Table of Contents
Last modified on August 3rd, 2023
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:
Let us now prove the above statements one by one.
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
Statement | Reason |
---|---|
∠ACB ≅ ∠CAD | Alternate interior angles |
∠BAC ≅ ∠ACD | Alternate interior angles |
AC ≅ AC | Common side (identity) |
ΔADC ≅ ΔABC | Angle-Angle-Side (AAS) postulate |
AD ≅ BC AB ≅ CD | Corresponding Parts of Congruent Triangles are Congruent (CPCTC) Hence Proved |
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
Statement | Reason |
---|---|
AC ≅ AC | Common side (identity) |
ΔADC ≅ ΔABC | SSS postulate |
∠ACB ≅ ∠CAD | CPCTC |
∠BAC ≅ ∠ACD | CPCTC |
AB ∥ CD | Alternate interior angles Hence Proved |
AD ∥ BC | Alternate interior angles Hence Proved |
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
Statement | Reason |
---|---|
AD ∥ BC | Definition of parallelogram |
∠DAE ≅ ∠BCE | Alternate interior angles |
ADE ≅ CBE | Alternate interior angles |
AD ≅ BC | Opposite sides of parallelogram |
ΔADC ≅ ΔABC | AAS postulate |
AE ≅ EC | CPCTC Hence Proved |
BE ≅ ED | CPCTC Hence Proved |
And the converse
2. To prove: ABCD is a parallelogram
Proof:
Given: AE ≅ EC, BE ≅ ED
Statement | Reason |
---|---|
∠AED ≅ ∠BEC | Vertical angles |
∠AEB ≅ ∠CED | Vertical angles |
ΔADC ≅ ΔABC | AAS |
AD ≅ BC | CPCTC |
AB ≅ CD | CPCTC |
ABCD is a parallelogram | Theorem 2 Hence Proved |
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
Statement | Reason |
---|---|
∠ACB ≅ ∠CAD | Alternate interior angles |
∠ADB ≅ ∠CBD | Alternate interior angles |
∠AED ≅ ∠BEC | Vertical angles |
∠AEB ≅ ∠CED | Vertical angles |
ΔADC ≅ ΔABC | AAS postulate |
ΔBAD ≅ ΔBCD | AAS postulate |
∠ADC ≅ ∠ABC | CPCTC postulate Hence Proved |
∠BAD ≅ ∠BCD | CPCTC postulate Hence Proved |
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
Statement | Reason |
---|---|
∠ABC ≅ ∠DCP | Corresponding angle |
∠BCD + ∠DCP = 180° | Linear angle |
∠ABC + ∠BCD = 180° | ∠ABC ≅ ∠DCP Hence Proved |
Last modified on August 3rd, 2023