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AQRS AQPT. Complete the proof that ZPTS = ZRST. S R O' Q P T

AQRS AQPT. Complete the proof that ZPTS = ZRST. S R O' Q P T

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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### Proof of Angle Congruence in Congruent Triangles #### Given: \(\triangle QRS \cong \triangle QPT\). Complete the proof that \(\angle PTS \cong \angle RST\). #### Diagram: The diagram consists of two congruent triangles, \(\triangle QRS\) and \(\triangle QPT\), sharing vertex \(Q\). Line segments are labeled with letters corresponding to points \(S\), \(R\), \(Q\), \(P\), and \(T\). #### Proof: The proof is organized in a two-column format displaying statements and reasons needed to establish the congruence of the angle pairs. | Statement | Reason | |---------------------------------|-----------------------------------------| | 1. \(\triangle QRS \cong \triangle QPT\) | Given | | 2. \(PT = RS\) | CPCTC (Corresponding Parts of Congruent Triangles are Congruent) | | 3. \(PQ = QR\) | CPCTC | | 4. \(QT = QS\) | CPCTC | | 5. \(PS = PQ + QS\) | Additive Property of Length | | 6. \(RT = QR + QT\) | Additive Property of Length | | 7. \(PS = QR + QT\) | Substitution | | 8. \(PS = RT\) | Transitive Property of Equality | | 9. \(ST = ST\) | Reflexive Property of Congruence | | 10. \(\triangle PST \cong \triangle RTS\) | SAS (Side-Angle-Side Congruence Postulate) | | 11. \(\angle PTS \cong \angle RST\) | CPCTC | This organized approach utilizes the properties of congruent triangles, the Additive Property of Length, Substitution, Transitive and Reflexive Properties, and the Side-Angle-Side Congruence Postulate to deduce the congruence of angles \(\angle PTS\) and \(\angle RST\).
Transcribed Image Text:### Proof of Angle Congruence in Congruent Triangles #### Given: \(\triangle QRS \cong \triangle QPT\). Complete the proof that \(\angle PTS \cong \angle RST\). #### Diagram: The diagram consists of two congruent triangles, \(\triangle QRS\) and \(\triangle QPT\), sharing vertex \(Q\). Line segments are labeled with letters corresponding to points \(S\), \(R\), \(Q\), \(P\), and \(T\). #### Proof: The proof is organized in a two-column format displaying statements and reasons needed to establish the congruence of the angle pairs. | Statement | Reason | |---------------------------------|-----------------------------------------| | 1. \(\triangle QRS \cong \triangle QPT\) | Given | | 2. \(PT = RS\) | CPCTC (Corresponding Parts of Congruent Triangles are Congruent) | | 3. \(PQ = QR\) | CPCTC | | 4. \(QT = QS\) | CPCTC | | 5. \(PS = PQ + QS\) | Additive Property of Length | | 6. \(RT = QR + QT\) | Additive Property of Length | | 7. \(PS = QR + QT\) | Substitution | | 8. \(PS = RT\) | Transitive Property of Equality | | 9. \(ST = ST\) | Reflexive Property of Congruence | | 10. \(\triangle PST \cong \triangle RTS\) | SAS (Side-Angle-Side Congruence Postulate) | | 11. \(\angle PTS \cong \angle RST\) | CPCTC | This organized approach utilizes the properties of congruent triangles, the Additive Property of Length, Substitution, Transitive and Reflexive Properties, and the Side-Angle-Side Congruence Postulate to deduce the congruence of angles \(\angle PTS\) and \(\angle RST\).
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