Question

complete the proof that \\(\overline{vw} \cong \overline{tz}\\).

(image of triangles with markings and a table of statements and reasons)

1. \\(\angle v \cong \angle t\\) - given
2. \\(\overline{yz} \cong \overline{wx}\\) - given
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 reason for statement 3

$\angle VYW \cong \angle TXZ$ (vertical angles are congruent)

Step2: Identify reason for statement 4

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

Step3: Identify reason for statement 5

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

Step4: Identify reason for statement 6

$XZ = WX + XY$ (substitution property, since $YZ \cong WX$)

Step5: Identify reason for statement 7

$WY = XZ$ (transitive property of equality)

Step6: Identify reason for statement 8

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

Step7: Identify reason for statement 9

$\overline{VW} \cong \overline{TZ}$ (corresponding parts of congruent triangles are congruent)

Answer:

The completed reasons are:

1. Vertical angles are congruent
2. Segment addition postulate
3. Segment addition postulate
4. Substitution property
5. Transitive property of equality
6. ASA congruence
7. Corresponding parts of congruent triangles are congruent

$\overline{VW} \cong \overline{TZ}$ is proven.