🔺 Triangle Congruence to Similarity Interactive Proof

Problem Statement:

Given: △ABE ≅ △ACD

To Prove: △ADE ~ △ABC

Interactive Diagram

AB 0

AC 0

AD 0

AE 0

△ADE is NOT similar to △ABC

🎛️ Controls - Adjust to Make △ABE ≅ △ACD

Position of Point D on AB: D at 40% of AB

Position of Point E on AC: E at 40% of AC

📝 Proof Steps

Step 1: Given that △ABE ≅ △ACD, by CPCTC we have:

• AB = AC (corresponding sides)
• AE = AD (corresponding sides)
• ∠BAE = ∠CAD (corresponding angles)

Step 2: Since AE = AD, we can write: AEAC = ADAB

Step 3: We also have ∠EAD = ∠EAD (common angle)

Step 4: By SAS similarity criterion:

• AEAC = ADAB (proportional sides)
• ∠EAD = ∠BAC (same angle)

Therefore, △ADE ~ △ABC

🔍 Key Insight:

The congruence condition △ABE ≅ △ACD forces the ratio ADAB to equal AEAC. Combined with the shared angle at A, this creates the SAS similarity condition for △ADE ~ △ABC!