Question

given ( p parallel q ), and ( r parallel s ). prove: ( angle 4 ) is supplementary to ( angle 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 structure. We know \( m\angle4 = m\angle8 \) (from congruent angles) and \( m\angle8 = m\angle16 \) (from congruent angles). The missing step should relate \( \angle4 \) and \( \angle16 \) using the Transitive Property (if \( a = b \) and \( b = c \), then \( a = c \)). Option D states \( m\angle4 = m\angle16 \) with the Transitive Property, which fits. Other options: A is incorrect (supplementary, not equal), B is not relevant to the flow, C is about \( \angle7 \) and \( \angle8 \) (not needed), E is incorrect (transitive doesn't apply to supplementary here).

Answer:

D. Statement: \( m\angle4 = m\angle16 \)
Reason: Transitive Property of Equality