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Line AB is parallel to EF. Transversal GJ crosses line AB at K and crosses line EF at L. The measure of angle KLF is 116 degrees.

Missy is constructing a fence that consists of parallel sides line AB and line EF. Complete the proof to explain how she can show that m∠AKL = 116° by filling in the missing justifications.

Statement Justification
line AB ∥ line EF
m∠KLF = 116° Given
m∠KLF+ m∠BKL = 180° 1.
m∠BKL + m∠AKL = 180° Linear Pair Postulate
m∠KLF + m∠BKL = m∠BKL + m∠AKL 2.
m∠KLF = m∠AKL Subtraction Property
m∠AKL = m∠KLF Symmetric Property
m∠AKL = 116° Substitution Property
1. Alternate Interior Angles Theorem; 2. Substitution Property
1. Definition of Complementary Angles; 2. Substitution Property
1. Definition of Supplementary Angles; 2. Transitive Property
1. Same-Side Interior Angles Theorem; 2. Transitive Property

1 answer

To complete the proof for Missy, we need to justify the steps in the proof correctly. We have the statement that line AB is parallel to line EF and that transversal GJ crosses them. We need to show that \( m∠AKL = 116° \).

Here's how we can fill in the justifications:

Proof

Statement Justification

  1. \( \text{line AB} \parallel \text{line EF} \)

    • Given

  2. \( m∠KLF = 116° \)

    • Given

  3. \( m∠KLF + m∠BKL = 180° \)

    • 1. Definition of Supplementary Angles (since K is on transversal GJ and lines AB and EF are parallel, angles KLF and BKL are consecutive interior angles)

  4. \( m∠BKL + m∠AKL = 180° \)

    • Linear Pair Postulate (angles BKL and AKL are a linear pair as they are on the same line)

  5. \( m∠KLF + m∠BKL = m∠BKL + m∠AKL \)

    • 2. Substitution Property (since both equations have \( m∠BKL \))

  6. \( m∠KLF = m∠AKL \)

    • Transitive Property (if two angles are equal to the same angle, they are equal to each other)

  7. \( m∠AKL = m∠KLF \)

    • Symmetric Property (rearranging the equality)

  8. \( m∠AKL = 116° \)

    • Substitution Property (substituting \( m∠KLF \) with 116°)

Final Justifications:

  1. Definition of Supplementary Angles; 2. Substitution Property

By using the definitions and properties stated above, Missy can successfully prove that \( m∠AKL = 116° \).