Question

in the diagram, (overline{mn} cong overline{lq}) and (overline{lq} cong overline{pn}). describe and correct the error in the reasoning. blue box with red x: \because (overline{mn} cong overline{lq}) and (overline{lq} cong overline{pn}), then (overline{mn} cong overline{pn}) by the reflexive property of segment congruence (thm. 2.1).\ diagram: rectangle with vertices l, q, p, n, m. options: 1. the transitive property of segment congruence should have been used. 2. the symmetric property of segment congruence should have been used. 3. you do not have enough information to conclude that (overline{mn}) and (overline{pn}) are congruent. 4. you need to know (qp) in order to use the reflexive property of segment congruence.

Explanation:

The original reasoning used the Reflexive Property incorrectly. The Reflexive Property states a segment is congruent to itself (e.g., $\overline{AB} \cong \overline{AB}$). Here, we have $\overline{MN} \cong \overline{LQ}$ and $\overline{LQ} \cong \overline{PN}$, so to conclude $\overline{MN} \cong \overline{PN}$, the Transitive Property of Segment Congruence (if $\overline{a} \cong \overline{b}$ and $\overline{b} \cong \overline{c}$, then $\overline{a} \cong \overline{c}$) should be used, not the Reflexive Property. So the error is using the wrong property, and the correct property is the Transitive Property. Among the options, "The Transitive Property of Segment Congruence should have been used" is correct.

Answer:

A. The Transitive Property of Segment Congruence should have been used