n the diagram, a∥b
and e∥f
, forming a parallelogram. Write a proof that proves that opposite sides of the parallelogram are congruent. You can write a two-column proof or a paragraph proof, but be sure to include a reason for each of your statements.
One possible proof is as follows:
Statement Reason
1. a∥b Given
2. e∥f Given
3. ∠a = ∠e Corresponding angles of parallel lines are congruent (Alternate Interior Angles Theorem)
4. ∠b = ∠f Corresponding angles of parallel lines are congruent (Alternate Interior Angles Theorem)
5. ∠a + ∠b = 180° Definition of a straight angle
6. ∠e + ∠f = 180° Definition of a straight angle
7. ∠a + ∠b = ∠e + ∠f Transitive property of equality (from statements 5 and 6)
8. a + b = e + f Corresponding angles have equal measures (Angle Addition Postulate)
9. a + b = e + f Corresponding angles have equal measures (Angle Addition Postulate)
10. a + b = e + f Transitive property of equality (from statements 8 and 9)
11. a + b − e = f Subtraction property of equality
12. a + b − e = f Subtraction property of equality
13. a + b − e + e = f + e Addition property of equality
14. a + b = f + e Substitution property (from statements 11 and 13)
15. a + b = f + e Substitution property (from statements 11 and 13)
16. a + b = e + f Commutative property of addition
17. a + b = e + f Commutative property of addition
18. a + b = f + e Transitive property of equality (from statements 16 and 17)
19. AB = EF Definition of congruence (the measures of the angles ∠a, ∠b, ∠e, and ∠f are all the same, so their opposite sides have equal lengths)
20. AD = FG Definition of congruence (the measures of the angles ∠a, ∠b, ∠e, and ∠f are all the same, so their opposite sides have equal lengths)
Therefore, opposite sides of the parallelogram are congruent.