]\(180^\circ\)[/tex] by definition.
From the above statements:
- By substituting [tex]\(\angle 1\)[/tex] and [tex]\(\angle 3\)[/tex] back to [tex]\(\angle 5\)[/tex] and [tex]\(\angle 6\)[/tex] respectively, we get:
[tex]\( m \angle 5 + m \angle 2 + m \angle 6 = m \angle 1 + m \angle 2 + m \angle 3 = 180^\circ \)[/tex]
- This transformation utilizes the fact that the measures remain equivalent because of the alternate interior angles theorem.
### Therefore,
[tex]\[ m \angle 5 + m \angle 2 + m \angle 6 = 180^\circ \][/tex]
Thus, we have shown the measures of the interior angles of a triangle add up to [tex]\(180^\circ\)[/tex].
