Chapter 4.6, Problem 2WE

To determine

To prove: $\overline{BE}\cong \overline{DF}$ by writing reasons for the given key steps.

Explanation of Solution

Given information: Following Steps of proof are given.

1. $\Delta ABC\cong \Delta CDA$
2. $\angle 1\cong \angle 2$
3. $\Delta ABE\cong \Delta CDF$
4. $\overline{BE}\cong \overline{DF}$

Proof:

  

Step a. $\Delta ABC\cong \Delta CDA$

Consider $\Delta ABC,\Delta CDA$

  $\overline{AB}\cong \overline{CD}$ (Given)

  $\overline{BC}\cong \overline{AD}$ (Given)

  $\overline{AC}\cong \overline{AC}$ (according to reflexive property of congruence, a line segment is congruent to itself)

So,

  $\Delta ABC\cong \Delta CDA$ (SSS congruence criteria)

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

Step b. $\angle 1\cong \angle 2$

  $\angle 1\cong \angle 2$ (corresponding parts of congruent triangles are congruent)

Step c. $\Delta ABE\cong \Delta CDF$

Consider $\Delta ABE,\Delta CDF$

  $\overline{AB}\cong \overline{CD}$ (Given)

  $\angle 1\cong \angle 2$ (From step b.)

  $\angle BEA=90°,\angle CFD=90°$ (Given)

So,

  $\angle BEA\cong \angle CFD$ (right angles are congruent to each other)

Hence,

  $\Delta ABE\cong \Delta CDF$

(According to AAS congruence criteria, two triangles are said to be congruent if two angles and the side not included between them are congruent to the two angles and the side not included between them of another triangle)

Step d. $\overline{BE}\cong \overline{DF}$

  $\overline{BE}\cong \overline{DF}$ (corresponding parts of congruent triangles are congruent)