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If ∠1≅∠2 or ∠3≅∠4, then ℓ1∥ℓ2.
The proof will be based on the given diagram, but it holds true for any pair of lines cut by a transversal. Consider only one pair of congruent alternate interior angles and one more angle.
Converse Corresponding Angles Theorem |
If two lines are cut by a transversal so that the corresponding angles are congruent, then the lines are parallel. |
Since ∠1 and ∠α are corresponding congruent angles, then ℓ1 and ℓ2 are parallel lines. To summarize, all of the steps will be described in a two-column proof.
0. Statement
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0. Reason
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1. ∠1≅∠2
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1. Given
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2. ∠2≅∠α
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2. Vertical Angles Theorem
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3. ∠1≅∠α
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3. Transitive Property of Congruence
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4. ℓ1∥ℓ2
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4. Converse Corresponding Angles Theorem
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