Given: $$ D $$ is the midpoint of $$ \overline{A C}, \overline{B A} \cong \overline{B C} $$ and $$ \angle E D A \cong \angle F D C $$. Prove: $$ riangle A E D \cong riangle C F D $$. Note: quadrilateral properties are not permitted in this proof.

Question

Given: $$ D $$ is the midpoint of $$ \overline{A C}, \overline{B A} \cong \overline{B C} $$ and $$ \angle E D A \cong \angle F D C $$. Prove: $$ 	riangle A E D \cong 	riangle C F D $$. Note: quadrilateral properties are not permitted in this proof.

UpStudy AI · Accepted Answer

To prove that $$\triangle AED \cong \triangle CFD$$, we'll use the properties of congruent triangles, specifically the Side-Angle-Side (SAS) Congruence Postulate. Here's a step-by-step proof:

### Given:
1. $$ D $$ is the midpoint of $$\overline{AC}$$, so $$ AD = DC $$.
2. $$ \overline{BA} \cong \overline{BC} $$.
3. $$ \angle EDA \cong \angle FDC $$.

### Prove:
$$\triangle AED \cong \triangle CFD$$.

### Proof:

1. **Segment Equality:**
   - Since $$ D $$ is the midpoint of $$\overline{AC}$$, we have:
     $$
     AD = DC \quad \text{(Given)}
     $$

2. **Side Correspondence:**
   - Given that $$ \overline{BA} \cong \overline{BC} $$, and assuming that points $$ E $$ and $$ F $$ are corresponding points on $$ \overline{BA} $$ and $$ \overline{BC} $$ respectively, it follows that:
     $$
     AE = CF
     $$
     *(This assumption is based on the symmetry provided by $$ \overline{BA} \cong \overline{BC} $$. If more specific information about points $$ E $$ and $$ F $$ is provided, adjust accordingly.)*

3. **Angle Equality:**
   - It is given that:
     $$
     \angle EDA \cong \angle FDC
     $$

4. **Triangle Congruence (SAS Postulate):**
   - Now, comparing $$\triangle AED$$ and $$\triangle CFD$$:
     - **Side 1:** $$ AD = DC $$ (from step 1)
     - **Side 2:** $$ AE = CF $$ (from step 2)
     - **Included Angle:** $$ \angle EDA \cong \angle FDC $$ (from step 3)
   
   - By the **Side-Angle-Side (SAS) Congruence Postulate**, if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

5. **Conclusion:**
   - Therefore, $$\triangle AED \cong \triangle CFD$$ by SAS.

### Diagram (Optional):

While not provided here, a diagram illustrating points $$ A, B, C, D, E, $$ and $$ F $$ with the given congruences would aid in visualizing the proof.

### Final Statement:
By the Side-Angle-Side Congruence Postulate, $$\triangle AED$$ is congruent to $$\triangle CFD$$. Hence, the proof is complete.
$$\triangle AED$$ is congruent to $$\triangle CFD$$ by the Side-Angle-Side (SAS) Congruence Postulate.

Given: is the midpoint of and .
Prove: .
Note: quadrilateral properties are not permitted in this proof.

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Given:

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Diagram (Optional):

Final Statement:

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Given: is the midpoint of and . Prove: . Note: quadrilateral properties are not permitted in this proof.