Prove:-
The opposite sides of a parallelogram are parallel and equal.
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What are the Properties of a Parallelogram?
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Properties of a parallelogram help us to identify a parallelogram from a given set of figures easily and quickly. Before we learn about the properties of a parallelogram, let us first know about parallelogram. It is a four-sided closed figure with opposite sides are equal and opposites angles are equal. The properties of a parallelogram mainly deal with its sides and angles.
We all know that a parallelogram is a convex polygon with 4 edges and 4 vertices. The opposite sides are equal and parallel; the opposite angles are also equal. Let's learn more about the properties of parallelograms in detail in this lesson.
What are the Properties of a Parallelogram?
A parallelogram is a type of quadrilateral in which the opposite sides are parallel and equal. There are four angles of a parallelogram at the vertices. Understanding the properties of parallelograms helps to easily relate the angles and sides of a parallelogram. Also, the properties are helpful for calculations in problems relating to sides and angles of a parallelogram. The four important properties of a parallelogram are as follows.
Opposites sides of a parallelogram are equal and parallel to each other.
Opposite angles are equal. ∠A= ∠C, and ∠B = ∠D
All the angles of a parallelogram add up to 360o. ∠A + ∠B + ∠C + ∠D = 360o.
The consecutive angles of a parallelogram are supplementary
∠A + ∠B = 180o
∠B + ∠C = 180o
∠C + ∠D = 180o
∠D + ∠A = 180o
Sides and Angles of a Parallelogram
Properties of Diagonal of a Parallelogram
First, we will recall the meaning of a diagonal. Diagonals are line segments that join the opposite vertices. In parallelogram PQRS, PR and QS are the diagonals. The properties of diagonals of a parallelogram are as follows:
Diagonals of a parallelogram bisect each other. OQ =OS and OR = OP
Each diagonal divides the parallelogram into two congruent triangles, so, ΔRSP ≅ ΔPQR and ΔQPS ≅ ΔSRQ.
Parallelogram Law: The sum of the squares of the sides is equal to the sum of the squares of the diagonals. PQ2+QR2+RS2+SP2 = QS2+PR2
Diagonals of a Parallelogram
Theorems on Properties of a Parallelogram
The theorems on properties of a parallelogram are helpful to define the rules for working across the problems on parallelograms. The properties relating to the sides and angles of a parallelogram can all be easily understood and applies to solve various problems. Further, these theorems are also supportive to understand the concepts in other quadrilaterals. Four important theorems relating to the properties of a parallelogram are given below:
Opposite sides of a parallelogram are equal
Opposite angles of a parallelogram are equal
Diagonals of a parallelogram bisect each other
One pair of opposite sides is equal and parallel in a parallelogram
Theorem 1: In a Parallelogram the Opposite Sides Are Equal. This means, in a parallelogram, the opposite sides are equal.
Given: ABCD is a parallelogram.
To Prove: The opposite sides are equal, AB=CD, and BC=AD. Proof: In parallelogram ABCB, compare triangles ABC and CDA. In these triangles AC = CA (common sides). Also ∠BAC =∠DCA (alternate interior angles), and ∠BCA = ∠DAC (alternate interior angles). Hence by the ASA criterion, both the triangles are congruent and the corresponding sides are equal. Therefore we have AB = CD, and BC = AD.
Converse of Theorem 1: If the opposite sides in a quadrilateral are equal, then it is a parallelogram. If AB = CD and BC = AD in the given quadrilateral ABCD, then it is a parallelogram.
Given: The opposite sides in a quadrilateral ABCD are equal, AB = CD, and BC = AD.
To Prove: ABCD is a parallelogram.
Proof: In the quadrilateral ABCD we are given that AB = CD, and AD = BC. Now compare the two triangles ABC, and CDA. Here we have AC = AC (Common sides), AB = CD (since alternate interior angles are equal), and AD = BC (given). Thus by the SSS criterion both the triangles are congruent, and the corresponding angles are equal. Hence we can conclude that ∠BAC = ∠DCA, and ∠BCA = ∠DAC. Therefore AB // CD, BC // AD, and ABCD is a parallelogram.
Theorem 2: In a Parallelogram, the Opposite Angles Are Equal.
Given: ABCD is a parallelogram, and ∠A, ∠B, ∠C, ∠D are the four angles.
To Prove: ∠A =∠C and ∠B=∠D
Hence, proved.
Answer refers to the attachment.
