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Parallelogram
Theorems

Prove that opposite sides and angles are congruent, diagonals bisect each other, and rectangles are parallelograms with congruent diagonals.

Geometry

What Is a Parallelogram?

A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. This single defining property gives rise to a rich collection of theorems about sides, angles, and diagonals.

A B C D \(AB \parallel DC\) \(AD \parallel BC\)

Why Parallelograms Matter

Parallelograms are everywhere: from the structural trusses in bridges and rooftops to the linkage mechanisms in scissors and car jacks. Understanding their theorems lets you prove congruence, find missing measurements, and classify special quadrilaterals.

Architecture
Steel beams form parallelogram braces that distribute loads evenly across structures.
Engineering
Pantograph linkages use parallelogram geometry to copy, enlarge, or reduce drawings.
Proof Skills
Parallelogram theorems build fluency with two-column and paragraph proofs for all quadrilaterals.

The Four Core Theorems

1
Opposite sides are congruent: \(\overline{AB} \cong \overline{DC}\) and \(\overline{AD} \cong \overline{BC}\).
2
Opposite angles are congruent: \(\angle A \cong \angle C\) and \(\angle B \cong \angle D\).
3
Diagonals bisect each other: If diagonals \(\overline{AC}\) and \(\overline{BD}\) meet at \(M\), then \(AM = MC\) and \(BM = MD\).
4
Rectangle diagonal theorem: A rectangle is a parallelogram whose diagonals are congruent: \(\overline{AC} \cong \overline{BD}\).

Theorem 1: Opposite Sides Are Congruent

A B C D \(\triangle ABC\) \(\triangle CDA\)

Theorem 2: Opposite Angles Are Congruent

\(\alpha\) \(\alpha\) \(\beta\) \(\beta\) A B C D

Theorem 3: Diagonals Bisect Each Other

M A B C D

Theorem 4: Rectangles Have Congruent Diagonals

≅ A B C D \(\overline{AC}\) \(\overline{BD}\)

Parallelogram vs. Rectangle vs. Rhombus

Property Parallelogram Rectangle Rhombus
Opposite sides \(\parallel\) ✓ ✓ ✓
Opposite sides \(\cong\) ✓ ✓ ✓
All angles \( = 90^\circ\) ✕ ✓ ✕
All sides \(\cong\) ✕ ✕ ✓
Diagonals \(\cong\) ✕ ✓ ✕
Diagonals \(\perp\) ✕ ✕ ✓
Diagonals bisect each other ✓ ✓ ✓

Converses: Proving It's a Parallelogram

Each theorem has a converse: any one of these conditions is enough to prove a quadrilateral is a parallelogram.

1
Both pairs of opposite sides congruent \(\Rightarrow\) parallelogram.
2
Both pairs of opposite angles congruent \(\Rightarrow\) parallelogram.
3
Diagonals bisect each other \(\Rightarrow\) parallelogram.
4
One pair of opposite sides both parallel and congruent \(\Rightarrow\) parallelogram.

How to Apply These Theorems

1
Identify the shape. Confirm it is a parallelogram (or prove it using a converse).
2
Select the right theorem. Need sides? Use Theorem 1. Angles? Theorem 2. Diagonals? Theorem 3.
3
Set up equations. Write congruence or bisection relationships as algebraic equations.
4
Solve and verify. Solve for unknowns, then check that the answer makes geometric sense.

Example 1: Finding a Side Length

\(3x + 2\) \(5x - 4\) \(2x + 1\) \(y\) A B C D

Example 2: Finding Angles

\(5x-10\) \(2x+50\) A B C D

Example 3: Diagonal Bisection

M \(x+3\) \(2x-1\) \(3y\) \(y+8\) A B C D

Common Mistakes

✕ Assuming Diagonals Are Congruent
In a general parallelogram, diagonals bisect each other but are not necessarily equal. Congruent diagonals only happen in a rectangle.
✕ Confusing Consecutive with Opposite
Opposite angles are congruent, but consecutive angles (sharing a side) are supplementary. Setting consecutive angles equal is a common error.
✕ Using One Condition Alone
Having only one pair of opposite sides congruent does not prove a parallelogram. You need both pairs, or one pair that is both parallel and congruent.

Question 1 of 5

In parallelogram \(PQRS\), \(\angle P = 72^\circ\). What is \(\angle Q\)?

Question 2 of 5

The diagonals of parallelogram \(ABCD\) intersect at \(M\). If \(AM = 7\), what is \(AC\)?

Question 3 of 5

Which additional property distinguishes a rectangle from a general parallelogram?

Question 4 of 5

In parallelogram \(WXYZ\), \(WX = 4a - 1\) and \(ZY = 2a + 7\). What is the length of \(\overline{WX}\)?

Question 5 of 5

A quadrilateral has diagonals that bisect each other. Which conclusion is always valid?