Chapter 3.5, Problem 8PSB

To determine

To prove:Two-line segments are congruent

Explanation of Solution

Given information In a given figure

Two angles are congruent, i.e., $\angle 9\cong \angle 10$

Two other angles are congruent, i.e., $\angle \text{GFH}\cong \angle \text{HJG}$

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

ASA criteria of congruency is satisfied

Congruent parts of congruent triangle are congruent

  $\text{corresponding angles}$ are always equal

Co interiors angles are equal

Proof

  $\begin{array}{l}\text{In }\Delta \text{FGJ and }\Delta \text{HJF, we have}\\ \angle 9\cong \angle \mathrm{10......}\text{corresponding angles}\mathrm{.....}\left(i\right)\\ \angle 11\cong \angle \mathrm{12.....}\text{co-interior angles}\mathrm{.........}\left(ii\right)\\ \overline{\text{FH}}=\overline{\text{GJ}}\mathrm{......}\text{segments opposite to equal angles}\mathrm{....}\left(iii\right)\\ From\left(i\right),\left(ii\right)\text{and}\left(iii\right),\text{we have}\\ \Delta \text{FGJ }\cong \Delta \text{HJF}\mathrm{......}\text{ASA criteria is satisfied}\\ \overline{\text{FG}}=\overline{\text{JH}}\mathrm{......}\text{Congruent parts of congruent triangle are congruent}\\ \text{Hence proved}\end{array}$