Chapter 3.4, Problem 11PSB

To determine

To prove:In the given triangle, $\angle \text{CDE \& }\angle \text{CED}$ are congruent.

Explanation of Solution

Given information

  $\begin{array}{l}\overrightarrow{\text{DF}}\text{ is a bisector of }\angle \text{CDE }\\ \overrightarrow{\text{EF}}\text{ is a bisector of }\angle \text{CED}\\ \text{G is mid point of }\overrightarrow{\text{DE}}\\ \text{Ray }\overrightarrow{\text{DF}}\text{ is equal to }\overrightarrow{\text{EF}}\\ \end{array}$

Formula used

Congruent Part of Congruent Triangles are Congruent

Common arms in triangle are always equal and congruent

SSS criteria of congruency

Proof

Statements	Reasons
1. In $\Delta \text{FGD and }\Delta \text{FGE}$
2. GD=GE	Given, G is the mid-point of DE
3. $\text{FG = FG}$	Common arm
4. DF=FE	Given
5. $\Delta \text{FGD }\cong \Delta \text{FGE}$	SSS criteria is satisfied
6. $\angle \text{FDG = }\Delta \text{FEG}$	Congruent Part of Congruent Triangles are Congruent
7. $\angle \text{FDC = }\Delta \text{FEC}$	DF & FE are the bisectors
8. $\angle \text{FDC+}\angle \text{FDG = }\angle \text{FEC + }\angle \text{FEG}$	All these angles are equal
9. $\angle \text{CDE = }\angle \text{CED}$	From pt. 8
10. $\angle \text{CDE }\cong \angle \text{CED}$	Hence, proved, From the definition of congruency