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m∠ 1 + m∠ 3 &= 180 ^(∘) or m∠ 2 + m∠ 4 &= 180 ^(∘) ⇒ l_1 ∥ l_2
The theorem is named the Converse Consecutive Interior Angles Theorem because the same thing holds true but in opposite order.
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 consecutive interior angles and one more angle.
It needs to be proven that l_1 and l_2 are parallel lines. It is already given that ∠ 1 is supplementary to ∠ 3. m∠ 1 + m∠ 3 = 180^(∘) The diagram shows that ∠ 3 and ∠ α form a linear pair, which means that these angles are supplementary. m∠ 3 + m∠ α = 180^(∘) Notice that ∠ 3 is common in both relationships. Subtracting the two equations gives a relation between ∠ 1 and ∠ α. m∠ 1 + m∠ 3 &= 180^(∘) ^- m∠ 3 + m∠ α &= 180^(∘) m∠ 1 - m∠ α &= 0^(∘) The last equation tells that ∠ 1 and ∠ α have the same measure and therefore, they are congruent. The diagram also shows that ∠ 1 and ∠ α are corresponding angles. Given that relation, the Converse Corresponding Angles Theorem can be applied.
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Converse Corresponding Angles Theorem |
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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 l_1 and l_2 are parallel lines. This completes the proof.