Question

1. what statement is missing from the 2 - column proof proving that δrus ≅ δsvt? statements: ∠u ≅ ∠v (given), ru ≅ sv (given), missing statement, 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, rs ≅ st, us ≅ vt. (diagram shows triangles rus and svt with markings.)

Explanation:

To prove \(\triangle RUS \cong \triangle SVT\) using the ASA (Angle - Side - Angle) Congruence Postulate, we already have \(\angle U\cong\angle V\) (given) and \(RU\cong SV\) (given). We need another pair of congruent angles.
Looking at the diagram, \(RS\) and \(ST\) seem to be parts of a line, and \(RU\) and \(VT\) are marked as parallel (from the arrow marks). So, \(\angle R\) and \(\angle S\) (or more precisely, \(\angle R\) and \(\angle T\)?) Wait, no. Wait, for ASA, we need two angles and the included side. We have \(\angle U\cong\angle V\), \(RU\cong SV\). The missing angle should be \(\angle R\cong\angle S\)? Wait, no, let's re - examine.
Wait, the ASA 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 triangles are congruent.
We have \(\angle U\cong\angle V\) (given), \(RU\cong SV\) (given). The included side between \(\angle U\) and \(\angle R\) in \(\triangle RUS\) is \(RU\), and the included side between \(\angle V\) and \(\angle S\) in \(\triangle SVT\) is \(SV\). Wait, no, maybe the alternate interior angles. Since \(RU\parallel VT\) (from the arrow marks), \(\angle R\cong\angle S\) (alternate interior angles). Wait, the options are \(\angle R\cong\angle S\), \(\angle S\cong\angle T\), \(RS\cong ST\), \(US\cong VT\).
To use ASA, we need two angles and the included side. We have \(\angle U\cong\angle V\), \(RU\cong SV\). The missing angle should be \(\angle R\cong\angle S\) because then we have \(\angle R\cong\angle S\), \(RU\cong SV\), \(\angle U\cong\angle V\), which satisfies ASA.

Step 1: 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.

Step 2: Analyze Given Information

We are given \(\angle U\cong\angle V\) (given) and \(RU\cong SV\) (given). We need to find the missing angle to apply ASA.

Step 3: Determine the Missing Angle

From the diagram, \(RU\) and \(VT\) are parallel (indicated by the arrow marks). By the Alternate Interior Angles Theorem, \(\angle R\) and \(\angle S\) are alternate interior angles and thus \(\angle R\cong\angle S\). With \(\angle R\cong\angle S\), \(RU\cong SV\), and \(\angle U\cong\angle V\), we can apply the ASA Congruence Postulate to prove \(\triangle RUS\cong\triangle SVT\).

Answer:

\(\boldsymbol{\angle R\cong\angle S}\) (the option corresponding to \(\angle R\cong\angle S\))