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Given: 41 24 Ris the midpoint of PT Prove: 2Q L 1/2 X P

Given: 41 24 Ris the midpoint of PT Prove: 2Q L 1/2 X P

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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**Transcription for Educational Website** ### Geometry Proof Exercise **Given:** \(\angle 1 \cong \angle 4\) \(R\) is the midpoint of \(PT\) **Prove:** \(\angle Q \cong \angle S\) #### Diagram Explanation: The accompanying diagram displays a line segment \(PT\) with point \(R\) as its midpoint. From \(P\), line \(PQ\) extends upwards to point \(Q\), forming angle \(1\) with the line \(PX\) which is parallel to \(PT\). Similarly, from \(T\), line \(TS\) extends upwards to point \(S\), forming angle \(3\) with the line \(TY\). Additionally, angles \(2\) and \(4\) are formed with lines \(PR\) and \(TR\), respectively. #### Two-Column Proof | **Statement** | **Reason** | |---------------------------------|-----------------------------| | 1) | 1) | | 2) | 2) | | 3) | 3) | | 4) | 4) | | 5) | 5) | | 6) | 6) | | 7) | 7) | Use the given information and the diagram to complete the proof of \(\angle Q \cong \angle S\). Apply the properties of congruent angles, midpoints, and logical deductions accordingly.
Transcribed Image Text:**Transcription for Educational Website** ### Geometry Proof Exercise **Given:** \(\angle 1 \cong \angle 4\) \(R\) is the midpoint of \(PT\) **Prove:** \(\angle Q \cong \angle S\) #### Diagram Explanation: The accompanying diagram displays a line segment \(PT\) with point \(R\) as its midpoint. From \(P\), line \(PQ\) extends upwards to point \(Q\), forming angle \(1\) with the line \(PX\) which is parallel to \(PT\). Similarly, from \(T\), line \(TS\) extends upwards to point \(S\), forming angle \(3\) with the line \(TY\). Additionally, angles \(2\) and \(4\) are formed with lines \(PR\) and \(TR\), respectively. #### Two-Column Proof | **Statement** | **Reason** | |---------------------------------|-----------------------------| | 1) | 1) | | 2) | 2) | | 3) | 3) | | 4) | 4) | | 5) | 5) | | 6) | 6) | | 7) | 7) | Use the given information and the diagram to complete the proof of \(\angle Q \cong \angle S\). Apply the properties of congruent angles, midpoints, and logical deductions accordingly.
This image contains a list of geometric statements and theorems useful for proofs. Below is the transcription of each item in the image: 1. ∠1 is supplementary to ∠2 2. ∠2 ≅ ∠3 3. ∠3 is supplementary to ∠4 4. ASA (Angle-Side-Angle) 5. Congruent Supplements Theorem 6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent) 7. Definition of midpoint 8. Definition of straight angle 9. Definition of supplementary angles 10. Definition of vertical angles 11. Given 12. Given 13. \( \overline{PR} \cong \overline{RT} \) 14. △PRQ ≅ △RST 15. △PRQ ≅ △SRT 16. ∠PRQ ≅ ∠TRS 17. ∠Q ≅ ∠S 18. SAS (Side-Angle-Side) 19. SSS (Side-Side-Side) 20. Vertical Angle Theorem This list can be used to understand the relationships between angles and sides in geometric figures, as well as to form logical proofs using these theorems and definitions.
Transcribed Image Text:This image contains a list of geometric statements and theorems useful for proofs. Below is the transcription of each item in the image: 1. ∠1 is supplementary to ∠2 2. ∠2 ≅ ∠3 3. ∠3 is supplementary to ∠4 4. ASA (Angle-Side-Angle) 5. Congruent Supplements Theorem 6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent) 7. Definition of midpoint 8. Definition of straight angle 9. Definition of supplementary angles 10. Definition of vertical angles 11. Given 12. Given 13. \( \overline{PR} \cong \overline{RT} \) 14. △PRQ ≅ △RST 15. △PRQ ≅ △SRT 16. ∠PRQ ≅ ∠TRS 17. ∠Q ≅ ∠S 18. SAS (Side-Angle-Side) 19. SSS (Side-Side-Side) 20. Vertical Angle Theorem This list can be used to understand the relationships between angles and sides in geometric figures, as well as to form logical proofs using these theorems and definitions.
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