To determine the correct remaining steps to complete Anna's proof of the Triangle Angle Sum Theorem, we need to analyze her steps and their corresponding conclusions.
Let's review what Anna worked on in her proof:
- She established that angles ∠1 and ∠5, and angles ∠2 and ∠4 are congruent by stating that they are alternate interior angles.
- As a result, she concluded that the measures of those corresponding angles are equal.
- She begins the next steps intended to show the sum of the angles in a triangle is 180 degrees.
Now let's analyze each option:
a. Step 3: m∠5 + m∠3 + m∠4 = 180 degrees by definition of a straight angle.
Step 4: m∠1 + m∠3 + m∠2 = 180 degrees by Triangle Angle of Sum theorem.
This option incorrectly refers to the Triangle Angle Sum theorem in Step 4 before the proof is established.
b. Step 3: m∠1 + m∠3 + m∠2 = 180 degrees by definition of a straight angle.
Step 4: m∠5 + m∠3 + m∠4 = 180 degrees by substitution.
In this option, Step 3 does not logically link the previous findings correctly to show the required sums for angles in the triangle.
c. Step 3: m∠2 + m∠3 + m∠4 = 180 degrees by definition of a straight angle.
Step 4: m∠1 + m∠3 + m∠2 = 180 degrees by substitution.
In this option, Step 3 uses angles that were not previously established as related or congruent, leading to confusion.
d. Step 3: m∠5 + m∠3 + m∠4 = 180 degrees by definition of a straight angle.
Step 4: m∠1 + m∠3 + m∠2 = 180 degrees by substitution.
This option correctly uses the straight angle property with ∠5, ∠3, and ∠4 in Step 3. Then in Step 4, it appropriately uses the substitution of the congruences established in the earlier steps to lead to the desired conclusion.
Therefore, the remaining steps that Anna should take to complete her proof are:
d.
Step 3: m∠5 + m∠3 + m∠4 = 180 degrees by definition of a straight angle.
Step 4: m∠1 + m∠3 + m∠2 = 180 degrees by substitution.
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