Question

Archie has been working on a proof of the Triangle Angle Sum Theorem. One part of his proof is missing. Which answer choice correctly completes his proof? Archie's Proof: Given the diagram shown,  ∠1≅∠5 ;  ∠2≅∠4  because alternate interior angles are congruent when lines are parallel. Then, I know that  m∠1=m∠5 ;  m∠2=m∠4  because congruent angles have equal measures. _________________ by the definition of a straight angle. Finally,  m∠1+m∠3+m∠2=180°  by substitution. (1 point)Responsesm∠5+m∠3+m∠4=180°m angle 5 plus m angle 3 plus m angle 4 equals 180 degrees m∠5+m∠3+m∠1=180°   m angle 5 plus m angle 3 plus m angle 1 equals 180 degrees   m∠2+m∠3+m∠4=180°   m angle 2 plus m angle 3 plus m angle 4 equals 180 degrees   m∠1+m∠3+m∠2=180° 

Answer 1

To complete Archie's proof of the Triangle Angle Sum Theorem, we need to express a relationship involving the angles in question that leads to the conclusion that their measures sum to 180 degrees.

In the context of the proof, we have established that \( m∠1 = m∠5 \) and \( m∠2 = m∠4 \) due to the congruence of alternate interior angles when the lines are parallel. We also know that \(∠1\), \(∠2\), and \(∠3\) form a straight angle (180 degrees), which gives us the measure equation for angles around point \(∠3\).

Given these relationships, Archie's proof can be completed by stating that:

\[ m∠5 + m∠3 + m∠4 = 180° \]

This statement reflects the fact that \(∠5\), \(∠3\), and \(∠4\) also form a straight angle.

So the correct choice to fill in the blank is:

m∠5 + m∠3 + m∠4 = 180°

This shows that the angles indeed sum up to 180 degrees as required.