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Homework 2 solutions 2.6, #3 Outline how SMSG Postulate 4 can be deduced from the other SMSG postulates. Outline: Let P and Q be points on a line. The ruler postulate guarantees that the point of the line can be placed in 1-1 correspondence with the real numbers. Let x P and x Q be the real numbers associated with P and Q under this correspondence. Define a 2nd correspondence of the points on the line with the real numbers by setting the real number associated with a point A, y A , to be x A x P for each point A on the line. Then this second correspondence assigns the number 0 to P. Finally, if the coordinate assigned to Q, y Q , is positive, we are done. Otherwise, replace each coordinate y A by its negative to obtain a third correspondence that satisfies Postulate 4. 2.6, # 5 Which of Birkhoff’s Axioms implies SMSG Postulate 15? Postulate IV with k 1. 3.2, # 8 An equilateral triangle has three congruent sides. Prove that an equilateral triangle is equiangular. Proof: Let ABC be equilateral. Consider angles A and B. By Theorem 3.2.7, these angles are congruent since the sides opposite these angles are congruent. Similarly, angles A and C are congruent. So all three angles are congruent. 3.2, # 9 Prove that if a point P is on the perpendicular bisector of a line segment AB, then it is equidistant from the endpoints of the line segment. Proof: Let P be on the perpendicular bisector of a line segment AB. Let M be the midpoint of the segment. Consider the triangles AMP and BMP. Side AM is congruent to side BM since M is the midpoint. Angles AMP and BMP are both right angles, so they are congruent. Side MP of the first triangle is congruent to side MP of the second. So by SAS, the triangles are congruent. So sides AP and BP are congruent. Thus P is equidistant from A and B. 1