There are five ways to confirm that a quadrilateral is a parallelogram.

Question

If both pairs of opposite sides are parallel, then the quadrilateral is a parallelogram.

If both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram.

If both pairs of opposite angles are congruent, then the quadrilateral is a parallelogram.

If the diagonals bisect each other, then the quadrilateral is a parallelogram.

If one pair of sides is both congruent and parallel, then the quadrilateral is a parallelogram.

(6 points)
Use the diagram to complete the following statements. ​When inputting answers use the proper math notation by selecting the Geometry keypad ( ∠ ); located above the number pad, to indicate that segments, angles, and triangles are used.

If AB¯¯¯¯¯¯¯¯≅

, and BC¯¯¯¯¯¯¯¯≅

, then ABCD is parallelogram.

If m∠A+m∠B=

, and m∠

+ m∠D=180
, then ABCD is parallelogram.

If ∠A≅​∠

, and ​∠

≅​∠D​
, then ABCD is parallelogram.

If AE¯¯¯¯¯¯¯¯≅

, and BE¯¯¯¯¯¯¯¯≅

, then ABCD is parallelogram.

If BC¯¯¯¯¯¯¯¯≅

, and BC¯¯¯¯¯¯¯¯∥

, then ABCD is parallelogram.

If CD¯¯¯¯¯¯¯¯≅

, and CD¯¯¯¯¯¯¯¯∥

, then ABCD is parallelogram.

Answer 1

To complete the statements based on the properties of a parallelogram, we can fill in the information as follows:

1. If \( AB \cong CD \), and \( BC \cong AD \), then ABCD is a parallelogram.

2. If \( m\angle A + m\angle B = 180^\circ \), and \( m\angle C + m\angle D = 180^\circ \), then ABCD is a parallelogram.

3. If \( \angle A \cong \angle C \), and \( \angle B \cong \angle D \), then ABCD is a parallelogram.

4. If \( AE \cong BE \), and \( DE \cong CE \), then ABCD is a parallelogram (assuming AE and BE are segments that cut the diagonal of the quadrilateral).

5. If \( BC \cong AD \), and \( BC \parallel AD \), then ABCD is a parallelogram.

6. If \( CD \cong AB \), and \( CD \parallel AB \), then ABCD is a parallelogram.

Just replace \( AB \), \( CD \), \( BC \), \( AD \), or their corresponding angle notations with any specified values from your diagram if needed.