Question

(1) Assume the axioms of metric geometry. Let A, B, C, D be distinct collinear points. Let f : l → R be a coordinate function for the line l that crosses all of A, B, C, D. Suppose f(A) < f(B) < f(C) < f(D). Prove that AD = AB ∪ BC ∪ CD. (2) Assume the axioms of metric geometry. Let A, B, C, D be distinct collinear points. Suppose A ∗ B ∗ C and B ∗ C ∗ D. Prove that AD = AB ∪ BC ∪ CD.

Question 2. (15pt) (1) Assume the axioms of metric geometry. Let A, B, C, D be distinct collinear points. Let f:1 + R be a coordinate function for the line l that crosses all of A, B, C, D. Suppose f(A) < f(B) < f(C) < f(D). Prove that AD = AB U BCUCD. (2) Assume the axioms of metric geometry. Let A, B, C, D be distinct collinear points. Suppose A * B *C and B*C * D. Prove that AD = ABU BCUCD.

Answer 1

Enswco Part (1): if {A, B, C, D } au bout distint points in a Meloic geometry no three of which are collinear and int (AB) int (BC) int (CD), and int (DA) are disjoint! Then we call a quadri lateral ABCO AB U BC U COUDA The Veeltes au the points - A, B, C, and D. → The sides are m The sides are AB, BC, co and y a B Tig(1) DA =) The diagonals are ÁC and Do 3 The angles ale CDAB ,ZABC, <BCA and <con Sides with a Common endpoints and Angles Containing a common side Then we can as adjacent and a Consecutive - Two sides or angles that are not adjacent are called opposite 0 ABCD = ĀB U BC U COUDA By using Segment Cut" Theorem, - We Can Say Lão AB U BC U TO CS Scanned with CamScanner

Ve Suppose Arbec and Be CoD > Applying Axiom Bi to An BRC, We get A#B, A# C and B #C > plying Aniom B, to By CoD, Ve get B # D and C#0 - finally Suppose that A=D By Substitution in An Ba C , We have Dr BR C This contradicts Gr (+D A#D. 2 A, B, C and D are distinct points. As lue prove the part (1) Thus We conclude that if a quadrilateral - . AD= ÄR U BC U CD CS Scanned with CamScanner