SOLUTION: https://ibb.co/M6MBW92 (please copy paste the link to view the figure) Given: AB||CD; BE = DF; AE and CF are perpendicular to DB. Prove: AD||BC

Question 1179587: https://ibb.co/M6MBW92
(please copy paste the link to view the figure)
Given: AB||CD; BE = DF; AE and CF are perpendicular to DB.
Prove: AD||BC
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Absolutely, let's break down this geometric proof step-by-step.
**Given:**
* AB || CD (Line segment AB is parallel to line segment CD)
* BE = DF (Line segment BE is equal in length to line segment DF)
* AE ⊥ DB (Line segment AE is perpendicular to line segment DB)
* CF ⊥ DB (Line segment CF is perpendicular to line segment DB)
**Prove:**
* AD || BC (Line segment AD is parallel to line segment BC)
**Proof:**
| Statement | Reason |
|---|---|
| 1. AB || CD | Given |
| 2. ∠ABE ≅ ∠CDF | Alternate interior angles are congruent (when parallel lines are cut by a transversal) |
| 3. BE = DF | Given |
| 4. AE ⊥ DB | Given |
| 5. CF ⊥ DB | Given |
| 6. ∠AEB and ∠CFD are right angles | Definition of perpendicular lines |
| 7. ∠AEB ≅ ∠CFD | All right angles are congruent |
| 8. △ABE ≅ △CDF | Angle-Side-Angle (ASA) Congruence Theorem (∠ABE ≅ ∠CDF, BE = DF, ∠AEB ≅ ∠CFD) |
| 9. AE = CF | Corresponding Parts of Congruent Triangles are Congruent (CPCTC) |
| 10. ∠AED and ∠CFB are right angles | Definition of perpendicular lines |
| 11. ∠AED ≅ ∠CFB | All right angles are congruent |
| 12. DE = DB - BE | segment subtraction postulate |
| 13. BF = DB - DF | segment subtraction postulate |
| 14. BE = DF | Given |
| 15. DE = BF | substitution property of equality |
| 16. △ADE ≅ △CBF | Side-Angle-Side (SAS) Congruence Theorem (AE = CF, ∠AED ≅ ∠CFB, DE = BF) |
| 17. ∠ADE ≅ ∠CBF | CPCTC |
| 18. AD || BC | Alternate interior angles are congruent (converse) |
**Explanation:**
1. **Parallel Lines and Alternate Interior Angles:** We use the given parallel lines to establish that the alternate interior angles ∠ABE and ∠CDF are congruent.
2. **Triangle Congruence (ASA):** We use the given information (BE = DF, perpendicular lines) and the alternate interior angles to prove that triangles △ABE and △CDF are congruent using the ASA congruence theorem.
3. **CPCTC:** We use CPCTC to show that AE = CF.
4. **Second Triangle Congruence (SAS):** We then prove that triangles △ADE and △CBF are congruent using the SAS congruence theorem (AE = CF, DE = BF, right angles).
5. **CPCTC and Parallel Lines:** Finally, we use CPCTC again to show that ∠ADE and ∠CBF are congruent, and then use the converse of the alternate interior angles theorem to prove that AD || BC.

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