DWG NO. 06.2 — Lesson 2 of 5

Properties of Parallelograms

Unit 6: Quadrilaterals and Polygons · ~15–30 min

Objective

Use the properties of parallelograms to find missing side lengths, angle measures, and diagonal segments.

What makes a parallelogram a parallelogram

A parallelogram is a quadrilateral with two pairs of parallel sides. That single condition — both pairs of opposite sides parallel — forces a whole chain of other relationships inside the figure. Once you know a quadrilateral is a parallelogram, all of the following are guaranteed true.

Opposite Sides Theorem: Opposite sides of a parallelogram are congruent.
Opposite Angles Theorem: Opposite angles of a parallelogram are congruent.
Consecutive Angles Theorem: Consecutive angles (angles that share a side) are supplementary — they add to 180°.
Diagonals Theorem: The diagonals of a parallelogram bisect each other, splitting one another into two congruent halves.

Every one of these follows from the parallel sides using alternate interior angles and triangle congruence — they aren't separate assumptions, just consequences worth memorizing because they come up so often.

Diagonals AC and BD cross at E, bisecting each other: AE ≅ EC and DE ≅ EB.

Worked Example 1 · Missing side and angle

ProblemIn parallelogram ABCD, AB = 14 and m∠A = 65°. Find CD and m∠B.

1.Opposite sides are congruent, so CD = AB = 14.

2.∠A and ∠B are consecutive, so they're supplementary: m∠B = 180° − 65° = 115°.

CD = 14, m∠B = 115°

Worked Example 2 · Solving with diagonals

ProblemDiagonals AC and BD of parallelogram ABCD meet at E. If AE = 3x − 1 and EC = 2x + 4, find x and AC.

1.The diagonals bisect each other, so AE = EC: 3x − 1 = 2x + 4.

2.Solving gives x = 5, so AE = 3(5) − 1 = 14 and EC = 14. AC = AE + EC = 28.

x = 5, AC = 28

Guided practice

1.In parallelogram JKLM, JK = 9.5. What is LM?
Show answer
Opposite sides are congruent, so LM = 9.5.
2.In parallelogram WXYZ, m∠W = 78°. Find m∠X and m∠Y.
Show answer
∠X is consecutive to ∠W: m∠X = 180° − 78° = 102°. ∠Y is opposite ∠W, so m∠Y = 78°.
3.Diagonals of parallelogram PQRS meet at T. PT = 4x, TR = x + 18. Find x and PR.
Show answer
PT = TR, so 4x = x + 18, giving 3x = 18, x = 6. PT = 24, TR = 24, so PR = 48.
4.Parallelogram ABCD has m∠A = (2x + 10)° and m∠C = (3x − 20)°. Find x.
Show answer
∠A and ∠C are opposite angles, so they're congruent: 2x + 10 = 3x − 20, giving x = 30.
5.Explain why a parallelogram's consecutive angles must be supplementary, using what you know about parallel lines.
Show answer
Two consecutive angles sit along the same side, which acts as a transversal cutting the pair of parallel sides. That makes them same-side interior angles, and same-side interior angles formed by a transversal across parallel lines are always supplementary.

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