Question

which of the following is the correct statement to complete the proof?

given: \\(\angle 14\\) and \\(\angle 15\\) are supplementary

prove: \\(f \parallel g\\)

statement | reason
--- | ---
\\(\angle 14\\) and \\(\angle 15\\) are supplementary | given
? | definition of consecutive (same-side) interior angles
\\(f \parallel g\\) | converse of the consecutive (same-side) interior angles theorem

a. \\(m\angle 14 + m\angle 15 = 180^\circ\\)
b. \\(\angle 14\\) and \\(\angle 15\\) are consecutive (same-side) interior angles
c. \\(\angle 14 \cong \angle 15\\)
d. \\(m\angle 14 + m\angle 15 = 90^\circ\\)

Explanation:

Analyze the missing step in the proof

Using the Two-Column Proofs knowledge point

• The given reason for the missing statement is: "Definition of consecutive (same-side) interior angles".
• The final step proves \(f \parallel g\) using the reason: "Converse of the Consecutive (Same-Side) Interior Angles Theorem".
• Therefore, the missing statement must establish that \(\angle 14\) and \(\angle 15\) are indeed consecutive (same-side) interior angles.

Match with the multiple-choice options

Using the Consecutive Interior Angles Theorem knowledge point

• Option A: \(m\angle 14 + m\angle 15 = 180^\circ\) (This is the definition of supplementary angles, not the definition of consecutive interior angles).
• Option B: \(\angle 14\) and \(\angle 15\) are consecutive (same-side) interior angles.
• Option C: \(\angle 14 \cong \angle 15\) (This is not true in general for supplementary angles unless they are right angles).
• Option D: \(m\angle 14 + m\angle 15 = 90^\circ\) (This is the definition of complementary angles).

Thus, Option B is the correct statement to complete the proof.

Answer:

• (A) \(m\angle 14 + m\angle 15 = 180^\circ\)
• (B) \(\angle 14\) and \(\angle 15\) are consecutive (same-side) interior angles (Correct answer)
• (C) \(\angle 14 \cong \angle 15\)
• (D) \(m\angle 14 + m\angle 15 = 90^\circ\)