Chapter 3.2, Problem 25PSB

To determine

To prove : The congruency of $\Delta CAD\cong \Delta CAB$ is to be proved

Explanation of Solution

Given information : The following information has been given

  $\angle 3\cong \angle 1$

  $\angle 4\cong \angle 2$

  $\overrightarrow{AD}\cong \overrightarrow{AB}$

  $\angle DAC\cong \angle BAC\cong \angle 3\cong \angle 1$

Formula used : If two angles of the triangles are congruent and the side containing them is also congruent, then we can say that the triangles are also congruent by ASA property

Proof : We know that for triangles CAD and CAB,

  $\overrightarrow{AD}\cong \overrightarrow{AB}$

  $\angle DAC\cong \angle BAC\cong \angle 3\cong \angle 1$

Also,

  $\angle ADC\cong \angle ABC$ , and

AC is the common chord

Thus, we can say that the two triangles are congruent by ASA method and SAS method as well

Hence, $\Delta CAD\cong \Delta CAB$ is proved