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14 1 Pappus’s Theorem: Nine proofs and three variations A B C Z Y X Fig. The alternate interior angles are congruent. If two parallel lines are cut by a transversal, then. Start studying Unit 3 Angle Relationships and Parallel Line Theorems. The alternate exterior angles are congruent. In the situation with zero slope both lines are parallel and the intersection point vanishes. A composition of reflections across two parallel lines is equivalent to a translation. If three parallel lines cut off equal segments on some transversal line, then they will cut off equal segments on any other transversal. read more. Theorem 3-10 includes the phrase in a plane.On the other hand,Theorem 3-9 is true for any three such lines,whether they are coplanar (Exercise 3) or noncoplanar. The interior angles … 1.3. Euclidean Version of Pappus’s Theorem. move further and further to the right (In fact it can be arbitrarily far away). basic proportionality theorem proof If a straight line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides in the same ratio. Given : In a triangle ABC, a straight line l parallel to BC, intersects AB at D and AC at E. The parallel line theorems are useful for writing geometric proofs. These three straight lines bisect the side AD of the trapezoid.Hence, they bisect any other transverse line, in accordance with the Theorem 1 of this lesson. The following proof of Theorem 3-12 relies on the idea that through a point not on a given line you can draw a line parallel to the given line. Theorem 6.6 :- Lines which are parallel to the same lines are parallel to each other. In particular, they bisect the straight line segment IJ. Parallel lines cut by transversal. 15 terms. ... Converse of the Same Side Interior Angles Theorem. ... Geometric Proofs. line.Theorems 3-9 and 3-10 guarantee that the lines you draw are indeed parallel. Thus the tree straight lines AB, DC and EF are parallel. For proving this theorem, let's look at a pair of parallel lines, line-1 and line-2 intersected by a transversal, forming an exterior angle A with line -1. For lines l & n with transversal t, corresponding angles are equal Hence l and n are parallel. It is what has to be proved. To Prove :- l n. Proof :- From (1) and (2) 1 = 3 But they are corresponding angles. Theorem 12 Proof: Line Parallel To One Side Of A Triangle. Theorems involving reflections in mathematics Parallel Lines Theorem. Given :- Three lines l, m, n and a transversal t such that l m and m n . Their corresponding angles are congruent. Unit 1 Lesson 13 Proving Theorems involving parallel and perp lines WITH ANSWERS!.notebook 3 October 04, 2017 Oct 3­1:08 PM note: You may not use the theorem … Learn vocabulary, terms, and more with flashcards, games, and other study tools. Sum Theorem. Proof of Theorem 3-12 Given: #ABC Prove: m&A + m&B + m&3 = 180 Proof: By the Protractor Postulate, you can draw so that m&1 = m&A.Then, &1 and &A are congruent alternate interior angles, so 6 .