Note there is another set of parallel lines in a quadrilateral. But K belongs to the line at infinity ( ab)( cd), implying that EF is indeed parallel to BC. By Pappus' theorem, the three points E, F, K are collinear. Then L and M are parallel if and only if corresponding angles of the intersection of L and T, and M and T are equal. If point B is in the interior of AOC, then mAOB+mBOCmAOC. Suppose that L, M and T are distinct lines. This chapter uses simple and fun videos that are.

divides the segment into two congruent segments. The Parallel Line Proofs in Geometry chapter of this Geometry Proofs: Help & Tutorials course is the simplest way to master parallel line proofs. Let K be the intersection of the third (Pappus) pair, viz., BC and ( ab)( cd). a line, segment, ray, or plane that intersects a segment at its midpoint the segment into two congruent segments. E is then the intersection of A cd and CD, while F is the intersection of AC and D ab. Instead of the quadrilateral ABCD, consider the hexagon BA abCD cd. Parallel Lines Proofs quiz for 9th grade students. Let ab, cd, and bc be the points at infinity defined by the directions of AB, CD, and BC, respectively. The plain reason why this figure looks so familiar is that it is nothingĮlse than Pappus configuration.' You may want to ponder this one before reading further. Michel's third message read: 'My last remark about this problem.

Michel has also observed that by a repeated application of one of Thales theorems (the one that claims proportions on parallel transversals which we already used several times above) it is a simple matter to present the same proof while omitting all mention of homotheties: Implying f(BC) = EF so that EF is parallel to BC. Since homotheties with a common center commute, we also have f = hg. Let g be the homothety with center I mapping BA onto DF, h the homothety with center I mapping DC into EA. Independent Practice: PROOFS OF PARALLEL LINES NAME: DATE: PERIOD: Geometry Unit 3 - Reasoning & Proofs w/Congruent Triangles For 1-3, given a b, state the postulate or theorem that justifies each conclusion. Michel Cabart came up with a shorter variant of the proof. Two transversals GH and BC induce equal ratios on sides BD and CD of angle BDC. In trapezoid ABMD, triangles ABE and MDE are similar. Triangles AEF and CHF are also similar, implying (2) Otherwise, let M be the intersection of AE and DF and G, H the intersections of EF with AB and CD, respectively. If AB||CD there is nothing to prove since, in this case, E coincides with B and F with C. Students will practice using theorems about parallel lines, including Alternate Interior Angles Theorem, Alternate Exterior Angles Theorem, Corresponding Angles. The line through A parallel to CD meets BD at E and the line through D parallel to AB meets AC at F.

Parallel line proofs