Chapter 3.5, Problem 5PSA

To determine

To prove: The two given triangles are congruent.

Explanation of Solution

Given information In a pair of triangles,

Two-line segments are congruent, i.e., $\overline{\text{JH}}\cong \overline{\text{KH}}$

Two-line segments are congruent, i.e., $\overline{\text{HG}}\cong \overline{\text{HM}}$

Two angles are congruent, i.e., $\angle \text{5}\cong \angle \text{6}$

Formula used Congruent means same size and same shape

SAS criteria of congruency.

Proof

  $\begin{array}{l}\text{In }\Delta \text{JHG and }\Delta \text{KHM, we have}\\ \overline{\text{JH}}\cong \overline{\text{KH}}\mathrm{......}\text{given}\\ \overline{\text{JH}}=\overline{\text{KH}}\mathrm{....}\text{by definition of congruency}\mathrm{......}\left(i\right)\\ \overline{\text{HG}}\cong \overline{\text{HM}}\mathrm{.......}\text{given}\\ \overline{\text{HG}}=\overline{\text{HM}}\mathrm{.......}\text{by definition of congruency}\mathrm{.....}\left(ii\right)\\ \angle 5\cong \angle \mathrm{6......}\text{given}\\ \angle 5\text{ = }\angle \mathrm{6.......}\text{by definition of congruency}\\ \angle \text{JHG = }\angle \text{KHM}\mathrm{..........}\left(iii\right)\\ \text{because}\\ \angle \text{KHG + }\angle 5\text{ = }\angle 6+\angle \text{KHG}\\ \angle 5\text{ = }\angle \mathrm{6......}\text{given}\\ From\left(i\right),\left(ii\right)\text{and}\left(iii\right),\text{we have}\\ \Delta \text{JHG }\cong \Delta \text{KHM}\\ \text{Hence proved}\end{array}$