To reorder the steps of the proof logically, let's carefully analyze what needs to be proven and the given information:

Given:

1. $E$ is the midpoint of $\overline{BD}$.
2. $\angle ABC \cong \angle CDF$.

To Prove:

• $\overline{BD} \perp \overline{CE}$.

Proof Steps

Step 1: Show $E$ as the midpoint of $\overline{BD}$

Since $E$ is the midpoint of $\overline{BD}$, we know that: $BE = ED.$

Step 2: Use congruent angles

Given $\angle ABC \cong \angle CDF$, we know that these angles have equal measure: $m\angle ABC = m\angle CDF.$

Step 3: Establish congruent triangles

Consider triangles $\triangle ABE$ and $\triangle CDE$.

• $BE = ED$ (from midpoint property).
• $\angle ABC \cong \angle CDF$ (given).
• $\overline{CE}$ is shared between the two triangles.

By SAS (Side-Angle-Side) Congruence, we conclude that: $\triangle ABE \cong \triangle CDE.$

Step 4: Conclude perpendicularity

Since $\triangle ABE \cong \triangle CDE$, corresponding parts of congruent triangles are equal. Therefore, $\angle BEC = 90^\circ$. This implies: $\overline{BD} \perp \overline{CE}.$

Final Proof Sequence

The reordered proof follows logically as:

1. Show $E$ as the midpoint of $\overline{BD}$.
2. Use the congruent angles $\angle ABC \cong \angle CDF$.
3. Prove $\triangle ABE \cong \triangle CDE$ by SAS.
4. Conclude $\overline{BD} \perp \overline{CE}$ by showing $\angle BEC = 90^\circ$.

Would you like further details on any of the steps?

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1. What is the definition of midpoint and how is it used in proofs?
2. How does congruence in triangles help in establishing perpendicularity?
3. What other methods could prove lines are perpendicular?
4. How does SAS congruence apply to different triangles in proofs?
5. What properties of right angles can help simplify proofs involving perpendicular lines?

Tip: In proofs involving congruent angles, look for opportunities to use triangle congruence criteria like SAS, ASA, or SSS to establish relationships.