Let’s go step by step to understand why the proof works.
We are given:
- Line p is parallel to line r (p || r)
- Angle 1 is congruent to angle 3 (∠1 ≅ ∠3)
We need to prove:
- Line ℓ is parallel to line m (ℓ || m)
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Step 1: We start with what’s given — p || r. That’s our starting point.
Step 2: Because p and r are parallel, and they’re cut by transversal m, we can use the
Alternate Exterior Angles Theorem. This tells us that ∠3 and ∠2 are congruent. So, ∠3 ≅ ∠2.
Step 3: We’re also told that ∠1 ≅ ∠3 — that’s given directly.
Step 4: Now, since ∠1 ≅ ∠3 and ∠3 ≅ ∠2, then by the
Transitive Property of Congruence, ∠1 must be congruent to ∠2. So, ∠1 ∠2.
Step 5: Look at angles 1 and 2 — they are corresponding angles formed when lines ℓ and m are cut by transversal p. If corresponding angles are congruent, then the lines are parallel. That’s the
Converse of the Corresponding Angles Postulate. So, ℓ || m.
✔ All steps check out logically.
Final Answer:
The proof correctly shows that ℓ || m using the given information and geometric theorems.
Parent Tip: Review the logic above to help your child master the concept of parallel line proofs worksheet.
