Given that lines l 𝑙 and n 𝑛 are perpendicular, which of the following is part of the proof that l 𝑙 and n 𝑛 have slopes that are opposite reciprocals? (1 point) Responses △ABC∼△EDA △ 𝐴 𝐵 𝐶 ∼ △ 𝐸 𝐷 𝐴 triangle cap A cap b cap c approximates triangle cap e cap d cap A ∠ABC≅∠DAE ∠ 𝐴 𝐵 𝐶 ≅ ∠ 𝐷 𝐴 𝐸 ∠ABC≅∠DAE ∠ 𝐴 𝐵 𝐶 ≅ ∠ 𝐷 𝐴 𝐸 The slope of line L is BC/CA=−DE/AD

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GPT-4o mini · Answer

In the proof that two lines $ l $ and $ n $ are perpendicular and that their slopes are opposite reciprocals, one of the key elements is showing the relationship between the angles formed by these lines. Given that $ \triangle ABC \sim \triangle EDA $, it indicates that the two triangles are similar, which would imply that their corresponding angles are congruent. Specifically, if we have $ \angle ABC $ and $ \angle DAE $ as corresponding angles, then: $$ \angle ABC \cong \angle DAE $$ This would help establish that the angle between line $ l $ and the x-axis (determining line $ l $'s slope) is complementary to the angle between line $ n $ and the x-axis (determining line $ n $'s slope). Thus, we can conclude that if the slopes of the lines are represented in terms of tangent (where slope $ m = \tan(\theta) $), then: $$ m_l \cdot m_n = -1 $$ This establishes the opposite reciprocal relationship. In this context, the provided statements regarding angle congruency are essential to show that the angles formed and subsequently the slopes are opposite reciprocals. Therefore, the statement that is part of the proof is: **$ \angle ABC \cong \angle DAE $** This highlights the angle relationship necessary to reach the conclusion about the slopes being opposite reciprocals when lines $ l $ and $ n $ are perpendicular.