To complete the proof that quadrilateral ABCD is a parallelogram, we can fill in the blank spaces as follows:
| 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: angle D A C is congruent to angle B C A | Reason: Alternate Interior Angles Theorem | | Blank space 2: angle D A C + angle D C A = 180 degrees | Reason: Linear Pair (if lines intersect, adjacent angles are supplementary) | | Blank space 3: angle B C A + angle A B D = 180 degrees | Reason: Linear Pair (if lines intersect, adjacent angles are supplementary) | | angle D A C is congruent to angle B C A, angle D C A is congruent to angle B A C | Reason: Given (from previous congruences) | | Blank space 4: angle A D C is congruent to angle B C A | Reason: Corresponding Angles Postulate | | Blank space 5: A B C D is a parallelogram | Reason: Definition of parallelogram |
With these completed blanks, the proof is structured to show that because pairs of alternate interior angles are congruent and both pairs of opposite sides are congruent, ABCD must be a parallelogram.