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

Solution Steps

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

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

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

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

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

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

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

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

From the previous steps, we have: $$QS = RT$$

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: $$\triangle QRT \cong \triangle RQS$$

Final Answer