Chapter 3.5, Problem 1PSA

To determine

To prove: The given Triangles are congruent.

Explanation of Solution

Given information:

In given triangles ABC and DBC,

Two corresponding sides in the given triangles are congruent, i.e., $\overline{\text{AB}}\cong \overline{\text{DC}}$

Two another corresponding sides in the given triangles are congruent, i.e., $\overline{\text{AC}}\cong \overline{\text{DB}}$

Formula used Definition of congruency means same size and same shape objects are congruent

Common side in two triangles is always congruent

SSS criteria of congruency

Proof

  $\begin{array}{l}\text{In }\Delta \text{ABC and }\Delta \text{DBC}\\ \overline{\text{AB}}\cong \overline{\text{DC}}\\ \therefore \overline{\text{AB}}=\overline{\text{DC}}\mathrm{....}\text{by definition of congruency}\mathrm{....}\left(i\right)\\ \overline{\text{AC}}\cong \overline{\text{DB}}\\ \therefore \overline{\text{AB}}=\overline{\text{DB}}\mathrm{....}\text{by definition of congruency}\mathrm{....}\left(ii\right)\\ \overline{\text{BC}}=\overline{\text{BC}}\mathrm{......}\text{common side, thus congruent}\mathrm{.....}\left(iii\right)\\ \text{From }\left(i\right),\left(ii\right)\text{ and }\left(iii\right),\text{we have}\\ \Delta \text{ABC }\cong \Delta \text{DBC, SSS criteria of congruency is satisfied}\\ \text{Hence proved}\end{array}$