Question

given: $overline{df}congoverline{af}$, $angle1congangle2$, $overline{dc}congoverline{ab}$. prove: $overline{ce}congoverline{bg}$. 2. $angle acongangle d$. 3. given. 4. $angle dec$ and $angle1$ are a linear - pair; $angle bga$ and $angle2$ are a linear pair. 4. definition of a linear pair. 5. $angle dec$ and $angle1$ are

Explanation:

Step1: Prove $\triangle DFC\cong\triangle AFB$

Given $\overline{DF}\cong\overline{AF}$, $\angle 1\cong\angle 2$, $\overline{DC}\cong\overline{AB}$, use SAS.

Step2: Find angle - congruence from linear - pairs

$\angle 1\cong\angle 2$, linear - pairs supplementary, so $\angle DEC\cong\angle BGA$.

Step3: Prove $\triangle DEC\cong\triangle BGA$

Use $\angle D\cong\angle A$, $\angle DEC\cong\angle BGA$, $\overline{DC}\cong\overline{AB}$ with AAS.

Step4: Prove $\overline{CE}\cong\overline{BG}$

Corresponding parts of $\triangle DEC\cong\triangle BGA$ are congruent.

Answer:

1. First, prove $\triangle DFC\cong\triangle AFB$:

• In $\triangle DFC$ and $\triangle AFB$, we have $\overline{DF}\cong\overline{AF}$ (given), $\angle 1\cong\angle 2$ (given), and $\overline{DC}\cong\overline{AB}$ (given).
• By the Side - Angle - Side (SAS) congruence criterion, $\triangle DFC\cong\triangle AFB$.
• So, $\angle D\cong\angle A$ (corresponding parts of congruent triangles are congruent).

1. Then, note the linear - pair relationships:

• $\angle DEC$ and $\angle 1$ are a linear pair; $\angle BGA$ and $\angle 2$ are a linear pair (given).
• Since $\angle 1\cong\angle 2$, and linear pairs are supplementary ($\angle DEC + \angle 1=180^{\circ}$ and $\angle BGA+\angle 2 = 180^{\circ}$), we can say $\angle DEC\cong\angle BGA$ (if two angles are supplementary to congruent angles, then they are congruent).

1. Now, prove $\triangle DEC\cong\triangle BGA$:

• In $\triangle DEC$ and $\triangle BGA$, we have $\angle D\cong\angle A$ (proven above), $\angle DEC\cong\angle BGA$ (proven above), and $\overline{DC}\cong\overline{AB}$ (given).
• By the Angle - Angle - Side (AAS) congruence criterion, $\triangle DEC\cong\triangle BGA$.

1. Finally, conclude that $\overline{CE}\cong\overline{BG}$:

• Since $\triangle DEC\cong\triangle BGA$, then $\overline{CE}\cong\overline{BG}$ (corresponding parts of congruent triangles are congruent).