Prove that the opposite sides and angles
of a parallelogram are equal.
Answers
Opposite Sides of a Parallelogram are Equal
In a parallelogram, opposite sides are equal.
Conversely, if the opposite sides in a quadrilateral are equal, then it is a parallelogram.
Consider the following figure:
Parallelogram theorems - In a parallelogram ABCD, opposite sides are equal.
Proof:
We will assume that
A
B
C
D
ABCD is a parallelogram.
Compare
Δ
A
B
C
ΔABC and
Δ
C
D
A
ΔCDA:
A
C
=
C
A
(
common sides
)
∠
1
=
∠
4
(
alternate
interior
angles
)
∠
2
=
∠
3
(
alternate
interior
angles
)
AC=CA(common sides)∠1=∠4(alternate interior angles)∠2=∠3(alternate interior angles)
Thus, by the ASA criterion, the two triangles are congruent, which means that the corresponding sides must be equal.
Thus,
A
B
=
C
D
and
A
D
=
B
C
AB=CDandAD=BC
Now, we will prove the converse of this.
Converse of the Theorem
If the opposite sides in a quadrilateral are equal, then it is a parallelogram.
Assume that
A
B
C
D
ABCD is a quadrilateral in which
A
B
=
C
D
AB=CD and
A
D
=
B
C
AD=BC,
Compare
Δ
A
B
C
ΔABC and
Δ
C
D
A
ΔCDA once again:
A
C
=
A
C
(
common sides
)
A
B
=
C
D
(
since alternate interior angles are equal
)
A
D
=
B
C
(
given
)
AC=AC(common sides)AB=CD(since alternate interior angles are equal )AD=BC(given)
A
C
=
A
C
(
common sides
)
A
B
=
C
D
(
since alternate interior
angles are equal
)
A
D
=
B
C
(
given
)
AC=AC(common sides)AB=CD(since alternate interiorangles are equal )AD=BC(given)
Thus, by the SSS criterion, the two triangles are congruent, which means that the corresponding angles are equal:
∠
1
=
∠
4
⇒
A
B
∥
C
D
∠
2
=
∠
3
⇒
A
D
∥
B
C
∠1=∠4⇒AB∥CD ∠2=∠3⇒AD∥BC
Hence,
A
B
∥
C
D
and
A
D
∥
B
C
AB∥CDandAD∥BC
Thus,
A
B
C
D
ABCD is a parallelogram.
Opposite Angles of a Parallelogram are Equal
In a parallelogram, opposite angles are equal.
Conversely, if the opposite angles in a quadrilateral are equal, then it is a parallelogram.
Consider the following figure:
Parallelogram theorems - In a parallelogram ABCD, opposite angles are equal.
Proof:
First, we assume that
A
B
C
D
ABCD is a parallelogram.
Compare
Δ
A
B
C
ΔABC and
Δ
C
D
A
ΔCDA once again:
A
C
=
A
C
(
common sides
)
∠
1
=
∠
4
(
alternate interior angles
)
∠
2
=
∠
3
(
alternate interior angles
)
AC=AC(common sides)∠1=∠4(alternate interior angles)∠2=∠3(alternate interior angles)
Thus, the two triangles are congruent, which means that
∠
B
=
∠
D
∠B=∠D
Similarly, we can show that
∠
A
=
∠
C
∠A=∠C
This proves that opposite angles in any parallelogram are equal.
Now, we prove the converse of this.
Converse of the Theorem
If the opposite angles in a quadrilateral are equal, then it is a parallelogram.
Assume that
∠
A
∠A =
∠
C
∠C and
∠
B
∠B =
∠
D
∠D.
We have to prove that
A
B
C
D
ABCD is a parallelogram.
Consider the following figure:
Parallelogram - If opposite angles in a quadrilateral are equal, it is a parallelogram.
We have:
∠
A
+
∠
B
+
∠
C
+
∠
D
=
360
∘
2
(
∠
A
+
∠
B
)
=
360
∘
∠
A
+
∠
B
=
180
∘
∠A+∠B+∠C+∠D=360∘2(∠A+∠B)=360∘∠A+∠B=180∘
This must mean that
A
D
∥
B
C
AD∥BC
Similarly, we can show that
A
B
∥
C
D
AB∥CD
Hence,
A
D
∥
B
C
and
A
B
∥
C
D
AD∥BCandAB∥CD
and thus,
A
B
C
D
ABCD is a parallelogram.
Note that the relation between two lines intersected by a transversal, when the angles on the same side of the transversal are supplementary, are parallel to each other.
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