document.write( "Question 1179587: https://ibb.co/M6MBW92 \r
\n" ); document.write( "\n" ); document.write( "(please copy paste the link to view the figure)\r
\n" ); document.write( "\n" ); document.write( "Given: AB||CD; BE = DF; AE and CF are perpendicular to DB.
\n" ); document.write( "Prove: AD||BC \n" ); document.write( "

Algebra.Com's Answer #850210 by CPhill(1959) $\"\"$ $\"About$

You can put this solution on YOUR website!
Absolutely, let's break down this geometric proof step-by-step.\r
\n" ); document.write( "\n" ); document.write( "**Given:**\r
\n" ); document.write( "\n" ); document.write( "* AB || CD (Line segment AB is parallel to line segment CD)
\n" ); document.write( "* BE = DF (Line segment BE is equal in length to line segment DF)
\n" ); document.write( "* AE ⊥ DB (Line segment AE is perpendicular to line segment DB)
\n" ); document.write( "* CF ⊥ DB (Line segment CF is perpendicular to line segment DB)\r
\n" ); document.write( "\n" ); document.write( "**Prove:**\r
\n" ); document.write( "\n" ); document.write( "* AD || BC (Line segment AD is parallel to line segment BC)\r
\n" ); document.write( "\n" ); document.write( "**Proof:**\r
\n" ); document.write( "\n" ); document.write( "| Statement | Reason |
\n" ); document.write( "|---|---|
\n" ); document.write( "| 1. AB || CD | Given |
\n" ); document.write( "| 2. ∠ABE ≅ ∠CDF | Alternate interior angles are congruent (when parallel lines are cut by a transversal) |
\n" ); document.write( "| 3. BE = DF | Given |
\n" ); document.write( "| 4. AE ⊥ DB | Given |
\n" ); document.write( "| 5. CF ⊥ DB | Given |
\n" ); document.write( "| 6. ∠AEB and ∠CFD are right angles | Definition of perpendicular lines |
\n" ); document.write( "| 7. ∠AEB ≅ ∠CFD | All right angles are congruent |
\n" ); document.write( "| 8. △ABE ≅ △CDF | Angle-Side-Angle (ASA) Congruence Theorem (∠ABE ≅ ∠CDF, BE = DF, ∠AEB ≅ ∠CFD) |
\n" ); document.write( "| 9. AE = CF | Corresponding Parts of Congruent Triangles are Congruent (CPCTC) |
\n" ); document.write( "| 10. ∠AED and ∠CFB are right angles | Definition of perpendicular lines |
\n" ); document.write( "| 11. ∠AED ≅ ∠CFB | All right angles are congruent |
\n" ); document.write( "| 12. DE = DB - BE | segment subtraction postulate |
\n" ); document.write( "| 13. BF = DB - DF | segment subtraction postulate |
\n" ); document.write( "| 14. BE = DF | Given |
\n" ); document.write( "| 15. DE = BF | substitution property of equality |
\n" ); document.write( "| 16. △ADE ≅ △CBF | Side-Angle-Side (SAS) Congruence Theorem (AE = CF, ∠AED ≅ ∠CFB, DE = BF) |
\n" ); document.write( "| 17. ∠ADE ≅ ∠CBF | CPCTC |
\n" ); document.write( "| 18. AD || BC | Alternate interior angles are congruent (converse) |\r
\n" ); document.write( "\n" ); document.write( "**Explanation:**\r
\n" ); document.write( "\n" ); document.write( "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.
\n" ); document.write( "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.
\n" ); document.write( "3. **CPCTC:** We use CPCTC to show that AE = CF.
\n" ); document.write( "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).
\n" ); document.write( "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.
\n" ); document.write( " \n" ); document.write( "

\n" );