Question

problem 24 given: $overline{df}congoverline{af}$, $angle1congangle2$, $overline{dc}congoverline{ab}$ prove: $overline{ce}congoverline{bg}$ statements 1. $overline{df}congoverline{af}$, $overline{dc}congoverline{ab}$ 1. given 2. $angle acongangle d$ 2. given 3. click here to insert 3. given 4. $angle dec$ and $angle1$ are a linear pair; 4. definition of a linear pair

Explanation:

Step1: List given information

$\overline{DF}\cong\overline{AF},\overline{DC}\cong\overline{AB},\angle A\cong\angle D,\angle 1\cong\angle 2$

Step2: Identify linear pairs

$\angle DEC$ and $\angle 1$, $\angle BGA$ and $\angle 2$ are linear pairs

Step3: Use linear - pair property

$\angle DEC+\angle 1 = 180^{\circ},\angle BGA+\angle 2 = 180^{\circ}$

Step4: Prove $\angle DEC\cong\angle BGA$

Since $\angle 1\cong\angle 2$, by substitution $\angle DEC\cong\angle BGA$

Step5: Prove triangle congruence

By ASA ($\angle D\cong\angle A,\overline{DC}\cong\overline{AB},\angle DEC\cong\angle BGA$), $\triangle DEC\cong\triangle BGA$

Step6: Prove segment congruence

By CPCTC, $\overline{CE}\cong\overline{BG}$

Answer:

1. Statements: $\overline{DF}\cong\overline{AF},\overline{DC}\cong\overline{AB}$; Reasons: Given
2. Statements: $\angle A\cong\angle D$; Reasons: Given
3. Statements: $\angle 1\cong\angle 2$; Reasons: Given
4. Statements: $\angle DEC$ and $\angle 1$ are a linear - pair; Reasons: Definition of a linear pair
5. Statements: $\angle BGA$ and $\angle 2$ are a linear - pair; Reasons: Definition of a linear pair
6. Statements: $\angle DEC+\angle 1 = 180^{\circ},\angle BGA+\angle 2=180^{\circ}$; Reasons: Linear pairs are supplementary
7. Statements: $\angle DEC\cong\angle BGA$; Reasons: Substitution (since $\angle 1\cong\angle 2$)
8. Statements: $\triangle DEC\cong\triangle BGA$; Reasons: Angle - Side - Angle (ASA) congruence criterion ($\angle D\cong\angle A,\overline{DC}\cong\overline{AB},\angle DEC\cong\angle BGA$)
9. Statements: $\overline{CE}\cong\overline{BG}$; Reasons: Corresponding parts of congruent triangles are congruent (CPCTC)