Properties of Parallelograms

Key Concepts

• Define a parallelogram.
• Find the relation between measures of sides of a parallelogram.
• Find the relation between the angles of a parallelogram.
• Explain other properties of parallelogram.

Parallelogram 

A quadrilateral whose two pairs of opposite sides are parallel is called a parallelogram. 

Theorem 

If a quadrilateral is a Properties of parallelograms, then its opposite sides are congruent. 

Proof: In △DCA and △BAC, 

∠1=∠3

AC=AC [Reflexive property] 

∠2=∠4

Then,

△DCA≅ △BAC by Angle-Side-Angle congruence criterion. 

If two triangles are congruent, their corresponding sides are equal. 

So, AD− = BC−and AB− = CD−.

Theorem 

If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. 

If

AB− ∥ CD−,  AD− ∥ BC−

then

m∠A+m∠B=180°

m∠B+m∠C=180°

m∠C+m∠D=180°

m∠D+m∠A=180°

Proof: The angles ∠A and ∠B are the consecutive interior angles formed when parallel lines

AB− and CD− are intersected by transversal AD−.

We know that when two parallel lines are intersected by a transversal, the consecutive interior angles formed are supplementary. 

So, ∠A+∠B=180°

Theorem 

If a quadrilateral is a parallelogram, then opposite angles are congruent. 

Proof: The angles ∠1 and ∠2 are the consecutive interior angles formed when parallel lines

PQ− and SR− are intersected by transversal PS−.

We know that when two parallel lines are intersected by a transversal, the consecutive interior angles formed are supplementary. 

So, ∠1+∠2=180° …(1) 

The angles ∠2 and ∠3 are the consecutive interior angles formed when parallel lines PS− and QR− are intersected by the transversal SR−

Therefore, ∠2 + ∠3 = 180°    …(2) 

From (1) and (2), we get, 

∠1 + ∠2 = ∠2 + ∠3

∠1 = ∠3

So, opposite angles are equal. 

Theorem 

If a quadrilateral is a parallelogram, then its diagonals bisect each other. 

Proof: In △PXS and △RXQ   

∠1=∠3 [Consecutive interior angles formed when PS- and QR−are intersected by PR-] 

PS=QR [If a quadrilateral is a parallelogram, then its opposite sides are congruent] 

∠2=∠4 [Consecutive interior angles formed when PS- and QR− are intersected by QS−] 

So, △PXS≅ △RXQ by Angle-Side-Angle criterion. 

We know that if two triangles are congruent, their corresponding sides are equal. 

Since △PXS≅ △RXQ then PX=XR and QX=XS

Exercise

• Find the length of BC.

• In parallelogram WXYZ, find the measure of

• Find the measure of

• Find the perimeter of the parallelogram.

• What is the angle measure of the point at the bottom?

Concept Map

What we have learned

• A quadrilateral whose two pairs of opposite sides are parallel is called a parallelogram.
• If a quadrilateral is a parallelogram, then its opposite sides are congruent.
• If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.
• If a quadrilateral is a parallelogram, then opposite angles are congruent.
• If a quadrilateral is a parallelogram, then its diagonals bisect each other.