in parallelogram abcd diagonals ac and bd

Proof: From Problem 1, we know that the diagonals of a parallelogram ABCD bisect each other. (b) Prove that a parallelogram with perpendicular diagonals is a rhombus. Proof: In the homework, it was proved that if a quadrilateral ABCD has opposite sides equal, then it is a parallelogram. A rhombus is defined to be a quadrilateral with four equal sides.Since there was nothing special about those two side, using the same argument, we can also conclude that BC and DA are parallel, so by definition ABCD is a parallelogram. Thus we see that two opposite sides of ABCD are parallel. Since AC is a transversal of lines AB and CD, these equal alternate interior angles imply that the lines AB and CD are parallel. Thus angle MAB (which is the same as angle CAB) and angle MCD (which is the same as angle ACD) are congruent. Thus we conclude that triangle AMB is congruent to triangle CMD by SAS.Ĭorresponding angles are congruent. We also know that angle AMB = angle CMD by vertical angles. We are given than M is the midpoint of AC and also of BD, so MA = MC and MB = MD. If ABCD is a quadrilateral such that the diagonals AC and BD bisect each other, then ABCD is a parallelogram.

These are two corresponding sides of the similar triangles, so the two triangles ABO and CDO are congruent.įrom the congruence, we conclude that AO = CO and BO = DO.Īssertion 2. We know from the homework (*) that opposite sides of ABCD, AB = CD. Next we show that these two triangles are congruent by showing the ratio of similitude is 1. Thus triangle ABO is similar to triangle CDO. Also, by vertical angles, angle AOB = angle COD. Since line AC is a transversal of the parallel lines AB and CD, then angle OAB = angle CAB = angle ACD = angle OCD. This is what we will prove using congruent triangles.įirst we show triangle ABO is similar to triangle CDO using Angle-Angle. Likewise, O is the midpoint of BD if BO = DO. Since O is on segment AC, O is the midpoint of AC if AO = CO.

The Assertion can be restated thus: O is the midpoint of AC and also the midpoint of BD. Let O be the intersection of the diagonals AC and BD. If ABCD is a parallelogram, then the diagonals of ABCD bisect each other. A quadrilateral ABCD is a parallelogram if AB is parallel to CD and BC is parallel to DA.Īssertion 1. (In other words, the diagonals intersect at a point M, which is the midpoint of each diagonal.)ĭefinition. Prove that a quadrilateral is a parallelogram if and only if the diagonals bisect each other. State the definition of a parallelogram (the one in B&B). Problem 2 was demonstrated quickly on the overhead and was not done as a group activity. This theorem is an if-and-only-if, so there are two parts to the solution. Problem 1 was given as an in-class group activity. Therefore the area of parallelogram ABCD is equal to the area of rectangle AEFD, or b×h.In-class Activity and Classroom Self-Assessment The area of a parallelogram is the product of the length of its base (b) and height (h). Rotating the parallelogram above 180° about point E will result in a parallelogram that is the same as the original shape. Parallelogram with four right angles and all four sides equal in lengthĪ parallelogram has rotational symmetry of order 2. Parallelogram with all sides equal in length There are three cases when a parallelogram is also another type of quadrilateral. In the figure below diagonals AC and BD bisect each other. The diagonals of a parallelogram bisect each other. Diagonals of a parallelogramĪ parallelogram has two diagonals. Therefore, ∠A and ∠ABC are supplementary. Since ∠ADB = ∠DBC and the sum of the angles of a triangle is 180°, ∠A + ∠B Similarly, it can be proven that ∠A = ∠C. In the diagram above, diagonal BD cuts parallelogram ABCD into two triangles. The consecutive angles are supplementary to each other. The opposite angles of a parallelogram are equal in measure. Interior angles of a parallelogramĪ parallelogram has 4 internal angles. According to the Angle-Side-Angle (ASA) postulate, △ABD ≅ △CDB. In the figure below, side BC is equal to AD in length and side AB is equal to CD in length.ĭiagonal BD cuts parallelogram ABCD into two triangles. The opposite sides of a parallelogram are equal in length. Sides of a parallelogramĪ parallelogram has 4 sides. In the figure below are 4 types of parallelograms. Home / geometry / shape / parallelogram ParallelogramĪ parallelogram is a quadrilateral with two pairs of opposite, parallel sides.

In parallelogram abcd diagonals ac and bd