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Points [tex]\( A, B, \)[/tex] and [tex]\( C \)[/tex] form a triangle. Complete the statements to prove that the sum of the interior angles of [tex]\( \triangle ABC \)[/tex] is [tex]\( 180^{\circ} \)[/tex].

| Statement | Reason |
| --- | --- |
| Points [tex]\( A, B, \)[/tex] and [tex]\( C \)[/tex] form a triangle. | Given |
| Let [tex]\(\overline{DE}\)[/tex] be a line passing through [tex]\( B \)[/tex] and parallel to [tex]\(\overline{AC}\)[/tex]. | Definition of parallel lines |
| [tex]\(\angle 3 \cong \angle 5\)[/tex] and [tex]\(\angle 1 \cong \angle 4\)[/tex]. | |
| [tex]\(m\angle 1 = m\angle 4\)[/tex] and [tex]\(m\angle 3 = m\angle 5\)[/tex]. | |
| [tex]\(m\angle 4 + m\angle 2 + m\angle 5 = 180^{\circ}\)[/tex]. | Angle addition and definition of a straight line |
| [tex]\(m\angle 1 + m\angle 2 + m\angle 3 = 180^{\circ}\)[/tex]. | Substitution |



Answer :

Okay, let's complete the statements step-by-step to prove that the sum of the interior angles of [tex]\(\triangle ABC\)[/tex] is [tex]\(180^{\circ}\)[/tex].

To prove this, let's complete each statement with the appropriate reason:

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 created by a transversal with parallel lines are congruent

4. Statement: [tex]\(m \angle 1 = m \angle 4\)[/tex] and [tex]\(m \angle 3 = m \angle 5\)[/tex]
Reason: Definition 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

So, the completed statements are as follows:

[tex]\[ \begin{array}{|l|l|} \hline \text{Statement} & \text{Reason} \\ \hline \text{Points } A, B, \text{ and } C \text{ form a triangle.} & \text{given} \\ \hline \text{Let } \overline{DE} \text{ be a line passing through } B \text{ and parallel to } \overline{AC} & \text{definition of parallel lines} \\ \hline \angle 3 \cong \angle 5 \text{ and } \angle 1 \cong \angle 4 & \text{corresponding angles created by a transversal with parallel lines are congruent} \\ \hline m \angle 1 = m \angle 4 \text{ and } m \angle 3 = m \angle 5 & \text{definition of congruent angles} \\ \hline m \angle 4 + m \angle 2 + m \angle 5 = 180^{\circ} & \text{angle addition and definition of a straight line} \\ \hline m \angle 1 + m \angle 2 + m \angle 3 = 180^{\circ} & \text{substitution} \\ \hline \end{array} \][/tex]

This completes the proof that the sum of the interior angles of [tex]\(\triangle ABC\)[/tex] is [tex]\(180^{\circ}\)[/tex].