Question

x is the midpoint of \\(\overline{vw}\\). complete the proof that \\(\triangle uvx \cong \triangle uwx\\).
(image of triangle with points v, w, u, x, right angle at x between ux and vw)

	statement	reason
1	x is the midpoint of \\(\overline{vw}\\)
2	\\(\overline{ux} \perp \overline{vw}\\)
3	\\(\angle uxv \cong \angle uxw\\)
4	\\(\overline{vx} \cong \overline{wx}\\)
5	\\(\overline{ux} \cong \overline{ux}\\)
6	\\(\triangle uvx \cong \triangle uwx\\)

Explanation:

Step1: Reason for Statement 1

The problem states "X is the midpoint of \(\overline{VW}\)", so the reason is "Given" (as it's the information provided to start the proof).

Step2: Reason for Statement 2

\(\overline{UX} \perp \overline{VW}\) is likely given by the diagram (the right angle symbol), so the reason is "Given (from diagram, perpendicular lines)".

Step3: Reason for Statement 3

\(\angle UXV \cong \angle UXW\) because they are right angles (since \(\overline{UX} \perp \overline{VW}\)), so the reason is "All right angles are congruent".

Step4: Reason for Statement 4

Since X is the midpoint of \(\overline{VW}\), by the definition of a midpoint, \(\overline{VX} \cong \overline{WX}\), so the reason is "Definition of midpoint".

Step5: Reason for Statement 5

\(\overline{UX} \cong \overline{UX}\) because a segment is congruent to itself, so the reason is "Reflexive property of congruence".

Step6: Reason for Statement 6

To prove \(\triangle UVX \cong \triangle UWX\), we have two sides and the included angle: \(\overline{VX} \cong \overline{WX}\), \(\angle UXV \cong \angle UXW\), \(\overline{UX} \cong \overline{UX}\), so by SAS (Side - Angle - Side) congruence criterion, \(\triangle UVX \cong \triangle UWX\). The reason is "SAS Congruence Postulate".

Answer:

1. Reason: Given
2. Reason: Given (from diagram, perpendicular lines)
3. Reason: All right angles are congruent
4. Reason: Definition of midpoint
5. Reason: Reflexive property of congruence
6. Reason: SAS Congruence Postulate