Question

1. what statement is missing from the 2 - column proof proving that △rus≅△svt? statements: ∠u≅∠v, ru≅sv, △rus≅△svt. reasons: given, given, asa congruence postulate. options: ∠r≅∠s, rs≅st, us≅vt, ∠s≅∠t

Explanation:

Step1: Recall ASA congruence postulate

The ASA (Angle - Side - Angle) congruence postulate states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. We are given that $\angle U\cong\angle V$ and $RU\cong SV$. We need the angle - side - angle combination. The missing part for ASA is the side between the given angles and the included angle with the given side. Since the angles are $\angle U$ and the side $RU$ in $\triangle RUS$ and $\angle V$ and the side $SV$ in $\triangle SVT$, the missing statement is $\angle RSU\cong\angle STV$ (corresponding angles on a transversal of 2 parallel lines are congruent would be the reason if there was a parallel - line context which is not fully shown here but in terms of ASA requirements). But looking at the options, we know that for ASA with the given $\angle U\cong\angle V$ and $RU\cong SV$, we need the included angles to be congruent. The included angles for the given side and angle pairs are such that we need to show that the sides adjacent to the given congruent angles and the given congruent sides are part of the congruence setup. The missing statement for the ASA congruence of $\triangle RUS$ and $\triangle SVT$ is $RS\cong ST$.

Answer:

RS ≅ ST