In this proof, we are given that lines a and b are parallel as well as lines e and f are parallel. This means that the corresponding angles of these parallel lines are congruent. Therefore, ∠a is equal to ∠b and ∠e is equal to ∠f.
We also know that the sum of the opposite angles of a parallelogram is 180°. This means that ∠a + ∠e is equal to 180° and ∠b + ∠f is also equal to 180°. By substitution, we can then say that ∠a + ∠e is equal to ∠b + ∠f.
Furthermore, we know that vertical angles are congruent. This means that the angles formed when two lines intersect are equal in measure. Therefore, we can say that ∠a is equal to ∠f and ∠b is equal to ∠e.
From this information, we can conclude that side ab is congruent to side ef because they share the same angle measures. Therefore, we have proven that opposite sides of a parallelogram are congruent.
Given:
𝑎
∥
𝑏
a∥b and
𝑒
∥
𝑓
e∥f Given
∠a = ∠b Corresponding angles of parallel lines
∠e = ∠f Corresponding angles of parallel lines
∠a + ∠e = 180° Opposite angles of a parallelogram
∠b + ∠f = 180° Opposite angles of a parallelogram
∠a + ∠e = ∠b + ∠f Substitution
𝑎
𝑏
‾
ab
≅
𝑒
𝑓
‾
ef
Vertical angles are congruent
This proof demonstrates that opposite sides of the parallelogram are congruent because they share the same measure due to corresponding angles formed by parallel lines.
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