Question

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

statement
1 wx ≅ vx
2 tx ≅ ux
3 ∠txw ≅ ∠uxv
4

reason
given
given
vertical angle theorem

Explanation:

Step1: Identify congruent triangles

We have two triangles \(\triangle TXW\) and \(\triangle UXV\).
In \(\triangle TXW\) and \(\triangle UXV\), we know that \(\overline{WX}\cong\overline{VX}\) (given), \(\overline{TX}\cong\overline{UX}\) (given), and \(\angle TXW\cong\angle UXV\) (Vertical Angle Theorem).
By the Side - Angle - Side (SAS) congruence criterion, 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.

Step2: State the congruence of triangles

So, \(\triangle TXW\cong\triangle UXV\) by the SAS (Side - Angle - Side) congruence rule.

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

Since \(\triangle TXW\cong\triangle UXV\), their corresponding sides \(\overline{UV}\) and \(\overline{TW}\) are congruent.

Answer:

1. \(\triangle TXW\cong\triangle UXV\); SAS (Side - Angle - Side) Congruence Rule.