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Theorems about
Parallelograms

Master the essential properties of parallelograms: opposite sides, opposite angles, consecutive angles, and diagonals that bisect each other.

Geometry

What is a Parallelogram?

A parallelogram is a quadrilateral with both pairs of opposite sides parallel. This simple definition leads to a rich set of properties that we can prove and use to solve problems.

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

Why Parallelograms Matter

Parallelograms are one of the most fundamental shapes in geometry. Understanding their properties helps us prove congruence, solve for unknown angles and lengths, and build toward more advanced topics like vectors and coordinate geometry.

Architecture
Structural beams and trusses use parallelogram properties for stability.
Engineering
Linkage mechanisms (like scissor lifts) rely on parallelogram geometry.
Proofs
Many advanced proofs about rectangles, rhombi, and squares build on these theorems.

Key Vocabulary

Opposite Sides
Sides that do not share a vertex. In \(\square ABCD\): \(\overline{AB}\) and \(\overline{DC}\), \(\overline{AD}\) and \(\overline{BC}\).
Opposite Angles
Angles at vertices that are not adjacent. In \(\square ABCD\): \(\angle A\) and \(\angle C\), \(\angle B\) and \(\angle D\).
Consecutive Angles
Angles that share a side. \(\angle A\) and \(\angle B\) are consecutive since they share \(\overline{AB}\).
Diagonals
Segments connecting opposite vertices: \(\overline{AC}\) and \(\overline{BD}\).

Theorem 1: Opposite Sides are Congruent

\(A\)
\(B\)
\(C\)
\(D\)
\( \overline{AB} \cong \overline{DC} \)
\( \overline{AD} \cong \overline{BC} \)
 diagonal AC

Proof sketch: Draw diagonal \(\overline{AC}\). By the Alternate Interior Angles Theorem (since \(\overline{AB} \parallel \overline{DC}\) and \(\overline{AD} \parallel \overline{BC}\)), triangles \(\triangle ABC\) and \(\triangle CDA\) are congruent by ASA. Therefore their corresponding sides are congruent.

Theorem 2: Opposite Angles are Congruent

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

This follows directly from the congruent triangles formed by a diagonal. Since \(\triangle ABC \cong \triangle CDA\), we get \(\angle B \cong \angle D\). Drawing the other diagonal proves \(\angle A \cong \angle C\).

Theorem 3: Consecutive Angles are Supplementary

\(A\)
\(B\)
\(C\)
\(D\)
\(110^\circ\)
\(70^\circ\)
\(110^\circ + 70^\circ = 180^\circ\) ✓

Since \(\overline{AD} \parallel \overline{BC}\), angles \(\angle A\) and \(\angle B\) are co-interior angles (same-side interior angles) formed by transversal \(\overline{AB}\). Co-interior angles between parallel lines always sum to \(180^\circ\).

Theorem 4: Diagonals Bisect Each Other

\(A\)
\(B\)
\(C\)
\(D\)
\(M\)
\(AM\)
\(MC\)
\(BM\)
\(MD\)
\( AM = MC \)
\( BM = MD \)

Proof sketch: In \(\triangle AMB\) and \(\triangle CMD\), alternate interior angles give two angle pairs, and \(\overline{AB} \cong \overline{CD}\) (Theorem 1). By ASA, the triangles are congruent, so \(AM = MC\) and \(BM = MD\).

The Four Key Properties

1
Opposite sides are congruent: \(\overline{AB} \cong \overline{DC}\), \(\overline{AD} \cong \overline{BC}\).
2
Opposite angles are congruent: \(\angle A \cong \angle C\), \(\angle B \cong \angle D\).
3
Consecutive angles are supplementary: \(\angle A + \angle B = 180^\circ\).
4
Diagonals bisect each other: each diagonal is cut into two equal parts at the intersection.

Parallelogram vs. Special Types

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

How to Apply These Theorems

1
Identify the parallelogram. Confirm the shape is a parallelogram (given or proven).
2
Determine what is unknown. Are you looking for a side length, angle measure, or segment?
3
Select the right theorem. Match the unknown to the property: opposite sides, opposite angles, consecutive angles, or diagonals.
4
Set up an equation. Use the property to write an equation (e.g., \(\angle A = \angle C\) or \(AM = MC\)).
5
Solve and verify. Solve for the unknown, then check that all properties remain consistent.

Example 1: Finding Unknown Angles

In \(\square PQRS\), \(\angle P = 65^\circ\). Find \(\angle Q\), \(\angle R\), and \(\angle S\).

\(P\)
\(Q\)
\(R\)
\(S\)
\(65^\circ\)
\(?\)
\(?\)
\(?\)

Example 2: Finding Side Lengths

In \(\square ABCD\), \(AB = 3x + 5\) and \(DC = 7x - 11\). Find the length of \(\overline{AB}\).

\(A\)
\(B\)
\(C\)
\(D\)
\(3x + 5\)
\(7x - 11\)

Example 3: Diagonal Bisection

In \(\square WXYZ\), the diagonals meet at \(M\). If \(WM = 2a + 3\) and \(MY = 5a - 9\), find \(WY\).

\(W\)
\(X\)
\(Y\)
\(Z\)
\(M\)
\(2a + 3\)
\(5a - 9\)

Common Mistakes to Avoid

✕ Assuming Diagonals are Congruent
Diagonals bisect each other, but are not necessarily equal in length (only true for rectangles).
✕ Confusing Consecutive with Opposite
Opposite angles are congruent. Consecutive angles are supplementary. Mixing these up reverses the equation.
✕ Assuming All Sides are Equal
Only opposite sides must be congruent. All four sides being equal makes it a rhombus.

Question 1 of 5

In parallelogram \(JKLM\), \(\angle J = 72^\circ\). What is \(\angle K\)?

Question 2 of 5

In \(\square ABCD\), \(AB = 5x - 2\) and \(DC = 3x + 10\). What is the value of \(x\)?

Question 3 of 5

The diagonals of parallelogram \(PQRS\) intersect at point \(T\). If \(PT = 9\), what is \(PR\)?

Question 4 of 5

Which of the following is NOT always true for a parallelogram?

Question 5 of 5

In \(\square EFGH\), \(\angle E = (4x + 10)^\circ\) and \(\angle G = (6x - 20)^\circ\). What is \(\angle E\)?