Sure! Here is a detailed, step-by-step solution to prove that the sum of the interior angles of [tex]\(\triangle ABC\)[/tex] is [tex]\(180^\circ\)[/tex]:
To start, we know that points [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex] form a triangle.
Statement 1: Points [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex] form a triangle.
Reason 1: given
Next, we construct a line [tex]\(\overline{DE}\)[/tex] passing through [tex]\(B\)[/tex] and parallel to [tex]\(\overline{AC}\)[/tex].
Statement 2: Let [tex]\(\overline{DE}\)[/tex] be a line passing through [tex]\(B\)[/tex] and parallel to [tex]\(\overline{AC}\)[/tex].
Reason 2: definition of parallel lines
By the properties of parallel lines, alternate interior angles formed by a transversal are congruent.
Statement 3: [tex]\(\angle 3 \cong \angle 5\)[/tex] and [tex]\(\angle 1 \cong \angle 4\)[/tex]
Reason 3: Alternate Interior Angles Theorem
Next, because these angles are congruent, the measures of these angles are equal.
Statement 4: [tex]\(m \angle 1 = m \angle 4\)[/tex] and [tex]\(m \angle 3 = m \angle 5\)[/tex]
Reason 4: Definition of Congruent Angles
The sum of the angles along the straight line [tex]\(DE\)[/tex] is exactly [tex]\(180^\circ\)[/tex].
Statement 5: [tex]\(m \angle 4 + m \angle 2 + m \angle 5 = 180^\circ\)[/tex]
Reason 5: Angle Addition Postulate and definition of a straight angle
We substitute the congruent angles back into the equation to show that the sum of the interior angles of [tex]\(\triangle ABC\)[/tex] is [tex]\(180^\circ\)[/tex].
Statement 6: [tex]\(m \angle 1 + m \angle 2 + m \angle 3 = 180^\circ\)[/tex]
Reason 6: Substitution
Therefore, we have now proven that the sum of the interior angles of [tex]\(\triangle ABC\)[/tex] is [tex]\(180^\circ\)[/tex].
