Questions: X is the midpoint of W. Complete the proof that triangle UWX is congruent to triangle UVX. - Statement - Reason 1. X is the midpoint of VW - Given 2. W perpendicular to UX - Given 3. angle UXV is congruent to angle UXW - All right angles are congruent 4. VX is congruent to WX - Definition of midpoint 5. UX is congruent to UX - Reflexive Property of Congruence 6. triangle UWX is congruent to triangle UVX -

X is the midpoint of W. Complete the proof that triangle UWX is congruent to triangle UVX.

- Statement - Reason
1. X is the midpoint of VW - Given
2. W perpendicular to UX - Given
3. angle UXV is congruent to angle UXW - All right angles are congruent
4. VX is congruent to WX - Definition of midpoint
5. UX is congruent to UX - Reflexive Property of Congruence
6. triangle UWX is congruent to triangle UVX -

Transcript text: $X$ is the midpoint of $\bar{W}$. Complete the proof that $\triangle U W X \cong \triangle U V X$. \begin{tabular}{|l|l|l|} \hline & Statement & Reason \\ \hline 1 & $X$ is the midpoint of $\overline{V W}$ & Given \\ 2 & $\overline{\bar{W}} \perp \overline{U X}$ & Given \\ \hline 3 & $\angle U X V \cong \angle U X W$ & All right angles are congruent \\ 4 & $\overline{\overline{V X}} \cong \overline{W X}$ & Definition of midpoint \\ 5 & $\overline{U X} \cong \overline{U X}$ & Reflexive Property of Congruence \\ 6 & $\triangle U W X \cong \triangle U V X$ & \\ \hline \end{tabular}

Solution

Solution Steps

Step 1: Analyze the given information

We are given that $X$ is the midpoint of $\overline{VW}$, which means $\overline{VX} \cong \overline{WX}$. Also, $\overline{UW} \perp \overline{UX}$, implying that $\angle UXW$ is a right angle. Since $\overline{VW}$ is a straight line, $\angle UXV$ must also be a right angle. Therefore, $\angle UXV \cong \angle UXW$. Lastly, $\overline{UX}$ is congruent to itself by the reflexive property.

Step 2: Identify the congruence theorem

We have two pairs of congruent sides: $\overline{VX} \cong \overline{WX}$ and $\overline{UX} \cong \overline{UX}$. We also have a pair of congruent angles: $\angle UXV \cong \angle UXW$. Since the congruent angles are between the two pairs of congruent sides, we can use the Side-Angle-Side (SAS) congruence theorem.

Step 3: Complete the proof

The reason for the congruence of the triangles in step 6 is the SAS Congruence Postulate.