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Prove that triangle AOT is congruent to triangle ABE by completing the two-column proof. T E Given: TO a BE, LTOA E LABE Prove: AAOT AABE Statement Reason 1. TO BE Given 2. TO BE Definition of congruency 3. DE = BE +OB Segment Addition Postulate TB = OB + TO 4. DE = OB + TO Substitution TB = OB + TO 5. TB = OE Transitive Property of Equality 6. TB DE Definition of congruence 7.LTOA = LABE Given 8. ZTOA = LABE Definition of congruent angles 9.LTOA and LAOB are supplementary. Linear pairs of angles are supplementary. ABE and LAB0 are supplementary. 10. LTOA + LAOB = 180° Definition of supplementary LABE + LABO = 180° 11. LTOA + LAOB = 180° Substitution TOA + LABO = 180° 12. LTOA + LABO = LTOA + LAOB Transitive Property of Equality 13. 14. 15. 16. 17. Use the paperclip button below to attach files.

Prove that triangle AOT is congruent to triangle ABE by completing the two-column proof. T E Given: TO a BE, LTOA E LABE Prove: AAOT AABE Statement Reason 1. TO BE Given 2. TO BE Definition of congruency 3. DE = BE +OB Segment Addition Postulate TB = OB + TO 4. DE = OB + TO Substitution TB = OB + TO 5. TB = OE Transitive Property of Equality 6. TB DE Definition of congruence 7.LTOA = LABE Given 8. ZTOA = LABE Definition of congruent angles 9.LTOA and LAOB are supplementary. Linear pairs of angles are supplementary. ABE and LAB0 are supplementary. 10. LTOA + LAOB = 180° Definition of supplementary LABE + LABO = 180° 11. LTOA + LAOB = 180° Substitution TOA + LABO = 180° 12. LTOA + LABO = LTOA + LAOB Transitive Property of Equality 13. 14. 15. 16. 17. Use the paperclip button below to attach files.

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 Triangle Congruence** Prove that triangle \( \triangle AOT \) is congruent to triangle \( \triangle ABE \) by completing the two-column proof. **Diagram:** The diagram shows two triangles, \( \triangle AOT \) and \( \triangle ABE \), with a shared vertex at \( A \). The other points are marked \( T, O, E, \) and \( B \). **Given:** - \( TO \cong BE \) - \( \angle TOA \cong \angle ABE \) **To Prove:** - \( \triangle AOT \cong \triangle ABE \) **Two-Column Proof:** | Statement | Reason | |------------------------------------------------------|------------------------------------------------| | 1. \( TO \cong BE \) | Given | | 2. \( TO = BE \) | Definition of congruency | | 3. \( OE = BE + \overline{OB} \) | Segment Addition Postulate | | \( TB = \overline{OB} + TO \) | | | 4. \( OE = \overline{OB} + TO \) | Substitution | | \( TB = \overline{OB} + TO \) | | | 5. \( TB = OE \) | Transitive Property of Equality | | 6. \( TO \cong BE \) | Definition of congruence | | 7. \( \angle TOA \cong \angle ABE \) | Given | | 8. \( \angle TOA = \angle ABE \) | Definition of congruent angles | | 9. \( \angle TOA \) and \( \angle AOB \) are | Linear pairs of angles are | | supplementary. | supplementary. | | \( \angle ABE \) and \( \angle \AOB \) are | | | supplemental. | | | 10. \( \angle TOA + \angle AOB = 180^\circ \) | Definition of supplementary | | \( \angle ABE + \angle AOB = 180^\circ \) | | | 11. \( \angle TOA + \angle
Transcribed Image Text:**Proof of Triangle Congruence** Prove that triangle \( \triangle AOT \) is congruent to triangle \( \triangle ABE \) by completing the two-column proof. **Diagram:** The diagram shows two triangles, \( \triangle AOT \) and \( \triangle ABE \), with a shared vertex at \( A \). The other points are marked \( T, O, E, \) and \( B \). **Given:** - \( TO \cong BE \) - \( \angle TOA \cong \angle ABE \) **To Prove:** - \( \triangle AOT \cong \triangle ABE \) **Two-Column Proof:** | Statement | Reason | |------------------------------------------------------|------------------------------------------------| | 1. \( TO \cong BE \) | Given | | 2. \( TO = BE \) | Definition of congruency | | 3. \( OE = BE + \overline{OB} \) | Segment Addition Postulate | | \( TB = \overline{OB} + TO \) | | | 4. \( OE = \overline{OB} + TO \) | Substitution | | \( TB = \overline{OB} + TO \) | | | 5. \( TB = OE \) | Transitive Property of Equality | | 6. \( TO \cong BE \) | Definition of congruence | | 7. \( \angle TOA \cong \angle ABE \) | Given | | 8. \( \angle TOA = \angle ABE \) | Definition of congruent angles | | 9. \( \angle TOA \) and \( \angle AOB \) are | Linear pairs of angles are | | supplementary. | supplementary. | | \( \angle ABE \) and \( \angle \AOB \) are | | | supplemental. | | | 10. \( \angle TOA + \angle AOB = 180^\circ \) | Definition of supplementary | | \( \angle ABE + \angle AOB = 180^\circ \) | | | 11. \( \angle TOA + \angle
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