Given points [tex]\( A, B, \)[/tex] and [tex]\( C \)[/tex] form a triangle. We are to prove that the sum of the interior angles of [tex]\( \triangle ABC \)[/tex] is [tex]\( 180^\circ \)[/tex].
Here is the detailed step-by-step solution:
1. Statement: Points [tex]\(A, B,\)[/tex] and [tex]\(C\)[/tex] form a triangle.
Reason: given
2. Statement: Let [tex]\(\overline{DE}\)[/tex] be a line passing through [tex]\(B\)[/tex] and parallel to [tex]\(\overline{AC}\)[/tex].
Reason: definition of parallel lines
3. Statement: [tex]\(\angle 3 \cong \angle 5\)[/tex] and [tex]\(\angle 1 \cong \angle 4\)[/tex].
Reason: corresponding angles of parallel lines
4. Statement: [tex]\(m\angle 1 = m\angle 4\)[/tex] and [tex]\(m\angle 3 = m\angle 5\)[/tex].
Reason: equal measures of congruent angles
5. Statement: [tex]\(m\angle 4 + m\angle 2 + m\angle 5 = 180^\circ\)[/tex].
Reason: angle addition and definition of a straight line
6. Statement: [tex]\(m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ\)[/tex].
Reason: substitution
Therefore, the sum of the interior angles of [tex]\(\triangle ABC\)[/tex] is [tex]\(180^\circ\)[/tex].
