in parallelogram abcd diagonals ac and bd

I talked to several groups, listening to their plan, asking a few questions to get them unstuck.Īnd then I got out colored paper on which to write the team proof. Some were obviously practicing look for and make use of structure. Then they worked with their team to talk about their ideas and to begin to plan a proof. Students worked alone for a few minutes, thinking about what was given and what could be implied. Corresponding sides are equal, so AB = CD and BC = DA.It was the day before the test on Polygons, and so I thought that writing a proof and then giving feedback on another team’s proof might be helpful. Thus by ASA, triangles ABC and CDA are congruent. Also line AC is a transversal of parallel lines BC and DA, so angle ACB is congruent to angle CAD. The line AC is a transversal of parallel lines AB and CD, so angle CAB is congruent to angle ACD. To prove the angles congruent, we use transversals. The two triangles have a common side AC = CA. We will prove that triangle ABC is congruent to triangle CDA by ASA. By definition, line AB is parallel to line CD and line BC is parallel to line DA. Proposition: If ABCD is a parallelogram, its opposite sides are equal. (Opposite sides of a parallelogram are equal.) This says ABCD is a rhombus, by definition. From this is follows that the hypotenuses are all congruent: AB = AD = CB = CD. Thus the triangles AMB, AMD, CMB, and CMD are congruent by SAS. If we also assume that AC is perpendicular to BC, then each of the angles AMB, AMD, CMB, and CMD are right angles. We know from this that MA = MC and MB = MD. Let M be the intersection of the diagonals. 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. In-class Activity and Classroom Self-Assessment

In parallelogram abcd diagonals ac and bd