![]() Now consider various proofs introduced in class. Given points non-collinear points A, B, C, D with AB = CD and angle ABC = angle BCD, prove that there is a circle passing through ABCD. From the equations we can see that the slopes of the lines. Is this quadrilateral a parallelogram To check if its a parallelogram we have to check that it has two pairs of parallel sides. A quadrilateral is defined by the four lines (y2x+1), (yx+5), (y2x4), and (yx5). Given points non-collinear points A, B, C, D, prove that there is a circle passing through ABCD. Determining if a Quadrilateral is a Parallelogram. Given points non-collinear points A, B, C, D with AB = BC = CD, prove that there is a circle passing through ABCD. Textbook solution for McDougal Littell Jurgensen Geometry: Student Edition 5th Edition Ray C. Given points non-collinear points A, B, C, D with AB = BC = CD and angle ABC = angle BCD, prove that there is a circle passing through ABCD.Ĭonsider alternate versions. Examine critically whether all the hypotheses are used in a potential proof. Look for consequences of all the hypotheses. Keeping this in mind helps keep the various angles straight. The equality of angles can be viewed as point symmetry (180-degree rotational symmetry). Thus triangles ABC and CDA are congruent by SAS.Ĭomment: Notice that the first figure forms a figure like the letter Z. Angle BAC = angle DCA segments AB = CD and AC = CA. Looking at the "given" figure, we see a pair of congruent angles and two pairs of equal sides. If these triangles are congruent, then angle ACB = angle CAD, which implies BC parallel to DA.Ĭan we prove they are congruent. This is what we will prove using congruent triangles. 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. In the combined figure, one can see two apparently congruent triangles, ABC and CDA. Let O be the intersection of the diagonals AC and BD. If these pictures are put together, it suggests an idea for making the link between givens and goal. Continuing to use the transversal AC, we draw the equal angles imply the parallelism. To satisfy the definition of parallelogram, it suffices to prove that the sides BC and AD are parallel. Now draw a separate figure showing what one need to prove. Since a major feature of parallel lines is their relation to transversals, we try drawing in a natural transversal for this figure and mark the equal angles.ĭRAW GOAL. ![]() In this case, we are given a couple of parallel segments of the same length.ĭRAW SOME IMMEDIATE CONSEQUENCES. If ABCD is a quadrilateral with AB parallel to CD and with AB = CD, prove that ABCD is a parallelogram.ĭRAW GIVENS. Use diagrams and notes to clarify what is given and what is to be proved (and the difference between the two). ![]() Some Strategies for Achieving Correct Proofs
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