Question

x is the midpoint of \\(\overline{uw}\\) and \\(\overline{tv}\\). complete the proof that \\(\angle t \cong \angle v\\).

(image of two triangles sharing a common vertex x, with vertices u, v, t, w as labeled)

	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 uxv\\)	definition of midpoint
6	\\(\triangle twx \cong \triangle vux\\)	sas
7	\\(\angle t \cong \angle v\\)	cpctc

Explanation:

Step1: Determine the reason for \(\overline{TX}\cong\overline{VX}\)

Since \(X\) is the midpoint of \(\overline{TV}\) (given in statement 2), by the definition of a midpoint, a midpoint divides a segment into two congruent segments. So the reason for \(\overline{TX}\cong\overline{VX}\) is "Definition of midpoint".

Step2: Determine the reason for \(\angle TXW\cong\angle UXV\)

\(\angle TXW\) and \(\angle UXV\) are vertical angles. Vertical angles are congruent. So the reason for \(\angle TXW\cong\angle UXV\) is "Vertical angles congruence theorem".

Answer:

For statement 4 (\(\overline{TX}\cong\overline{VX}\)), the reason is "Definition of midpoint". For statement 5 (\(\angle TXW\cong\angle UXV\)), the reason is "Vertical angles congruence theorem".