Use the image to answer the question.

Question

A quadrilateral upper A upper B upper C upper D. Two diagonals connect opposing vertices upper A to upper C and upper B to upper D. The point of intersection is upper O.

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)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
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
Response area
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°
Response area
10. 2m∠BAD=180°
Combine like terms
11. Response area 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

responses: substitution property of equality, transitive property of congrunce, CPCTC theorem of congruence,

Answer 1

Here’s how to complete the two-column proof that parallelogram ABCD with congruent diagonals \( \overline{DB} \) and \( \overline{AC} \) is a rectangle:

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

You would place the responses in the correct areas as indicated above.