Question

Use the image to answer the 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
Two-Column Proof
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
the reason that justifies the statement angle upper B upper A upper D is congruent to angle upper A upper B upper C
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°
the reason that justifies the statement the measure of angle upper B upper A upper D plus the measure of angle upper B upper A upper D equals 180 degrees
10. 2m∠BAD=180°
Combine like terms.
11. the statement that aligns to the Division Property of Equality 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 for showing that parallelogram ABCD with congruent diagonals \( \overline{DB} \) and \( \overline{AC} \) is a rectangle. The statements and reasons have been matched appropriately:

| 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. \( \angle BAD \cong \angle ABC \) | Corresponding parts of congruent triangles are congruent. |
| 7. \( m\angle BAD = m\angle ABC \) | Definition of congruence |
| 8. \( m\angle BAD + m\angle ABC = 180^\circ \) | Consecutive angles of a parallelogram are supplementary. |
| 9. \( m\angle BAD + m\angle BAD = 180^\circ \) | Substituting \( m\angle ABC \) with \( m\angle BAD \) from step 6. |
| 10. \( 2m\angle BAD = 180^\circ \) | Combine like terms. |
| 11. \( m\angle BAD = 90^\circ \) | Division Property of Equality |
| 12. \( m\angle ABC = 90^\circ \) | Transitive Property of Equality |
| 13. \( m\angle ADC = m\angle DAB = 90^\circ \) | 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 can complete the proof by dragging and dropping each statement and reason to their respective locations in the two columns.