Question

△uvw and △stw are equilateral. complete the proof that △svw ≅ △tuw.1 △uvw is equilateral given2 △stw is equilateral given3 \overline{uw} ≅ \overline{vw} definition of equilateral triangle4 \overline{sw} ≅ \overline{tw} definition of equilateral triangle5

Explanation:

Step1: Find the included angle

Since \(\triangle UVW\) is equilateral, \(\angle UVW=\angle VWU = \angle WUV=60^{\circ}\). Since \(\triangle STW\) is equilateral, \(\angle STW=\angle TWS=\angle WST = 60^{\circ}\).
\(\angle VWU+\angle VWS=\angle TWS+\angle VWS\) (Add \(\angle VWS\) to both sides). So \(\angle UWT=\angle VWS\)

Step2: Use the Side - Angle - Side (SAS) congruence criterion

In \(\triangle SVW\) and \(\triangle TUW\):

• \(VW = UW\) (From step 3, definition of equilateral triangle)
• \(\angle VWS=\angle UWT\) (Proved in step 1)
• \(SW = TW\) (From step 4, definition of equilateral triangle)

By the SAS (Side - Angle - Side) congruence criterion, \(\triangle SVW\cong\triangle TUW\)

Answer:

\(\angle VWS\cong\angle UWT\) (Reason: Angle addition property and substitution, since \(\angle VWU = \angle TWS=60^{\circ}\) and \(\angle VWU+\angle VWS=\angle TWS+\angle VWS\)); \(\triangle SVW\cong\triangle TUW\) (Reason: SAS (Side - Angle - Side) congruence criterion)