Questions: Geometry ≥+2.3 Proving triangles congruent by SSS and SAS WZ Y is the midpoint of UW and VX. Complete the proof that triangle UXY is congruent to triangle WVY. 1. Y is the midpoint of UW - Given 2. Y is the midpoint of VX - Given 3. VW is congruent to UX - Given 4. UY is congruent to WY - Definition of midpoint

Geometry ≥+2.3 Proving triangles congruent by SSS and SAS WZ
Y is the midpoint of UW and VX. Complete the proof that triangle UXY is congruent to triangle WVY.

1. Y is the midpoint of UW - Given
2. Y is the midpoint of VX - Given
3. VW is congruent to UX - Given
4. UY is congruent to WY - Definition of midpoint
Transcript text: Geometry $\geqslant+2.3$ Proving triangles congruent by SSS and SAS WZ $Y$ is the midpoint of $\overline{U W}$ and $\overline{V X}$. Complete the proof that $\triangle U X Y \cong \triangle W V Y$. \begin{tabular}{|l|l|l|} \hline & Statement & Reason \\ \hline 1 & $Y$ is the midpoint of $\overline{U W}$ & Given \\ 2 & $Y$ is the midpoint of $\overline{V X}$ & Given \\ 3 & $\overline{V W} \cong \overline{U X}$ & Given \\ \hline 4 & $\overline{U Y} \cong \overline{W Y}$ & Definition of midpoint \\ 5 & & \\ \hline & & \\ \hline \end{tabular}
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Solution

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Solution Steps

Step 1: Identify the given information

We are given that YY is the midpoint of both UW\overline{UW} and VX\overline{VX}. This means that UYWY\overline{UY} \cong \overline{WY} and VYXY\overline{VY} \cong \overline{XY}. We are also given that VWUX\overline{VW} \cong \overline{UX}.

Step 2: Determine the congruent triangles

We want to prove that UXYWVY\triangle UXY \cong \triangle WVY.

Step 3: Complete the proof

We already have two pairs of congruent sides: UYWY\overline{UY} \cong \overline{WY} and UXWV\overline{UX} \cong \overline{WV}. We need one more piece of information. Notice that UYX\angle UYX and WYV\angle WYV are vertical angles. Therefore, UYXWYV\angle UYX \cong \angle WYV.

Now we have enough information to prove the triangles congruent using Side-Angle-Side (SAS) congruence.

\begin{tabular}{|l|l|l|} \hline & Statement & Reason \\ \hline 1 & YY is the midpoint of UW\overline{U W} & Given \\ 2 & YY is the midpoint of VX\overline{V X} & Given \\ 3 & VWUX\overline{V W} \cong \overline{U X} & Given \\ \hline 4 & UYWY\overline{U Y} \cong \overline{W Y} & Definition of midpoint \\ 5 & XYVY\overline{X Y} \cong \overline{V Y} & Definition of midpoint \\ 6 & UYXWYV\angle UYX \cong \angle WYV & Vertical Angles are Congruent \\ 7 & UXYWVY\triangle U X Y \cong \triangle W V Y & SAS Congruence \\ \hline \end{tabular}

Final Answer

\\(\boxed{\triangle UXY \cong \triangle WVY \text{ by SAS Congruence}}\\)

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