Drag and drop the statements and reasons into their correct locations in the two-column proof that parallelogram ABCD with congruent diagonals DB¯¯¯¯¯¯¯¯ and AC¯¯¯¯¯¯¯¯ is a rectangle. (2 points) Statement Reason 1. ABCD is a parallelogram. Given 2. DB¯¯¯¯¯¯¯¯≅AC¯¯¯¯¯¯¯¯ Given 3. DA¯¯¯¯¯¯¯¯≅CB¯¯¯¯¯¯¯¯ Opposite sides of a parallelogram are congruent. 4. AB¯¯¯¯¯¯¯¯≅BA¯¯¯¯¯¯¯¯ Reflexive Property of Congruence 5. △BAD≅△ABC SSS Congruence Theorem 6. ∠BAD≅∠ABC _____ 7. m∠BAD=m∠ABC Definition of congruence 8. m∠BAD+m∠ABC=180° Consecutive angles of a parallelogram are supplementary 9. m∠BAD+m∠BAD=180° _____ 10. 2m∠BAD=180° Combine like terms 11. _____ Division Property of Equality 12. m∠ABC=90° Transitive Property of Equality 13. m∠ADC=m∠DAB=90° Opposite angles of a parallelogram are congruent 14. ∠BAD , ∠DCB , ∠ABC , and ∠ADC are right angles Definition of a right angle 15. ABCD is a rectangle Definition of a rectangle

Answer 1

Here is the completed two-column proof with the statements and reasons placed correctly:

| Statement | Reason | |------------------------------------------------------------------|-----------------------------------------------------| | 1. ABCD is a parallelogram. | Given | | 2. DB¯¯¯¯¯¯¯¯≅AC¯¯¯¯¯¯¯¯ | Given | | 3. DA¯¯¯¯¯¯¯¯≅CB¯¯¯¯¯¯¯¯ | Opposite sides of a parallelogram are congruent. | | 4. AB¯¯¯¯¯¯¯¯≅BA¯¯¯¯¯¯¯¯ | Reflexive Property of Congruence | | 5. △BAD≅△ABC | SSS Congruence Theorem | | 6. ∠BAD≅∠ABC | Corresponding parts of congruent triangles are congruent. | | 7. m∠BAD=m∠ABC | Definition of congruence | | 8. m∠BAD+m∠ABC=180° | Consecutive angles of a parallelogram are supplementary | | 9. m∠BAD+m∠BAD=180° | Substitute m∠ABC for m∠BAD from step 7 | | 10. 2m∠BAD=180° | Combine like terms | | 11. m∠BAD=90° | Division Property of Equality | | 12. m∠ABC=90° | Transitive Property of Equality | | 13. m∠ADC=m∠DAB=90° | Opposite angles of a parallelogram are congruent | | 14. ∠BAD , ∠DCB , ∠ABC , and ∠ADC are right angles | Definition of a right angle | | 15. ABCD is a rectangle | Definition of a rectangle |

This sequence lists the statements and their corresponding reasons in the proper order for the proof that parallelogram ABCD with congruent diagonals is indeed a rectangle.

Answer 2

Options to use to fill in the blanks are, m<BAD = 90 degrees, substitution property of equality, cpctc theorem of congruence, transitive property of congruence, M<BAD/2 = 90 degrees

Answer 3

Thank you for the clarifications and the options to fill in the blanks. Based on the provided options, here’s the revised two-column proof with the appropriate statements and reasons:

| Statement | Reason | |------------------------------------------------------------------|-----------------------------------------------------| | 1. ABCD is a parallelogram. | Given | | 2. DB¯¯¯¯¯¯¯¯≅AC¯¯¯¯¯¯¯¯ | Given | | 3. DA¯¯¯¯¯¯¯¯≅CB¯¯¯¯¯¯¯¯ | Opposite sides of a parallelogram are congruent. | | 4. AB¯¯¯¯¯¯¯¯≅BA¯¯¯¯¯¯¯¯ | Reflexive Property of Congruence | | 5. △BAD≅△ABC | SSS Congruence Theorem | | 6. ∠BAD≅∠ABC | CPCTC Theorem of Congruence | | 7. m∠BAD=m∠ABC | Definition of congruence | | 8. m∠BAD+m∠ABC=180° | Consecutive angles of a parallelogram are supplementary | | 9. m∠BAD+m∠BAD=180° | Substitution Property of Equality | | 10. 2m∠BAD=180° | Combine like terms | | 11. m∠BAD=90° | Division Property of Equality | | 12. m∠ABC=90° | Transitive Property of Congruence | | 13. m∠ADC=m∠DAB=90° | Opposite angles of a parallelogram are congruent | | 14. ∠BAD , ∠DCB , ∠ABC , and ∠ADC are right angles | Definition of a right angle | | 15. ABCD is a rectangle | Definition of a rectangle |

In this proof:

• For Step 6, we now use "CPCTC Theorem of Congruence" as it provides a basis for saying the angles are congruent because the triangles are congruent.
• For Step 9, we substitute "m∠ABC" for "m∠BAD" using the substitution property of equality.

