Question

complete the proof that \\(\overline{vw} \cong \overline{tz}\\).
(image of triangles with angles and segments marked)

	statement	reason
1	\\(\angle v \cong \angle t\\)
2	\\(\overline{yz} \cong \overline{wx}\\)
3	\\(\angle vyw \cong \angle txz\\)
4	\\(xz = yz + xy\\)
5	\\(wy = wx + xy\\)
6	\\(xz = wx + xy\\)
7	\\(wy = xz\\)
8	\\(\triangle txz \cong \triangle vyw\\)
9	\\(\overline{vw} \cong \overline{tz}\\)

Explanation:

Step1: Identify given angle congruence

$\angle V \cong \angle T$ (marked in diagram)

Step2: Identify given segment congruence

$\overline{YZ} \cong \overline{WX}$ (marked in diagram)

Step3: Identify vertical angles

$\angle VYW \cong \angle TXZ$ (vertical angles theorem)

Step4: Segment addition for $XZ$

$XZ = YZ + XY$ (segment addition postulate)

Step5: Segment addition for $WY$

$WY = WX + XY$ (segment addition postulate)

Step6: Substitute congruent segments

$XZ = WX + XY$ (substitute $\overline{YZ} \cong \overline{WX}$ into Step4)

Step7: Equate $WY$ and $XZ$

$WY = XZ$ (transitive property from Steps5 and 6)

Step8: Prove triangle congruence

$\triangle TXZ \cong \triangle VYW$ (ASA: $\angle T \cong \angle V$, $XZ \cong WY$, $\angle TXZ \cong \angle VYW$)

Step9: Corresponding parts of congruent triangles

$\overline{VW} \cong \overline{TZ}$ (CPCTC)

Answer:

$\overline{VW} \cong \overline{TZ}$ is proven via ASA congruence and CPCTC.