Question

1. what statement is missing from the 2 - column proof proving that δrus ≅ δsvt?

statements:
∠u ≅ ∠v (given),
ru ≅ sv (given),
?
δrus ≅ δsvt (asa congruence postulate)
reasons:
given,
given,
corresponding angles on a transversal of 2 parallel lines are congruent,
asa congruence postulate
options:
∠r ≅ ∠s,
∠s ≅ ∠t,
us ≅ vt,
rs ≅ st

Explanation:

To prove \(\triangle RUS \cong \triangle SVT\) using ASA (Angle - Side - Angle) congruence postulate, we already have \(\angle U\cong\angle V\) (given) and \(RU\cong SV\) (given). The ASA postulate requires two angles and the included side to be congruent. We need the pair of congruent angles that are alternate interior angles (since \(RU\) and \(SV\) seem to be parallel, as indicated by the markings) formed by the transversal \(RS\) (or the line containing \(R, S, T\)). So, \(\angle R\cong\angle S\) (corresponding angles on a transversal of two parallel lines are congruent) is the missing statement to satisfy ASA, as \(\angle R\) and \(\angle U\) with included side \(RU\) in \(\triangle RUS\) and \(\angle S\) and \(\angle V\) with included side \(SV\) in \(\triangle SVT\) will then be congruent.

Answer:

\(\boldsymbol{\angle R\cong\angle S}\)