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}

Solution

Solution Steps

Step 1: Identify the given information

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

Step 2: Determine the congruent triangles

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

Step 3: Complete the proof

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

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 & $Y$ 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 & $\overline{X Y} \cong \overline{V Y}$ & Definition of midpoint \\ 6 & $\angle UYX \cong \angle WYV$ & Vertical Angles are Congruent \\ 7 & $\triangle U X Y \cong \triangle W V Y$ & SAS Congruence \\ \hline \end{tabular}