Question

x is the midpoint of \\(\overline{uw}\\) and \\(\overline{tv}\\). complete the proof that \\(\angle t \cong \angle v\\).
image of two triangles connected at point x, forming a vertical angle configuration: triangle t-w-x and triangle u-v-x

statement	reason
2. x is the midpoint of \\(\overline{tv}\\)	definition of midpoint
3. \\(\overline{ux} \cong \overline{wx}\\)	given
4. \\(\overline{tx} \cong \overline{vx}\\)
5. \\(\angle txw \cong \angle ux v\\)
6. \\(\triangle twx \cong \triangle vux\\)	sas
7. \\(\angle t \cong \angle v\\)	cpctc

Explanation:

Step1: Use the definition of midpoint

Since \(X\) is the midpoint of \(\overline{UW}\), by the definition of midpoint, \(UX = WX\) (i.e., \(\overline{UX}\cong\overline{WX}\)).
Since \(X\) is the midpoint of \(\overline{TV}\), by the definition of midpoint, \(TX = VX\) (i.e., \(\overline{TX}\cong\overline{VX}\)).

Step2: Consider vertical angles

\(\angle TXW\) and \(\angle UXV\) are vertical angles. By the property of vertical angles, \(\angle TXW\cong\angle UXV\).

Step3: Use the SAS (Side - Angle - Side) congruence criterion

In \(\triangle TWX\) and \(\triangle VUX\), we have \(\overline{UX}\cong\overline{WX}\), \(\angle TXW\cong\angle UXV\), \(\overline{TX}\cong\overline{VX}\). So, by the SAS congruence criterion, \(\triangle TWX\cong\triangle VUX\).

Step4: Use CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \(\triangle TWX\cong\triangle VUX\), by CPCTC, \(\angle T\cong\angle V\).

Answer:

\(\angle T\cong\angle V\) is proved as above.