Sylvie has started a proof of the Triangle Angle Sum Theorem. Which answer choice correctly completes her proof?

Question

Sylvie's Proof: Given the diagram shown, <1 ≥ L5; 22 * 24 because alternate interior angles are congruent when lines are
parallel. Then, I know that mL1 = mL5; mL2 = m<4 because congruent angles have equal measures.
(1 point)
• m/2 + m/3 + m/4 = 180° by the definition of a straight angle. Finally, m/1 + m/3 + m/2 = 180° by substitution.
• m/1 + m/3 + m/2 = 180° by the definition of a straight angle. Finally, mL5 + m/3 + m/4 = 180° by substitution.
0 m/5 + m/3 + m/4 = 180° by the definition of a straight angle. Finally, m/1 + m/3 + m/2 = 180° by substitution.
m/5 + m/3 + m/4 = 180° by the definition of a straight angle. Finally, m/1 + m/3 + m/2 - 180° by the Triangle Angle Sum Theorem.

Answer 1

To complete Sylvie's proof of the Triangle Angle Sum Theorem, we need to correctly use the relationships established in her argument.

From what you provided:

• Sylvie has shown that angle 1 (\(mL_1\)) is equal to angle 5 (\(mL_5\)) and that angle 2 (\(mL_2\)) is equal to angle 4 (\(mL_4\)), based on the properties of parallel lines and congruent angles.

The crucial part of the proof is showing that the sum of the angles in a triangle equals 180°. According to the given statements, we can derive an equation related to angles.

Among the answer choices:

• The option stating "mL5 + mL3 + mL4 = 180° by the definition of a straight angle. Finally, mL1 + mL3 + mL2 = 180° by substitution." is correct. This demonstrates that the angles \(L_1\), \(L_2\), and \(L_3\) within the triangle add up to 180°, following the proof's preceding steps.

So, the correct completion of Sylvie's proof is:

mL5 + mL3 + mL4 = 180° by the definition of a straight angle. Finally, mL1 + mL3 + mL2 = 180° by substitution.