Chapter 4.5, Problem 16PPS

(a)

To determine

To prove: $\Delta ABC\cong \Delta ABD$

Explanation of Solution

Given information: The segment $\overline{AB}$ bisects $\angle CBD$ and $\angle CAD$ .

  

Proof: It is given that $\overline{AB}$ bisects $\angle CBD$ and $\angle CAD$ ,

By the definition of angle bisector,

  $\angle CBA\cong \angle DBA$ and $\angle CAB\cong \angle DAB$

By using Reflective property,

  $\overline{AB}\cong \overline{AB}$

Now, by Angle-Side-Angle congruence,

  $\Delta ABC\cong \Delta ABD$

Hence, proved.

(b)

To determine

To prove: $\Delta CAF\cong \Delta DAE$

Explanation of Solution

Given information: $\Delta ABC\cong \Delta ABD$ and $\angle FCA\cong \angle EDA$

  

Proof: It is given that $\Delta ABC\cong \Delta ABD$ , so, corresponding parts of congruent triangles are congruent

  $\overline{AC}\cong \overline{AD}$

Since $\angle FCA\cong \angle EDA$ ,

Hence, $\angle FAC\cong \angle EAD$ because vertical angles are congruent.

Now, by Angle-Side-Angle congruence,

  $\Delta CAF\cong \Delta DAE$

Hence, proved.

(c)

To determine

To prove: $\Delta BHG\cong \Delta BEA$

Explanation of Solution

Given information: Following condition has been given:

  $\overline{HB}\cong \overline{EB}$, $\angle BHG\cong \angle BEA$ , $\angle HGJ\cong \angle EAD$ and $\angle JGB\cong \angle DAB$

  

Proof: It is given that $\overline{HB}\cong \overline{EB}$ , $\angle BHG\cong \angle BEA$ , $\angle HGJ\cong \angle EAD$ and $\angle JGB\cong \angle DAB$ .

Applying Addition postulate of angle congruence,

  $\angle HGJ+\angle JGB\cong \angle EAD+\angle DAB$

Substitution postulate of angle congruence,

  $\angle HGB\cong \angle EAB$

Now, by Angle-Side-Angle congruence,

  $\Delta BHG\cong \Delta BEA$

Hence, proved.