Question

proving a theorem in exercises 7 - 9, write a two - column proof for the property. (see example 3.) 7. reflexive property of segment congruence 8. transitive property of angle congruence 9. transitive property of segment congruence 10. error analysis in the diagram, $overline{mn}congoverline{lq}$ and $overline{lq}congoverline{pn}$. describe and correct the error in the reasoning. because $overline{mn}congoverline{lq}$ and $overline{lq}congoverline{pn}$, then $overline{mn}congoverline{pn}$ by the reflexive property of segment congruence.

Explanation:

Step1: Identify the error

The error is mis - using the Reflexive Property. The Reflexive Property states that a segment is congruent to itself (e.g., $AB\cong AB$). Here, we have two congruence statements $\overline{MN}\cong\overline{LQ}$ and $\overline{LQ}\cong\overline{PN}$, and we should use the Transitive Property.

Step2: Correct the reasoning

The Transitive Property of Segment Congruence states that if $\overline{a}\cong\overline{b}$ and $\overline{b}\cong\overline{c}$, then $\overline{a}\cong\overline{c}$. Since $\overline{MN}\cong\overline{LQ}$ and $\overline{LQ}\cong\overline{PN}$, by the Transitive Property of Segment Congruence, $\overline{MN}\cong\overline{PN}$.

Answer:

The error is using the Reflexive Property instead of the Transitive Property. The correct reasoning is that since $\overline{MN}\cong\overline{LQ}$ and $\overline{LQ}\cong\overline{PN}$, then $\overline{MN}\cong\overline{PN}$ by the Transitive Property of Segment Congruence.