Download Q1: Which of the following would be considered a "line" by Euclid`s

Propositions of the Elements - parallel lines, triangle, congruence, and
similarity (GR2 – part four)
For all questions, please explain your answer, and cite the relevant definition/common
notion/postulate from the Elements that supports it. We’re mostly taking a look at Book
I of the Elements, and that’s where most of this will be found – however, the propositions
on similarity come from Book VI.
When you cite a Proposition or Definition, Book 1 will be assumed unless otherwise
noted.
Q1: Go into Joyce's online Elements and dissect Book One, Proposition Three and the given
proof of it in the same manner that Proposition One was dissected in the lecture, and
Proposition Two was dissected in the suggested problem. This is the first chunk of material
listed under Module two, appearing as:
Proposition one
1.
2.
3.
4.
Download notes
Watch lecture
Suggested problems
Solutions to suggested problems
Please consult that to see what I'm asking for here: a writeup with pictures similar to the
solution to the suggested problem.
Q2: The following is the sort of proof you probably did in your high school geometry class. The
point to this section is largely a history lesson – all that stuff you learned as “base angles of an
isosceles triangle” or “side-angle-side” has its origins in the Elements, and there’s a specific
definition or proposition in Euclid associated with each of these familiar things. Find them – the
proof is already done in the language of high school geometry – locate and cite each definition
and proposition as it appears in Euclid to fill in the third column of
the table.
Given  BCD isosceles with base BD , C the midpoint of
AE , BD || AE , prove  A   E .
Statement
1)  BCD isosceles with
Reason
Cite from the Elements
Given
---------------------------------
2) C is the midpoint of AE
Given
---------------------------------
3)
BD || AE
Given
---------------------------------
4)
BC  DC
Definition of isosceles
[Fill in (1)]
5)  DBC   BDC
Base angles of an isosceles
triangle
[Fill in (2)]
6)  ACB   DBC
 ECD   BDC
Alternate interior angles
[Fill in (3)]
7)  ACB   ECD
Substitution
[Fill in (4)]
Definition of midpoint
[Fill in (5)]
9)  BAC   DEC
Side-angle-side
[Fill in (6)]
10)  A   E
CPCTC (or however you
learned it – something like
“corresponding parts of
congruent triangles are
congruent”)
[Fill in (7)]
base BD
8)
AC  EC
Q3: Given l || m , s and t transversals, m1  50 , and
m9  75 , find the measures of the angles in the diagram. Find
them sequentially; i.e. proceed in order from 1 to 2 , 2 to
3 , and so on. As you move through the angles, justify your
answer by describing the relationship of the calculated angle to
the previous angle. (I'll do a few to illustrate what I mean here.)
angle
measure
relationship
justification
1
50
-----
given
2
50
m2  m1
3
130
m3  180  m2
parallel lines → alternate interior angles are
equal (prop 29)
parallel lines → same side interior angles are
supplementary (sum to 180 ) (prop 29)
9
75
-----
given
10
75
m10  m9
vertical angles are equal (prop 15)
m11 180  (m1  m10)
angle sum of triangle is 180 (prop 32)
4
5
6
7
8
11
12
13
14
15
16
17
18
Q4: In the diagram to the right
l1 is the line FG
l2 is the line HI
t1 is the line DK
t 2 is the line EJ
mEAG  (6 x  y )
mFAB  (2 x  y )
mABC  (3x  10)
a) What values of x and y are needed so that l1 || l2 ? Don’t forget, for every equation you
set up in order to solve, be sure to CITE the relevant proposition from the Elements that
justifies that equation – be clear on how you’re obtaining your values.
b) Given additionally that mDAE  80 , what is the value of mACI ? (as always,
justify/cite).
Q5: Given EC  6 , EA  14 , ED  9 , EB  21
a) Prove  ECD ~  EAB (citing Elements).
b) Prove CD || AB .
c) Given further that CD  11, calculate AB .