Sure! Let's fill in the missing reasons step-by-step:
1. First, we know Points [tex]\(A, B\)[/tex], and [tex]\(C\)[/tex] form a triangle.
- Statement: Points [tex]\(A, B\)[/tex], and [tex]\(C\)[/tex] form a triangle.
- Reason: given
2. Let [tex]\(\overline{D E}\)[/tex] be a line passing through [tex]\(B\)[/tex] and parallel to [tex]\(\overline{A C}\)[/tex].
- Statement: Let [tex]\(\overline{D E}\)[/tex] be a line passing through [tex]\(B\)[/tex] and parallel to [tex]\(\overline{A C}\)[/tex].
- Reason: definition of parallel lines
3. [tex]\(\angle 3 \simeq \angle 5\)[/tex] and [tex]\(\angle 1 \approx \angle 4\)[/tex]
- Statement: [tex]\(\angle 3 \simeq \angle 5\)[/tex] and [tex]\(\angle 1 \approx \angle 4\)[/tex]
- Reason: corresponding angles are equal when a transversal intersects parallel lines
4. [tex]\(m \angle 1= m \angle 4\)[/tex] and [tex]\(m \angle 3= m \angle 5\)[/tex]
- Statement: [tex]\(m \angle 1= m \angle 4\)[/tex] and [tex]\(m \angle 3= m \angle 5\)[/tex]
- Reason: because they are corresponding angles
5. [tex]\(m \angle 4+ m \angle 2+ m \angle 5=180^{\circ}\)[/tex]
- Statement: [tex]\(m \angle 4+ m \angle 2+ m \angle 5=180^{\circ}\)[/tex]
- Reason: sum of angles on a straight line is 180 degrees
6. [tex]\(m \angle 1+ m \angle 2+ m \angle 3=180^{\circ}\)[/tex]
- Statement: [tex]\(m \angle 1+ m \angle 2+ m \angle 3=180^{\circ}\)[/tex]
- Reason: angle addition and definition of a straight line
Thus, completing the steps proves that the sum of the interior angles of [tex]\(\triangle ABC\)[/tex] is [tex]\(180^{\circ}\)[/tex].
