Question

determine the missing information in the paragraph proof. given: lines a and c intersect at point s, creating 4 angles. prove: corresponding angles are congruent. diagram: line c (with points r, q, p), line a (with points u, s, t) we are given that lines a and c intersect at point s. translate line a down line c until point s reaches point q. call the new line through q line b. because translations preserve orientation, lines a and b are ______. because translations preserve angle measure, ∠rsu ≅ ∠rqu. for the same reason, ∠rst ≅ ∠rqt, ∠psu ≅ ∠pqu, and ∠pst ≅ ∠pqt. each of the angle pairs are corresponding angles. therefore, corresponding angles are congruent. options: parallel, perpendicular, congruent, reflected.

Explanation:

A translation is a rigid transformation that slides a figure without rotating or reflecting it. When we translate line \( a \) down line \( c \) to get line \( b \), the orientation of line \( a \) is preserved. Parallel lines are lines in a plane that do not intersect and have the same orientation (or slope, in coordinate geometry). Since translation preserves the direction (orientation) of the line, lines \( a \) and \( b \) will not intersect and will have the same direction, so they are parallel. Perpendicular lines intersect at right angles, which is not the case here. Congruent is a term for angles or figures, not lines in terms of their relationship (we say lines are parallel, not congruent). Reflected would mean a reflection transformation, but we did a translation, so reflected is incorrect.

Answer:

A. parallel