Proving the Sum of the Interior Angle Measures of a Triangle is [tex]$180^{\circ}$[/tex]

Given: [tex]y \parallel z[/tex]
Prove: [tex]m \angle 5 + m \angle 2 + m \angle 6 = 180^{\circ}[/tex]

Statements
1. [tex]y \parallel z[/tex] (given)
2. [tex]\angle 3 \cong \angle 5[/tex] (alternate interior angles theorem)
3. [tex]\angle 1 \cong \angle 6[/tex] (alternate interior angles theorem)
4. [tex]m \angle 3 + m \angle 2 + m \angle 1 = 180^{\circ}[/tex] (angle addition postulate)
5. [tex]m \angle 5 + m \angle 2 + m \angle 6 = 180^{\circ}[/tex] (substitution)

Reasons
1. Given
2. Alternate interior angles theorem
3. Alternate interior angles theorem
4. Angle addition postulate
5. Substitution



Answer :

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].

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