5 Ways To Prove A Quadrilateral Is A Parallelogram - Math Monks

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This article will help us learn how to prove something is a parallelogram. More precisely, how to prove a quadrilateral is a parallelogram.

There are 5 basic ways to prove a quadrilateral is a parallelogram. They are as follows:

  1. Proving opposite sides are congruent
  2. Proving opposite sides are parallel
  3. Proving the quadrilateral’s diagonals bisect each other
  4. Proving opposite angles are congruent
  5. Proving consecutive angles are supplementary (adding to 180°)

Let us now prove the above statements one by one.

Parallelogram Proof 1

1) Proving Opposite Sides are Congruent

Prove that opposite sides of a parallelogram are congruent

To prove: AB ≅ CD, AD ≅ BC

Proof:

Given: AB ∥ CD, AD ∥ BC

Draw in a diagonal AC

StatementReason
∠ACB ≅ ∠CADAlternate interior angles
∠BAC ≅ ∠ACDAlternate interior angles
AC ≅ ACCommon side (identity)
ΔADC ≅ ΔABCAngle-Angle-Side (AAS) postulate
AD ≅ BC AB ≅ CDCorresponding Parts of Congruent Triangles are Congruent (CPCTC) Hence Proved

2) Proving Opposite Sides are Parallel

Prove that opposite sides of a parallelogram are parallel

To prove: AB ∥ CD, AD ∥ BC

Proof:

Given: AB ≅ CD, AD ≅ BC

Draw in a diagonal AC

StatementReason
AC ≅ ACCommon side (identity)
ΔADC ≅ ΔABCSSS postulate
∠ACB ≅ ∠CADCPCTC
∠BAC ≅ ∠ACDCPCTC
AB ∥ CDAlternate interior angles Hence Proved
AD ∥ BCAlternate interior angles Hence Proved

3) Proving Diagonals Bisect Each Other

Prove that the diagonals of a parallelogram bisect each other.

This is an “if and only if” proof, so there are two things to prove.

1. To prove: AE ≅ EC, BE ≅ ED

Proof:

Given: ABCD is a parallelogram

StatementReason
AD ∥ BCDefinition of parallelogram
∠DAE ≅ ∠BCEAlternate interior angles
ADE ≅ CBEAlternate interior angles
AD ≅ BCOpposite sides of parallelogram
ΔADC ≅ ΔABCAAS postulate
AE ≅ ECCPCTC Hence Proved
BE ≅ EDCPCTC Hence Proved

And the converse

2. To prove: ABCD is a parallelogram

Proof:

Given: AE ≅ EC, BE ≅ ED

StatementReason
∠AED ≅ ∠BECVertical angles
∠AEB ≅ ∠CEDVertical angles
ΔADC ≅ ΔABCAAS
AD ≅ BCCPCTC
AB ≅ CDCPCTC
ABCD is a parallelogramTheorem 2 Hence Proved

4) Proving Opposite Angles are Congruent

Prove that the opposite angles of a parallelogram are congruent.

To prove: ∠ADC ≅ ∠ABC, ∠BAD ≅ ∠BCD

Proof:

Given: AB ∥ CD, AD ∥ BC

Draw in a diagonal AC

StatementReason
∠ACB ≅ ∠CADAlternate interior angles
∠ADB ≅ ∠CBDAlternate interior angles
∠AED ≅ ∠BECVertical angles
∠AEB ≅ ∠CEDVertical angles
ΔADC ≅ ΔABCAAS postulate
ΔBAD ≅ ΔBCDAAS postulate
∠ADC ≅ ∠ABCCPCTC postulate Hence Proved
∠BAD ≅ ∠BCDCPCTC postulate Hence Proved

5) Proving Consecutive Angles are Supplementary

Prove that the consecutive angles of a parallelogram are supplementary (add up to 180°).

To prove: ∠ABC + ∠BCD = 180°

Proof:

Given: AB ∥ CD, AD ∥ BC

Extend BC till P

StatementReason
∠ABC ≅ ∠DCPCorresponding angle
∠BCD + ∠DCP = 180°Linear angle
∠ABC + ∠BCD = 180°∠ABC ≅ ∠DCP Hence Proved

Tag » How To Prove A Parallelogram