Questions: angle TVX congruent to angle TWU, overline TW congruent to overline TV, and angle WTX congruent to angle UTV. Complete the proof that overline UW congruent to overline VX. Statement Reason --------- ------ angle TVX congruent to angle TWU Given overline TW congruent to overline TV Given angle WTX congruent to angle UTV Given m angle VTX = m angle VTW + m angle WTX Additive Property of Angle Measure m angle UTW = m angle UTV + m angle VTW m angle VTX = m angle VTW + m angle UTV Substitution m angle UTW = m angle VTX Transitive Property of Equality triangle TVX congruent to triangle TWU ASA overline UW congruent to overline VX

angle TVX congruent to angle TWU, overline TW congruent to overline TV, and angle WTX congruent to angle UTV. Complete the proof that overline UW congruent to overline VX.

Statement Reason
--------- ------
angle TVX congruent to angle TWU Given
overline TW congruent to overline TV Given
angle WTX congruent to angle UTV Given
m angle VTX = m angle VTW + m angle WTX Additive Property of Angle Measure
m angle UTW = m angle UTV + m angle VTW
m angle VTX = m angle VTW + m angle UTV Substitution
m angle UTW = m angle VTX Transitive Property of Equality
triangle TVX congruent to triangle TWU ASA
overline UW congruent to overline VX

Transcript text: $\angle T V X \cong \angle T W U, \overline{T W} \cong \overline{T V}$, and $\angle W T X \cong \angle U T V$. Complete the proof that $\overline{U W} \cong \overline{V X}$. \begin{tabular}{|l|l|l|} \hline & Statement & Reason \\ \hline 1 & $\angle T V X \cong \angle T W U$ & Given \\ 2 & $\overline{T W} \cong \overline{T V}$ & Given \\ 3 & $\angle W T X \cong \angle U T V$ & Given \\ 4 & $m \angle V T X=m \angle V T W+m \angle W T X$ & Additive Property of Angle Measure \\ 5 & $m \angle U T W=m \angle U T V+m \angle V T W$ & \\ 6 & $m \angle V T X=m \angle V T W+m \angle U T V$ & Substitution \\ 7 & $m \angle U T W=m \angle V T X$ & Transitive Property of Equality \\ -8 & $\triangle T V X \cong \triangle T W U$ & ASA \\ 9 & $\overline{U W} \cong \overline{V X}$ & \\ \hline \end{tabular}

Solution

Solution Steps

Step 1: Angle Addition Postulate

The measure of angle UTW is the sum of the measures of angles UTV and VTW. This justifies statement 5: $m\angle UTW = m\angle UTV + m\angle VTW$.

Step 2: Substitution Property of Equality

Since $\angle WTX \cong \angle UTV$, their measures are equal. Substituting $m\angle UTV$ for $m\angle WTX$ in statement 4 ($m\angle VTX = m\angle VTW + m\angle WTX$) yields $m\angle VTX = m\angle VTW + m\angle UTV$, justifying statement 6.

Step 3: Congruence by ASA

We are given $\angle TVX \cong \angle TWU$, $\overline{TW} \cong \overline{TV}$, and we proved that $\angle VTX \cong \angle UTW$ (from statements 5, 6 and 7). Therefore, $\triangle TVX \cong \triangle TWU$ by the Angle-Side-Angle (ASA) Congruence Postulate. This justifies statement 8.

Step 4: Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

Since $\triangle TVX \cong \triangle TWU$, their corresponding parts are congruent. Therefore, $\overline{UW} \cong \overline{VX}$. This justifies statement 9.