Chapter 4.5, Problem 20WE

To determine

To prove: $\overline{AM}\cong \overline{LN}$

Explanation of Solution

Given information: $\overline{FL}\cong \overline{AK},\overline{SF}\cong \overline{SK}$ , M is the midpoint of $\overline{SF}$ and N is the midpoint of $\overline{SK}$

Proof:

  

First prove $\Delta AMF\cong \Delta LNK$

Consider $\Delta AMF,\Delta LNK$

  $\overline{FL}\cong \overline{AK}(i)$ (Given)

  $\overline{AL}\cong \overline{AL}(ii)$ (A line segment is congruent to itself according to reflexive property of congruence)

Add both $(i)$ and $(ii)$

  $\begin{array}{l}\overline{FL}+\overline{AL}\cong \overline{AK}+\overline{AL}\\ \Rightarrow \overline{AF}\cong \overline{LK}(iii)\end{array}$

As $\overline{SF}\cong \overline{SK}$ ,

  $\angle F\cong \angle K(iv)$ (Angles opposite to congruent sides are congruent)

As M is the midpoint of $\overline{SF}$ ,

  $\overline{FM}=\frac{1}{2}\left(\overline{SF}\right)$

As N is the midpoint of $\overline{SK}$ ,

  $\overline{KN}=\frac{1}{2}\left(\overline{SK}\right)$

As $\overline{SF}\cong \overline{SK}$ ,

  $\begin{array}{l}\frac{1}{2}\left(\overline{SF}\right)\cong \frac{1}{2}\left(\overline{SK}\right)\\ \Rightarrow \overline{FM}\cong \overline{KN}(v)\end{array}$

From $(iii),(iv),(v)$ ,

  $\Delta AMF\cong \Delta LNK$ (SAS congruence criteria)

(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 other triangle then the triangles are congruent.)

The corresponding angles and sides of congruent triangles are congruent to each other.

Therefore, $\overline{AM}\cong \overline{LN}$