Question

question 15 (5 points) listen d c p a q b in the figure, \\(\overline{bp} \cong \overline{bq}\\) and \\(\overline{pc} \cong \overline{qa}\\). which of the following postulates will be used to prove \\(\overline{ab} \cong \overline{bc}\\)? a) segment addition postulate b) angle addition postulate c) corresponding angles postulate d) all sides of a square are congruent.

Explanation:

To prove \(\overline{AB} \cong \overline{BC}\), we analyze the segments. We know \(AB = AQ + QB\) (by segment addition postulate, as \(Q\) is on \(AB\)) and \(BC = BP + PC\) (by segment addition postulate, as \(P\) is on \(BC\)). Given \(\overline{BP} \cong \overline{BQ}\) and \(\overline{PC} \cong \overline{QA}\), we can substitute: \(AB = QA + QB\) and \(BC = BP + PC\), and since \(QA = PC\) and \(QB = BP\), the segment addition postulate allows us to combine these congruent parts to show \(AB \cong BC\). The angle addition postulate is for angles, corresponding angles postulate is for parallel lines/transversals, and option D is a property of squares but we are proving using segment addition here (not assuming square properties directly from the proof's approach with given congruent segments). So the segment addition postulate is used.

Answer:

A) Segment addition postulate