Download Problems for Chapter 1

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.]