Question

x is the midpoint of \\(\overline{uw}\\) and \\(\overline{tv}\\). complete the proof that \\(\angle t \cong \angle v\\).
(image of two triangles with vertices t, w and u, v intersecting at x, and a table with statements 1 - 7 and empty reason column)

Explanation:

Step1: Definition of midpoint

If \(X\) is the midpoint of \(\overline{UW}\), then \(UX = WX\) (by the definition of a midpoint: a midpoint divides a segment into two equal - length segments). So, for statement \(2\): \(UX\cong WX\), the reason is "Definition of midpoint".

Step2: Vertical angles

\(\angle TXW\) and \(\angle UXV\) are vertical angles. Vertical angles are congruent. So, for statement \(4\): \(\angle TXW\cong\angle UXV\), the reason is "Vertical angles are congruent".

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

We have \(UX\cong WX\) (from step 1), \(\angle TXW\cong\angle UXV\) (from step 2), and if \(X\) is the midpoint of \(\overline{TV}\), then \(TX = VX\) (by the definition of a midpoint). So, for statement \(6\): \(\triangle TWX\cong\triangle VUX\), the reason is "SAS (Side - Angle - Side) congruence criterion".

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

Since \(\triangle TWX\cong\triangle VUX\), then \(\angle T\cong\angle V\). So, for statement \(7\): \(\angle T\cong\angle V\), the reason is "CPCTC (Corresponding Parts of Congruent Triangles are Congruent)".

Statement	Reason
\(2\). \(UX\cong WX\)	Definition of midpoint
\(3\). \(X\) is the midpoint of \(\overline{TV}\)	Given
\(4\). \(\angle TXW\cong\angle UXV\)	Vertical angles are congruent
\(5\). \(TX\cong VX\)	Definition of midpoint
\(6\). \(\triangle TWX\cong\triangle VUX\)	SAS (Side - Angle - Side)
\(7\). \(\angle T\cong\angle V\)	CPCTC

Answer:

Statement	Reason
\(2\). \(UX\cong WX\)	Definition of midpoint
\(3\). \(X\) is the midpoint of \(\overline{TV}\)	Given
\(4\). \(\angle TXW\cong\angle UXV\)	Vertical angles are congruent
\(5\). \(TX\cong VX\)	Definition of midpoint
\(6\). \(\triangle TWX\cong\triangle VUX\)	SAS (Side - Angle - Side)
\(7\). \(\angle T\cong\angle V\)	CPCTC