$\delta qtu \cong \delta rsu$. complete the proof that $\delta qrt \cong \delta rqs$.

statement | reason
1. $\delta qtu \cong \delta rsu$ |
2. $\overline{qt} \cong \overline{rs}$ |
3. $\overline{qu} \cong \overline{ru}$ |
4. $\overline{su} \cong \overline{tu}$ |
5. $\angle qtu \cong \angle rsu$ |
6. $qs = qu + su$ |
7. $rt = ru + tu$ |
8. $qs = ru + tu$ |
9. $qs = rt$ |
10. $\delta qrt \cong \delta rqs$ |

Question

$\delta qtu \cong \delta rsu$. complete the proof that $\delta qrt \cong \delta rqs$.

statement | reason
1. $\delta qtu \cong \delta rsu$ |
2. $\overline{qt} \cong \overline{rs}$ |
3. $\overline{qu} \cong \overline{ru}$ |
4. $\overline{su} \cong \overline{tu}$ |
5. $\angle qtu \cong \angle rsu$ |
6. $qs = qu + su$ |
7. $rt = ru + tu$ |
8. $qs = ru + tu$ |
9. $qs = rt$ |
10. $\delta qrt \cong \delta rqs$ |

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Geometric Proofs", "Triangle Congruence" ], "new_concepts": [ "Segment Addition Postulate", "CPCTC" ], "current_concepts": [ "Geometric Proofs", "Triangle Congruence", "CPCTC", "Segment Addition Postulate" ] } ## Analyze the given information Using the *Geometric Proofs* and *Triangle Congruence* knowledge points $$ \begin{aligned} &\text{Given: } \Delta QTU \cong \Delta RSU\ &\text{Target: Complete the reasons for the 10-step proof.} \end{aligned} $$ ## Determine reasons for CPCTC steps Using the *Triangle Congruence* knowledge point $$ \begin{aligned} &\text{Step 1: } \Delta QTU \cong \Delta RSU \implies \text{Given}\ &\text{Step 2: } \overline{QT} \cong \overline{RS} \implies \text{CPCTC}\ &\text{Step 3: } \overline{QU} \cong \overline{RU} \implies \text{CPCTC}\ &\text{Step 4: } \overline{SU} \cong \overline{TU} \implies \text{CPCTC}\ &\text{Step 5: } \angle QTU \cong \angle RSU \implies \text{CPCTC} \end{aligned} $$ ## Determine reasons for segment addition Using the *Geometric Proofs* knowledge point $$ \begin{aligned} &\text{Step 6: } QS = QU + SU \implies \text{Segment Addition Postulate}\ &\text{Step 7: } RT = RU + TU \implies \text{Segment Addition Postulate} \end{aligned} $$ ## Determine reasons for substitution and congruence Using the *Geometric Proofs* and *Triangle Congruence* knowledge points $$ \begin{aligned} &\text{Step 8: } QS = RU + TU \implies \text{Substitution Property of Equality}\ &\text{Step 9: } QS = RT \implies \text{Substitution Property of Equality}\ &\text{Step 10: } \Delta QRT \cong \Delta RQS \implies \text{SAS Congruence Theorem} \end{aligned} $$ | Step | Statement | Reason | | :--- | :--- | :--- | | 1 | $\Delta QTU \cong \Delta RSU$ | Given | | 2 | $\overline{QT} \cong \overline{RS}$ | CPCTC (Corresponding Parts of Congruent Triangles are Congruent) | | 3 | $\overline{QU} \cong \overline{RU}$ | CPCTC (Corresponding Parts of Congruent Triangles are Congruent) | | 4 | $\overline{SU} \cong \overline{TU}$ | CPCTC (Corresponding Parts of Congruent Triangles are Congruent) | | 5 | $\angle QTU \cong \angle RSU$ | CPCTC (Corresponding Parts of Congruent Triangles are Congruent) | | 6 | $QS = QU + SU$ | Segment Addition Postulate | | 7 | $RT = RU + TU$ | Segment Addition Postulate | | 8 | $QS = RU + TU$ | Substitution Property of Equality | | 9 | $QS = RT$ | Substitution Property of Equality | | 10 | $\Delta QRT \cong \Delta RQS$ | SAS (Side-Angle-Side) Congruence Theorem | { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Geometric Proofs" ] }

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QUESTION IMAGE

Question

Explanation:

Analyze the given information

Determine reasons for CPCTC steps

Determine reasons for segment addition

Determine reasons for substitution and congruence

Answer:

Step	Statement	Reason
2	\(\overline{QT} \cong \overline{RS}\)	CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
3	\(\overline{QU} \cong \overline{RU}\)	CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
4	\(\overline{SU} \cong \overline{TU}\)	CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
5	\(\angle QTU \cong \angle RSU\)	CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
6	\(QS = QU + SU\)	Segment Addition Postulate
7	\(RT = RU + TU\)	Segment Addition Postulate
8	\(QS = RU + TU\)	Substitution Property of Equality
9	\(QS = RT\)	Substitution Property of Equality
10	\(\Delta QRT \cong \Delta RQS\)	SAS (Side-Angle-Side) Congruence Theorem