e Sides of a Parallelogram are EqualIn 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 ABCDABCD is a parallelogram.Compare ΔABCΔABC and ΔCDAΔCDA:AC=CA(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,AB=CDandAD=BCAB=CDandAD=BCNow, we will prove the converse of this.Converse of the TheoremIf the opposite sides in a quadrilateral are equal, then it is a parallelogram.Assume that ABCDABCD is a quadrilateral in which AB=CDAB=CD and AD=BCAD=BC,Compare ΔABCΔABC and ΔCDAΔCDA once again:AC=AC(common sides)AB=CD(SINCE alternate interior angles are equal )AD=BC(given)AC=AC(common sides)AB=CD(since alternate interior angles are equal )AD=BC(given) AC=AC(common sides)AB=CD(since alternate interiorangles are equal )AD=BC(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⇒AB∥CD ∠2=∠3⇒AD∥BC ∠1=∠4⇒AB∥CD ∠2=∠3⇒AD∥BC HENCE,AB∥CDandAD∥BCAB∥CDandAD∥BCThus, ABCDABCD is a parallelogram.Opposite Angles of a Parallelogram are EqualIn 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 ABCDABCD is a parallelogram.Compare ΔABCΔABC and ΔCDAΔCDA once again:AC=AC(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=∠DSimilarly, we can show that∠A=∠C∠A=∠CThis proves that opposite angles in any parallelogram are equal.Now, we prove the converse of this.Converse of the TheoremIf 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 ABCDABCD 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 AD∥BCAD∥BC Similarly, we can show that AB∥CDAB∥CDHence,AD∥BCandAB∥CDAD∥BCandAB∥CDand thus, ABCDABCD 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.please mark me brilliant and give thanks....please