In today's lesson, we will show that the opposite sides of a parallelogram are equal to each other.
A parallelogram is defined as a quadrilateral where the two opposite sides are parallel. We will show that in that case, they are also equal to each other.
Problem
ABCD is a parallelogram, AD||BC and AB||DC. Show that AD = BC and that AB = DC.
Strategy
Once again, since we are trying to show line segments are equal, we will use congruent triangles. Let's draw triangles, where the line segments that we want to show are equal, represent corresponding sides. We will do it by drawing one of the diagonals of the parallelogram, as above.
If we can show that ΔABD and ΔCDB are congruent, we'll have what we need.
So, what can we use to show these two triangles are congruent?
We know this is a parallelogram so the two opposite sides are parallel, and the diagonal acts as a transversal line, intersecting both pairs of parallel lines - hinting we should use the Alternate Interior Angles Theorem.
Proof
Here's how you prove that opposite sides of a parallelogram are equal:
(1) ABCD is a parallelogram //Given
(2) AD || BC //From the definition of a parallelogram
(3) ∠ADB ≅ ∠CBD //Alternate Interior Angles Theorem
(4) AB || DC //From the definition of a parallelogram
(5) ∠ABD ≅ ∠CDB //Alternate Interior Angles Theorem
(6) BD= BD // Common side, reflexive property of equality
(7) ΔABD ≅ ΔCDB // (3), (6), (5) Angle-Side-Angle postulate
(8) AD=BC // Corresponding sides in congruent triangles (CPCTC)
(9) AB=DC // Corresponding sides in congruent triangles (CPCTC)