Question

x is the midpoint of vw and vw ⊥ ux. complete the proof that △uvx ≅ △uwx.

1 x is the midpoint of \\( \overline { vw } \\)
2 \\( \overline { vw } \perp \overline { ux } \\)
3 \\( \angle uxv \cong \angle uxw \\)
4 \\( \overline { vx } \cong \overline { wx } \\)
5 \\( \overline { ux } \cong \overline { ux } \\)
6 \\( \triangle u v x \cong \triangle u w x \\)

Explanation:

Step1: Given

Statement 1: \(X\) is the midpoint of \(\overline{VW}\) (Given).
Statement 2: \(\overline{VW}\perp\overline{UX}\) (Given).

Step2: Definition of perpendicular lines

Since \(\overline{VW}\perp\overline{UX}\), by the definition of perpendicular lines, \(\angle UXV\) and \(\angle UXW\) are right angles. And all right angles are congruent, so \(\angle UXV\cong\angle UXW\) (Statement 3).

Step3: Definition of midpoint

Because \(X\) is the midpoint of \(\overline{VW}\), by the definition of midpoint, \(VX = WX\), so \(\overline{VX}\cong\overline{WX}\) (Statement 4).

Step4: Reflexive property

For any segment \(AB\), \(AB = AB\). So for segment \(\overline{UX}\), \(\overline{UX}\cong\overline{UX}\) (Statement 5) by the reflexive property of congruence.

Step5: SAS (Side - Angle - Side) congruence criterion

In \(\triangle UVX\) and \(\triangle UWX\), we have \(\overline{VX}\cong\overline{WX}\), \(\angle UXV\cong\angle UXW\), and \(\overline{UX}\cong\overline{UX}\). By the SAS (Side - Angle - Side) congruence criterion, \(\triangle UVX\cong\triangle UWX\) (Statement 6).

Answer:

1. Given
2. Given
3. Definition of perpendicular lines (all right angles are congruent)
4. Definition of midpoint
5. Reflexive property of congruence
6. SAS (Side - Angle - Side) congruence criterion