Question

∠4 is supplementary to ∠15.
what is the missing step in the given proof?
a. statement: ( m angle 4 = m angle 15 )
reason: substitution property of equality
b. statement: ( m angle 4 + m angle 8 = 180^{circ} )
reason: definition of supplementary angles
c. statement: ( angle 7 ) and ( angle 8 ) are supplementary.
reason: linear pair theorem
d. statement: ( m angle 4 = m angle 16 )
reason: transitive property of equality
e. statement: ( angle 15 ) and ( angle 8 ) are supplementary.
reason: transitive property of equality

Explanation:

To find the missing step, we analyze the proof flow. We know \( m\angle4 = m\angle8 \) (from congruent angles) and \( m\angle8 = m\angle15 \) (from congruent angles). By the Substitution Property of Equality, we can substitute \( m\angle8 \) with \( m\angle4 \) in \( m\angle8 = m\angle15 \), giving \( m\angle4 = m\angle15 \). Let's check the options:

• Option A: Statement \( m\angle4 = m\angle15 \), Reason: Substitution Property (matches the substitution of \( m\angle8 \) with \( m\angle4 \) in \( m\angle8 = m\angle15 \)).
• Option B: \( m\angle4 + m\angle8 = 180^\circ \) is not relevant here as \( \angle4 \) and \( \angle8 \) are congruent, not supplementary.
• Option C: \( \angle7 \) and \( \angle8 \) supplementary is a linear pair, but not the missing step.
• Option D: \( m\angle4 = m\angle16 \) is incorrect as \( \angle4 \) and \( \angle16 \) don't have that relationship.
• Option E: \( \angle15 \) and \( \angle8 \) supplementary is wrong (they are congruent, not supplementary).

Answer:

A. Statement: \( m\angle4 = m\angle15 \)
Reason: Substitution Property of Equality