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Problems for Chapter 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 There are six conditions in the definition of congruent triangles, Labeled (1),(2),(3),(4),(5),(6).make a list of all (20) three-element subsets of (1),…,(6) and, for each one, tell whether it as hypothesis implies ∆ABC≅∆DEF. In our proof of SSS we choose G such that ∠GBC≅ ∠E, ∠GCB≅∠F and such that G was on the same side of BC as A. Find a proof of SSS with G on the opposite side of BC . Prove SSA for right triangles: If AB ≅ DE . AC ≅ DF and if ∠B=90° and ∠E=90° then ∆ABC ≅ ∆DEF. Prove SAA : If ∠A ≅ ∠D , ∠B ≅ ∠E and BC ≅ EF , then ∠ABC ≅ ∠DEF. Find the mistake in the following proof that all triangles are isosceles. See Fig. Let ∆ABC Be any triangles. We will prove that AB ≅ AC . Draw the perpendicular bisector of BC and the angle bisector of ∠A. Let these two lines intersect at the point E. From E draw lines EF and EG perpendicular to AB and AC . Now, ∆BED ≅ ∆CED by SAS, so BE ≅ CE . Also, ∆AEF ≅ ∆AEG. By SAA (see exercise 4), So EF ≅ EG . Since EF ≅ EG and BE ≅ CE .∆BEF ≅ ∆CEG by SSA for right triangles (see exercise 3). From ∆AEF ≅ ∆AEG we conclude that EF ≅ AG ; from ∆BEF ≅ ∆CEG we conclude that BF ≅ CG ; and by addition we get AB ≅ AC Let L be a line and p a point on L construct a line that contains p and that is perpendicular to L. Let L be a line and p a point not on L. construct a line that contains p and that meets L at a 45° angle. Construct a line that contains p and that meets L at a 30° angle. (To do this exercise you need to use the fact that the sum of the angles in any triangle is 180°.) 1.8 Given triangle ∆ABC and ∆DEF such that AB ≅ DE , AC ≅ DF and ∠A > ∠D, prove that BC > EF. 1.9 Given BC and A1 , A2 ,..., An Prove that BC < BA1 + A1 A2 + A2 A3 + ... + AnC. 1.10 This exercise extends the definition of congruence from triangles to more general polygons and explores some consequences. Let 2n points (n >2,a whole number) A1 , A2 ,..., An and B1 , B2 ,..., Bn be given. Think of these points as vertices and A1 A2 ... An as the polygon formed by the union of the segments A1 A2 , A2 A3 ,..., An A1 . Define A1 A2 ... An ≅ B1 B2 ...Bn if for all distinct vertices Ai , Aj and Ak the two triangles ∆Ai Aj Ak and ∆Bi B j Bk are congruent. (a) Prove ABCD ≅ EFGH if AB=EF , BC=FG , AD= EH , ∠A =∠E and ∠B = ∠F. (b) Prove ABCD ≅ EFGH if AB=EF, BC=FG,CD=GH,AD=EH and ∠A=∠E. c) Prove That two pentagons are congruent if SASASAS holds.[ hint :use (a).] (d) Prove that two polygons are congruent if all corresponding sides and angles are congruent. [hint : use induction on the number of sides.]