Please let me know if you need further adjustments or assistance!

Answer 4

There are three steps, 6, 9, and 11

Answer 5

Thank you for your patience! Here’s the corrected proof that matches the three specific steps you referenced (6, 9, and 11) using the options provided:

| Statement | Reason | |------------------------------------------------------------------|-----------------------------------------------------| | 1. ABCD is a parallelogram. | Given | | 2. DB¯¯¯¯¯¯¯¯≅AC¯¯¯¯¯¯¯¯ | Given | | 3. DA¯¯¯¯¯¯¯¯≅CB¯¯¯¯¯¯¯¯ | Opposite sides of a parallelogram are congruent. | | 4. AB¯¯¯¯¯¯¯¯≅BA¯¯¯¯¯¯¯¯ | Reflexive Property of Congruence | | 5. △BAD≅△ABC | SSS Congruence Theorem | | 6. ∠BAD≅∠ABC | CPCTC Theorem of Congruence | | 7. m∠BAD=m∠ABC | Definition of congruence | | 8. m∠BAD+m∠ABC=180° | Consecutive angles of a parallelogram are supplementary | | 9. m∠BAD+m∠BAD=180° | Substitution Property of Equality | | 10. 2m∠BAD=180° | Combine like terms | | 11. m∠BAD=90° | Division Property of Equality | | 12. m∠ABC=90° | Transitive Property of Equality | | 13. m∠ADC=m∠DAB=90° | Opposite angles of a parallelogram are congruent | | 14. ∠BAD , ∠DCB , ∠ABC , and ∠ADC are right angles | Definition of a right angle | | 15. ABCD is a rectangle | Definition of a rectangle |

The specific reasons for the three steps are:

• Step 6: CPCTC Theorem of Congruence (referring to corresponding angles being congruent as part of the congruence of triangles).
• Step 9: Substitution Property of Equality (substituting m∠ABC for m∠BAD).
• Step 11: Division Property of Equality (dividing both sides of the equation by 2).

This layout ensures that the proof is coherent and follows logically. If there are any further modifications you need, please let me know!

Answer 6

That's not one of the options to choose from, for step 11

Answer 7

Thank you for your clarification, and I apologize for the oversight. Based on your specific options, here’s the corrected proof with the appropriate step filled correctly for step 11.

The proof is arranged with the given options:

| Statement | Reason | |------------------------------------------------------------------|-----------------------------------------------------| | 1. ABCD is a parallelogram. | Given | | 2. DB¯¯¯¯¯¯¯¯≅AC¯¯¯¯¯¯¯¯ | Given | | 3. DA¯¯¯¯¯¯¯¯≅CB¯¯¯¯¯¯¯¯ | Opposite sides of a parallelogram are congruent. | | 4. AB¯¯¯¯¯¯¯¯≅BA¯¯¯¯¯¯¯¯ | Reflexive Property of Congruence | | 5. △BAD≅△ABC | SSS Congruence Theorem | | 6. ∠BAD≅∠ABC | CPCTC Theorem of Congruence | | 7. m∠BAD=m∠ABC | Definition of congruence | | 8. m∠BAD+m∠ABC=180° | Consecutive angles of a parallelogram are supplementary | | 9. m∠BAD+m∠BAD=180° | Substitution Property of Equality | | 10. 2m∠BAD=180° | Combine like terms | | 11. m∠BAD=90° | M∠BAD/2 = 90° | | 12. m∠ABC=90° | Transitive Property of Equality | | 13. m∠ADC=m∠DAB=90° | Opposite angles of a parallelogram are congruent | | 14. ∠BAD, ∠DCB, ∠ABC, and ∠ADC are right angles | Definition of a right angle | | 15. ABCD is a rectangle | Definition of a rectangle |

Summary of the Options Used

• Step 6: CPCTC Theorem of Congruence (corresponding angles are congruent as a result of the congruent triangles).
• Step 9: Substitution Property of Equality (substituting m∠ABC for m∠BAD).
• Step 11: \( m∠BAD/2 = 90° \) (from dividing both sides of the equation by 2).

Thank you for your understanding, and I hope this updated proof satisfies your requirements! If you need further changes or have more questions, feel free to ask!