Question

anna wants to prove the triangle angle sum theorem. what would be the remaining steps to complete her proof? annas proof: step 1: the diagram shows that ∠1≅∠5 and ∠2≅∠4 because alternate interior angles are congruent in parallel lines. step 2: then, m∠1≅m∠5 and m∠2≅m∠4 because congruent angles have equal angle measures. (1 point) step 3: m∠1 + m∠3 + m∠2 = 180° by definition of a straight angle. step 4: m∠5 + m∠3 + m∠4 = 180° by substitution. step 3: m∠2 + m∠3 + m∠4 = 180° by definition of a straight angle. step 4: m∠1 + m∠3 + m∠2 = 180° by substitution. step 3: m∠5 + m∠3 + m∠4 = 180° by definition of a straight angle.

Explanation:

Step1: Recall the goal

The goal is to prove the Triangle Angle - Sum Theorem, which states that the sum of the interior angles of a triangle is 180°. In the given triangle, the interior angles are ∠1, ∠2, and ∠3.

Step2: Analyze the parallel - line properties

We know from Step 1 and 2 that ∠1≅∠5 and ∠2≅∠4 (alternate - interior angles in parallel lines) and congruent angles have equal measures.

Step3: Use the straight - angle property

A straight angle has a measure of 180°. In the diagram, ∠5 + ∠3+∠4 form a straight angle. So, m∠5 + m∠3 + m∠4=180° by the definition of a straight angle.

Step4: Substitute equal - angle measures

Since m∠1 = m∠5 and m∠2 = m∠4 (from congruent angles), we substitute ∠5 with ∠1 and ∠4 with ∠2 in the equation m∠5 + m∠3 + m∠4 = 180°. We get m∠1 + m∠3 + m∠2 = 180°.

Answer:

The correct steps are:
Step 3: m∠1 + m∠3 + m∠2 = 180° by definition of a straight angle.
Step 4: m∠5 + m∠3 + m∠4 = 180° by substitution.