Question

wx ≅ vx and tx ≅ ux. complete the proof that uv ≅ tw.

Explanation:

Step1: Recall the Side - Angle - Side (SAS) Congruence Theorem

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.
In \(\triangle TWX\) and \(\triangle UVX\), we have \(\overline{WX}\cong\overline{VX}\) (given), \(\overline{TX}\cong\overline{UX}\) (given), and \(\angle TXW\cong\angle UXV\) (Vertical Angle Theorem).

Step2: Apply the SAS Congruence Theorem

By the SAS (Side - Angle - Side) congruence theorem, \(\triangle TWX\cong\triangle UVX\).

Step3: Use the Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

If two triangles are congruent, then their corresponding parts (sides and angles) are congruent.
Since \(\triangle TWX\cong\triangle UVX\), then \(\overline{UV}\cong\overline{TW}\) (CPCTC).

Answer:

\(\triangle TWX\cong\triangle UVX\) (by SAS); \(\overline{UV}\cong\overline{TW}\) (by CPCTC)