Question
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.
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.
Answers
GPT-4o mini
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.
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.