Chapter 4.6, Problem 6WE

To determine

To prove: the angle $\angle CBA\cong \angle DBA$ with the help of given information.

Explanation of Solution

Given information:

The system information:

  $\overline{OC}\cong \overline{OD}$

  $\text{Δ}CAO\cong \text{Δ}DAO$

  $\angle \text{CAO}\cong \angle D\text{AO}$

  $\text{Δ}CAB\cong \text{Δ}DAB$

  $\angle CBA\cong \angle DBA$

Given image:

  

Calculation:

Give a reason for each key step of the proof.

To prove that $\angle CBA\cong \angle DBA$

The key steps of the proof are:

a. $\overline{OC}\cong \overline{OD}$ base angles of an isosceles triangle are equal.

b. $\text{Δ}CAO\cong \text{Δ}DAO$ (SSS postulate)

Since three sides of one triangle are equal to three sides of another triangle,

Therefore, by side-side-side congruency

It is concluded that, two triangles are congruent.

c. $\angle \text{CAO}\cong \angle D\text{AO}$

  $\left(\text{Corresponding parts of }\cong △\text{ are}\cong \right)$

d. $\text{Δ}CAB\cong \text{Δ}DAB$ (SAS postulate)

e. $\angle CBA\cong \angle DBA$ (Corresponding parts of $\cong \text{Δ}$ are $\cong )$

Hence it is concluded that $\angle CBA\cong \angle DBA$