Questions: Triangle Q T U is congruent to triangle R S U. Complete the proof that triangle Q R T is congruent to triangle R Q S. 1. Triangle Q T U is congruent to triangle R S U - Given 2. Line segment Q T is congruent to line segment R S 3. Line segment Q U is congruent to line segment R U - CPCTC 4. Line segment S U is congruent to line segment T U 5. Angle Q T U is congruent to angle R S U - CPCTC 6. Q S equals Q U plus S U - Additive Property of Length 7. R T equals R U plus T U - Additive Property of Length 8. Q S equals R U plus T U - Substitution 9. Q S equals R T - Transitive Property of Equality 10. Triangle Q R T is congruent to triangle R Q S

Triangle Q T U is congruent to triangle R S U. Complete the proof that triangle Q R T is congruent to triangle R Q S.

1. Triangle Q T U is congruent to triangle R S U - Given
2. Line segment Q T is congruent to line segment R S
3. Line segment Q U is congruent to line segment R U - CPCTC
4. Line segment S U is congruent to line segment T U
5. Angle Q T U is congruent to angle R S U - CPCTC
6. Q S equals Q U plus S U - Additive Property of Length
7. R T equals R U plus T U - Additive Property of Length
8. Q S equals R U plus T U - Substitution
9. Q S equals R T - Transitive Property of Equality
10. Triangle Q R T is congruent to triangle R Q S

Transcript text: $\triangle Q T U \cong \triangle R S U$. Complete the proof that $\triangle Q R T \cong \triangle R Q S$. \begin{tabular}{|l|l|l} \hline & Statement & Reason \\ \hline 1 & $\triangle Q T U \cong \triangle R S U$ & Given \\ \hline 2 & $\overline{Q T} \cong \overline{R S}$ & \\ 3 & $\overline{Q U} \cong \overline{R U}$ & CPCTC \\ 4 & $\overline{S U} \cong \overline{T U}$ & \\ 5 & $\angle Q T U \cong \angle R S U$ & CPCTC \\ 6 & $Q S=Q U+S U$ & Additive Property of Length \\ 7 & $R T=R U+T U$ & Additive Property of Length \\ 8 & $Q S=R U+T U$ & Substitution \\ 9 & $Q S=R T$ & Transitive Property of Equality \\ 10 & $\Delta Q R T \cong \triangle R Q S$ & \\ \hline \end{tabular}

Solution

Solution Steps

Step 1: Given

The problem states that $ \triangle QTU \cong \triangle RSU $.

Step 2: Corresponding Sides

Since $ \triangle QTU \cong \triangle RSU $, the corresponding sides are congruent: \[ \overline{QT} \cong \overline{RS} \]

Step 3: Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

From the congruence of the triangles, the corresponding parts are congruent: \[ \overline{QU} \cong \overline{RU} \]

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

Similarly, the other corresponding parts are congruent: \[ \overline{SU} \cong \overline{TU} \]

Step 5: Corresponding Angles

The corresponding angles of the congruent triangles are also congruent: \[ \angle QTU \cong \angle RSU \]

Step 6: Additive Property of Length

The length of $ \overline{QS} $ can be expressed as the sum of the lengths of $ \overline{QU} $ and $ \overline{SU} $: \[ QS = QU + SU \]

Step 7: Additive Property of Length

Similarly, the length of $ \overline{RT} $ can be expressed as the sum of the lengths of $ \overline{RU} $ and $ \overline{TU} $: \[ RT = RU + TU \]

Step 8: Substitution

Since $ \overline{QU} \cong \overline{RU} $ and $ \overline{SU} \cong \overline{TU} $, we can substitute these into the previous equations: \[ QS = RU + TU \]

Step 9: Transitive Property of Equality

From the previous steps, we have: \[ QS = RT \]

Step 10: Conclusion

Since $ \overline{QT} \cong \overline{RS} $, $ \overline{QS} \cong \overline{RT} $, and $ \angle QTU \cong \angle RSU $, by the definition of congruent triangles, we have: \[ \triangle QRT \cong \triangle RQS \]