Chapter 4.6, Problem 1WE

To determine

To prove: $\overline{NE}\cong \overline{OS}$

Explanation of Solution

Given information: Key steps of the proof

Proof:

  

Key step a.

Consider $\Delta RNX,\text{}\text{}\Delta LOY$

  $\begin{array}{l}\overline{RN}\cong \overline{LO}\text{(Given)}\\ \overline{NX}\cong \overline{OY}\text{(Given)}\\ \overline{RX}\cong \overline{LY}\text{(Given)}\end{array}$

So, $\Delta RNX\cong \Delta LOY\text{(SSS}\text{congruence}\text{criteria)}$

(According to SSS congruence criteria, if all the sides of one triangle are congruent to the corresponding sides of the other triangle then the two triangles are said to be congruent)

Key step b.

  $\Delta RNX\cong \Delta LOY$ (from part a.)

So,

  $\angle X\cong \angle Y$ (corresponding parts of congruent triangles are congruent)

Key step c.

Consider $\Delta NEX,\Delta OSY$

  $\begin{array}{l}\overline{NX}\cong \overline{OY}\text{(Given)}\\ \angle X\cong \angle Y\text{(Proved}\text{}\text{}\text{}\text{in}\text{part}\text{b}\text{.)}\\ \overline{EX}\cong \overline{SY}\text{(Given)}\\ \therefore \Delta NEX\cong \Delta OSY\text{(SAS}\text{congruence}\text{criteria)}\end{array}$

(According to SAS congruence criteria, if two sides and the included angle of one triangle are congruent to the two sides and the included angle of the other triangle then the triangles are said to be congruent)

Key step d.

  $\Delta NEX\cong \Delta OSY$ (Proved in part c.)

  $\overline{NE}\cong \overline{OS}$ (Corresponding parts of congruent triangles are congruent)