Chapter 3.5, Problem 14PSC

To determine

To prove: that One of the pair of given angles is congruent.

Explanation of Solution

Given information:

One given pair of line segments is congruent, i.e., $\overline{\text{YD}}\cong \overline{\text{ZD}}$

Other given pair of line segments is congruent, i.e., $\overline{\text{BD}}\cong \overline{\text{CD}}$

Point (P) given is midpoint of line segment (YZ)

Formula used:

Congruency means same size and same shape

Congruent parts of congruent triangle are congruent

RHS criteria of congruency.

Proof:

  $\begin{array}{l}\text{In }\Delta \text{DYE and }\Delta \text{DYZ, we have}\\ \overline{\text{YD}}\cong \overline{\text{ZD}}\mathrm{........}\text{given}\\ \overline{\text{YD}}=\overline{\text{ZD}}\mathrm{.......}\text{by definition}\mathrm{........}\left(i\right)\\ \overline{\text{BD}}\cong \overline{\text{CD}}\mathrm{.......}\text{given}\\ \overline{\text{BD}}=\overline{\text{CD}},\mathrm{......}\text{by definition}\mathrm{.........}\left(ii\right)\\ \angle \text{DEY=}\angle \text{DEZ=90}°\\ \therefore \Delta \text{DYE }\cong \Delta \text{DYZ}\mathrm{.....}\text{RHS criteria of congruency is satisfied}\\ \text{⇒}\Delta \text{BYZ }\cong \Delta \text{CZY}\mathrm{.......}\text{Congruent parts of congruent triangle are congruent}\\ \text{Hence proved}\end{array}$