Chapter 3.6, Problem 7PSB

To determine

To Prove: In the given figure $\angle A\cong \angle C$

Explanation of Solution

Given information : We are given a figure in which $\overline{AD}$ and $\overline{CD}$ are legs of isosceles triangle ADC. Also, B is the midpoint of AC

  

Formula used :

Congruency criteria of SSS and CPCT (corresponding parts of congruent triangles) rule.

Proof: According to the given condition $\overline{AD}$ and $\overline{CD}$ are legs of isosceles triangle ADC. We know that the legs of an isosceles triangle represent its equal sides. Which means:

  $\overline{AD}\cong \overline{CD}\mathrm{...................}(\text{i)}$

Also, B is the midpoint of AC , so $\overline{AB}\cong \overline{BC}\mathrm{....................}\text{(ii)}$

Joining B and D we get two triangles $\Delta ABD,\Delta CBD$ considering these triangles we get:

  $\begin{array}{l}\overline{AD}\cong \overline{CD}\mathrm{.................................................}(\text{from i)}\\ \\ \overline{AB}\cong \overline{BC}\mathrm{................................................}\text{(from ii)}\\ \\ \overline{BD}\cong \overline{BD}\mathrm{............................................}\text{( Common in both triangles)}\end{array}$

Thus, by SSS(Side-Side-Side) rule we can say:

  $\Delta ABD\cong \Delta CBD$

As the corresponding parts of congruent triangles are also congruent thus:

  $\angle \text{A }\cong \angle C$