![]() Corollary 2: If P is a point not on, then the perpendicular dropped from P to is unique. By Proposition 8.14 all right angles are congruent, so the Alternate Interior Angle Theorem applies. ![]() The alternate interior angles are right angles. Corollary 1: If m and n are distinct lines both perpendicular to the line, then m and n are non-intersecting. Thus, and we have shown that or that is more that one point, contradicting Proposition 6.1. Since the angles share a side, they are themselves supplementary. Therefore, is congruent to the supplement of. Now, the supplement of is congruent to the supplement of, by Proposition 8.5. Since, (by Axiom C-2), we may apply the SAS Axiom to prove thatįrom the definition of congruent triangles, it follows that. By Congruence Axiom 1 there is a unique point so that. D is on one side of, so by changing the labeling, if necessary, we may assume that D lies on the same side of as C and C'. Let us denote this point of intersection by D. Assume that the lines m and n are not non-intersecting i.e., they have a nonempty intersection. Hence, it is on the opposite side of from A', by the Plane Separation Axiom. Likewise, choose on the opposite side of from A. ![]() Choose a point A on m on one side of, and choose on the same side of as A. Let the points of intersection be B and B', respectively. Proof: Let m and n be two lines cut by the transversal. Theorem 9.1: If two lines cut by a transversal have a pair of congruent alternate interior angles, then the two lines are non-intersecting. We want to reserve the word parallel for later. Definition: If k and are lines so that, we shall call these lines non-intersecting. Two of these angles are corresponding angles if their transversal sides have like directions and their non-transversal sides lie on the same side of. Definition:An angle of intersection of m and k and one of n and k are alternate interior angles if their transversal sides are opposite directed and intersecting, and if their non-transversal sides lie on opposite sides of. We say that each of the angles of intersection of and m and of and n has a transversal side in and a non-transversal side not contained in. Let be transversal to m and n at points A and B, respectively. A line k is transversal of if #, and # for all.
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