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If two lines and a transversal form same-side exterior angles that are supplementary, then the two lines are parallel.
Based on the characteristics of the diagram, the following relation holds true.
m∠ 1 + m∠ 2 &= 180 ^(∘) or m∠ 3 + m∠ 4 &= 180 ^(∘) ⇒ l_1 ∥ l_2
The theorem is named the Converse Same-Side Exterior Angles Theorem because the same thing holds true but in opposite order.
It needs to be proven that l_1 and l_2 are parallel lines. It is already given that ∠ 1 is supplementary to ∠ 2. m∠ 1 + m∠ 2 = 180^(∘) The diagram shows that ∠ 1 and ∠ α form a linear pair, which means that these angles are supplementary. m∠ 1 + m∠ α = 180^(∘) Notice that ∠ 1 is common in both relationships. Subtracting the two equations gives a relation between ∠ 2 and ∠ α. m∠ 1 + m∠ 2 &= 180^(∘) ^- m∠ 1 + m∠ α &= 180^(∘) m∠ 2 - m∠ α &= 0^(∘) The last equation tells that ∠ 2 and ∠ α have the same measure and therefore, they are congruent. The diagram also shows that ∠ 2 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 ∠ 2 and ∠ α are corresponding congruent angles, the lines l_1 and l_2 are parallel, which completes the proof.