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Given: Quadrilateral A B C D with segment A D is congruent to segment B C, segment A B is congruent to segment D C

Prove: A B C D is a parallelogram

Quadrilateral A B C D shows diagonal A C.

Question
Move statements and reasons to the table to complete the proof.

Response area with 5 blank spaces
Statements Reasons
Quadrilateral A B C D with segment A D is congruent to segment B C, segment A B is congruent to segment D C Given
Blank space 1 empty

Reflexive Property

Blank space 2 empty

Blank space 3 empty

angle D A C is congruent to angle B C A, angle D C A is congruent to angle B A C
Blank space 4 empty

Blank space 5 empty

Converse of Alternate Interior Angles Theorem
A B C D is a parallelogram Definition of parallelogram

Answer options with 9 options.

1 answer

To complete the proof that quadrilateral ABCD is a parallelogram, we can fill in the blanks in the proof table using geometric properties and theorems.

Here’s how you can complete the table:

| Statements | Reasons | |-------------------------------------------------------------------|------------------------------------------------------| | Quadrilateral A B C D with segment A D is congruent to segment B C, segment A B is congruent to segment D C | Given | | Angle DAB is congruent to angle BCA | Blank space 1: (We'll fill this in shortly) | | Segment AC is congruent to segment AC | Reflexive Property | | Angle DAC is congruent to angle BCA | Blank space 2: (We'll fill this in shortly) | | Angle DCA is congruent to angle BAC | Blank space 3: (We'll fill this in shortly) | | Angle DAC is congruent to angle BCA and angle DCA is congruent to angle BAC | ACB is an alternate interior angle | | Since angle DAB is congruent to angle BCA and angle DCA is congruent to angle BAC, | Angle properties; we can infer that lines are parallel| | A B C D is a parallelogram | Definition of parallelogram |

Now, we can fill out the previous blank spaces:

Fill in the blanks

  1. Blank space 1: "Angle D A B is congruent to angle B C A"
  2. Blank space 2: "Angle D A C is congruent to angle B A C"
  3. Blank space 3: "By the Alternate Interior Angles Theorem, lines AD and BC are parallel"

This means the filled table looks like the following:

| Statements | Reasons | |-------------------------------------------------------------------|------------------------------------------------------| | Quadrilateral A B C D with segment A D is congruent to segment B C, segment A B is congruent to segment D C | Given | | Angle D A B is congruent to angle B C A | Angle congruence due to congruent sides | | Segment A C is congruent to segment A C | Reflexive Property | | Angle DAC is congruent to angle BCA | Corresponding angles from the congruent triangles | | Angle DCA is congruent to angle BAC | Corresponding angles from the congruent triangles | | By the Alternate Interior Angles Theorem, AD is parallel to BC | As they are equal angles | | Therefore, since AD is parallel to BC and AB is congruent to DC, A B C D is a parallelogram | Definition of parallelogram |

This completed proof demonstrates that quadrilateral ABCD is indeed a parallelogram based on the given congruences and properties of angles.