Two parllelogram have a common diagonal.show that the angular points are those of parellogram
Answers
State the definition of a parallelogram (the one in B&B). Prove that a quadrilateral is a parallelogram if and only if the diagonals bisect each other. (In other words, the diagonals intersect at a point M, which is the midpoint of each diagonal.)
Definition. A quadrilateral ABCD is a parallelogram if AB is parallel to CD and BC is parallel to DA.
Assertion 1. If ABCD is a parallelogram, then the diagonals of ABCD bisect each other.
Proof of Assertion 1.
Let O be the intersection of the diagonals AC and BD. The Assertion can be restated thus: O is the midpoint of AC and also the midpoint of BD.
Since O is on segment AC, O is the midpoint of AC if AO = CO. Likewise, O is the midpoint of BD if BO = DO. This is what we will prove using congruent triangles.
First we show triangle ABO is similar to triangle CDO using Angle-Angle. Since line AC is a transversal of the parallel lines AB and CD, then angle OAB = angle CAB = angle ACD = angle OCD. Also, by vertical angles, angle AOB = angle COD. Thus triangle ABO is similar to triangle CDO.
Next we show that these two triangles are congruent by showing the ratio of similitude is 1. We know from the homework (*) that opposite sides of ABCD, AB = CD. These are two corresponding sides of the similar triangles, so the two triangles ABO and CDO are congruent.
From the congruence, we conclude that AO = CO and BO = DO.