Chapter 12, Problem 9CT

To determine

To find: Whether more than one theorem that could be used to prove that $\Delta ABD\cong \Delta CDB$ , if so then list all possible theorems.

Answer to Problem 9CT

The possible theorems are SAS Congruence theorem, AAS Congruence theorem, ASA Congruence theorem, and HL Congruence Theorem.

Explanation of Solution

Given:

The given figure is as follows.

  

Calculation:

From the figure it can be observed that,

By SAS Congruence theorem, it can be proved that $\Delta ABD\cong \Delta CDB$ .

From the figure it can be observed that,

By AAS Congruence theorem, it can be proved that $\Delta ABD\cong \Delta CDB$ .

From the figure it can be observed that,

By ASA Congruence theorem, $\Delta ABD$ , and $\Delta CDB$ are right angles so, $\overline{AB}\cong \overline{CD}$ , $\overline{DB}\cong \overline{DB}$ , and $\angle ADB$ , and $\angle CBD$ are right angles.

By HL Congruence Theorem, $\Delta ABD\cong \Delta CDB$ .

Therefore, the possible theorems are SAS Congruence theorem, AAS Congruence theorem, ASA Congruence theorem, and HL Congruence Theorem.