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Geometry > K.8 Proofs involving corresponding parts of congruent triangles AKL AQRS AQPT. Complete the proof that ZPTS = ZRST. S R 1 2 3 4 5 6 7 8 9 10 Ơ P 11 T Statement AQRS AQPT PT RS PQ = QR QT = QS PS= PQ+ QS RT = QR + QT PS= QR + QT PS= RT ST ST APSTARTS ZPTS ZRST Reason Given CPCTC CPCTC Additive Property of Length Additive Property of Length Substitution Transitive Property of Equality Reflexive Property of Congruence CPCTC

Geometry > K.8 Proofs involving corresponding parts of congruent triangles AKL AQRS AQPT. Complete the proof that ZPTS = ZRST. S R 1 2 3 4 5 6 7 8 9 10 Ơ P 11 T Statement AQRS AQPT PT RS PQ = QR QT = QS PS= PQ+ QS RT = QR + QT PS= QR + QT PS= RT ST ST APSTARTS ZPTS ZRST Reason Given CPCTC CPCTC Additive Property of Length Additive Property of Length Substitution Transitive Property of Equality Reflexive Property of Congruence CPCTC

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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**Interactive Proof: Proving Angles in Congruent Triangles** **Objective:** To prove that \(\angle PTS \cong \angle RST\) using the given congruent triangles. **Diagram Description:** The image presents two triangles, \(\triangle QRS\) and \(\triangle QPT\), with shared vertex \(Q\). The triangles are outlined in green. **Proof Structure:** | Statement | Reason | |-----------------------------------------------|------------------------------------------| | 1. \(\triangle QRS \cong \triangle QPT\) | Given | | 2. \(PT \cong RS\) | CPCTC (Corresponding Parts of Congruent Triangles are Congruent) | | 3. \(PQ \cong QR\) | CPCTC | | 4. \(QT = QS\) | Given | | 5. \(PS = PQ + QS\) | Additive Property of Length | | 6. \(RT = QR + QT\) | Additive Property of Length | | 7. \(PS = RT\) | Substitution | | 8. \(PS = QR + QT\) | Transitive Property of Equality | | 9. \(ST \cong ST\) | Reflexive Property of Congruence | | 10. \(\triangle PST \cong \triangle RST\) | CPCTC | | 11. \(\angle PTS \cong \angle RST\) | CPCTC | **Explanation:** The proof begins by using the given condition that \(\triangle QRS \cong \triangle QPT\). Using the properties of congruent triangles, it establishes the congruences of the sides and then verifies the angles are congruent through a series of logical steps. **Note:** Throughout the proof, concepts such as Corresponding Parts of Congruent Triangles are Congruent (CPCTC), the Additive Property of Length, and the Reflexive Property of Congruence are applied to derive the necessary conclusions. Students can apply these principles to understand the congruence relationship between parts of geometric figures and how to establish equality through proofs.
Transcribed Image Text:**Interactive Proof: Proving Angles in Congruent Triangles** **Objective:** To prove that \(\angle PTS \cong \angle RST\) using the given congruent triangles. **Diagram Description:** The image presents two triangles, \(\triangle QRS\) and \(\triangle QPT\), with shared vertex \(Q\). The triangles are outlined in green. **Proof Structure:** | Statement | Reason | |-----------------------------------------------|------------------------------------------| | 1. \(\triangle QRS \cong \triangle QPT\) | Given | | 2. \(PT \cong RS\) | CPCTC (Corresponding Parts of Congruent Triangles are Congruent) | | 3. \(PQ \cong QR\) | CPCTC | | 4. \(QT = QS\) | Given | | 5. \(PS = PQ + QS\) | Additive Property of Length | | 6. \(RT = QR + QT\) | Additive Property of Length | | 7. \(PS = RT\) | Substitution | | 8. \(PS = QR + QT\) | Transitive Property of Equality | | 9. \(ST \cong ST\) | Reflexive Property of Congruence | | 10. \(\triangle PST \cong \triangle RST\) | CPCTC | | 11. \(\angle PTS \cong \angle RST\) | CPCTC | **Explanation:** The proof begins by using the given condition that \(\triangle QRS \cong \triangle QPT\). Using the properties of congruent triangles, it establishes the congruences of the sides and then verifies the angles are congruent through a series of logical steps. **Note:** Throughout the proof, concepts such as Corresponding Parts of Congruent Triangles are Congruent (CPCTC), the Additive Property of Length, and the Reflexive Property of Congruence are applied to derive the necessary conclusions. Students can apply these principles to understand the congruence relationship between parts of geometric figures and how to establish equality through proofs.
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