Final answer:
The missing reason for the 3rd step in the proof is that corresponding sides of similar triangles are proportional, which is essential for establishing that the slopes of AB and CD are equal because the lines are parallel.
Step-by-step explanation:
To find the missing reason for the 3rd step in the proof, we need to understand the relationship between the given parallel lines and the slopes of the lines AB and CD. Since AX is parallel to CY and BX is parallel to DY, and also AB is parallel to CD, by the Corresponding Angles Postulate, we know that the angles formed by these lines are congruent, which implies that the corresponding sides are proportional. This leads us to similar triangles. From the information given, we can infer that triangles ABD and CBD are similar due to the AA (Angle-Angle) Similarity Postulate, since two pairs of angles are congruent (due to the parallel lines and their transversals).
The ratio of the corresponding sides of similar triangles is equal, therefore, the sides of triangle ABD over the sides of triangle CBD will be equal, which leads us to AX/BX = DY/CY. This gives us the proportions of the sides, allowing us to calculate the slopes.
We know that the slope of a line is the ratio of the vertical change (rise) to the horizontal change (run). Given that AB and CD are parallel, their slopes must be equal. Therefore, the reason for the 3rd step (according to the numbering provided) is that corresponding sides of similar triangles are proportional, which relates directly to the concept of slope.