Download Systems of Geometry Test File Spring 2010 Test 1 1.) Consider a

Systems of Geometry
Test File
Spring 2010
Test 1
1.)
Consider a geometry with undefined terms point, line, and contains (or belongs)
and the following axioms.
Axiom 1.
Axiom 2.
Axiom 3.
Axiom 4.
a.)
b.)
Given any two distinct points, there is exactly one line containing
both of them
Each line contains at least two points.
There exist at least three distinct points.
Not all points are contained on the same line.
Prove that the intersection of two distinct lines contains at most one point.
Prove that there exist three distinct lines that do not contain a common
point.
2.)
Discuss what we mean by the concepts of independence and consistency in the
context of an axiomatic system.
3.)
Discuss the contributions to geometry of two of the following men: Thales,
Pythagoras, Hippocrates, Plato, Eudoxus, Euclid, Clint Eastwood.
4.)
Suppose point, line, contains (belongs to), end point and congruence (of
segments and angles) are taken as undefined terms. Define the following terms.
a.)
A is between B and C .
____
b.)
AB

c.)
d.)
AB
BAC
5.)
If a and b are the lengths of the legs of a right triangle and c is the length of the
hypotenuse, Babylonian geometers approximated the value of c by the formula
 a2 
c  b    . Is this formula correct? If not, how does it compare with the
 2b 
correct value for c?
6.)
The undefined terms for a geometry are point, line, and on. The axioms are:
Axiom 1.
Axiom 2.
Axiom 3.
There exist exactly four points.
Any two distinct points have exactly one line on both of them.
Each line is on exactly two points.
a.)
b.)
c.)
Construct a model for this geometry.
Based on this geometry, define parallel lines.
Prove that each distinct line has exactly one line parallel to it. (You may
use the theorem that, in this geometry, each point has exactly three lines
on it.)
Test #2
1.)
(Assume the Euclidean Parallel Postulate.) Prove: The opposite sides of a
parallelogram are congruent (definition of parallelogram - quadrilateral with both pairs of
opposite sides parallel).
2.)
Prove that if two lines are intersected by a transversal such that a pair of alternate
interior angles is congruent, then the lines have a common perpendicular.
3.)
Vertical angles are congruent.
4.)
(SASAS) If the vertices of two convex quadrilaterals are in one-to-one correspondence
such that three sides and the two included interior angles of one quadrilateral are
congruent to the corresponding parts of a second quadrilateral, then the quadrilaterals
are congruent.
Hilbert
Exercises
1.)
Every line contains an infinite number of points.
2.)
Prove that two lines that are parallel to the same line are parallel to each other.
These all use SMSG and Euclid's first four postulates.
1.)
Prove that the Euclidean parallel postulate is equivalent to the following
statement: If two parallel lines are intersected by a transversal, then the pairs of
interior angles on the same side of the transversal are supplementary.
2.)
Prove that the Euclidean parallel postulate is equivalent to the following: The
diagonals of a parallelogram bisect each other.
3.)
If we define the angle sum of triangle ABC as m∠A + m∠B + m∠C, prove that the
Euclidean parallel postulate implies that every triangle has an angle sum of 180.
4.)
An equilateral triangle has three congruent sides. Prove that an equilateral
triangle is also equiangular.
Test #3
Euclidean Parallel Postulate - Given a line and a point not on the line, there exists exactly one line on
the point parallel to the given line.
SMSG - (also assume Euclid's first four postulates)
Note: If a problem is labeled "Chapter 3," then you may only assume Euclid's first four
postulates and SMSG postulates 1-15.
1.)
Define the following:
a.)
Saccheri quadrilateral
b.)
c.)
Lambert quadrilateral
Similar polygons
2.)
Which of the following statements are true in neutral geometry:
_____ The Alternate Interior Angle Theorem.
_____ The Converse of the Alternate Interior Angle Theorem.
_____ A Saccheri quadrilateral exists.
_____ A rectangle exists.
_____ The angle sum of any triangle is 180.
3.)
Prove that the diagonals of a parallelogram bisect each other.
4.)
Prove that the altitude to the hypotenuse of a right triangle divides the triangle into two
smaller triangles, each of which is similar to the original right triangle.
5.)
State and prove the Pythagorean Theorem. (You may assume without proof that the
altitude to the hypotenuse of a right triangle divides the triangle into two smaller
triangles, each of which is similar to the original right triangle.)
6.)
State and prove the Alternate Interior Angle Theorem.
7.)
(Chapter 3) Prove: If ABCD is a quadrilateral, where A and B are right angles
and C  D , then ABCD is a Saccheri quadrilateral.
Test #4
1.)
2.)
3.)
4.)
5.)
Prove: If a diameter of a circle is perpendicular to a chord of the circle, then the
diameter bisects the chord.
(Assume the Euclidean Parallel Postulate.) Prove: The opposite sides of a
parallelogram are congruent.
Prove that in neutral geometry, the angle sum of any convex quadrilateral is less
than or equal to 3600 .
Prove that if a convex quadrilateral ABCD is such that opposite sides are
congruent, then it is a parallelogram. (You may assume the Euclidean Parallel
Postulate.)
The following picture shows a circle with center, A. For each of the following,
identify an example from the picture.
D
a.)
chord ________________
b.)
tangent _______________
c.)
diameter ______________
d.)
secant _______________
e.)
central angle __________
f.)
semicircle ____________
g.)
minor arc _____________
h.)
major arc _____________
i.)
inscribed angle _________
j.)
radius ________________
E
F
C
A
B
G
I
H
J
6.)
Define the angle of parallelism. (Feel free to use the picture on #7.)
7.)
Prove: If T-R-S and RPQ and RPQ' are the angles of parallelism for P and
and P and
, respectively, then mRPQ = mRPQ'. (Assume PR is
perpendicular to TS)
P
Q'
T
Final Exam
Q
R
S
1.)
Define the following :
a.)
A is between B and C
____
b.)
AB

2.)
3.)
4.)
5.)
6.)
7.)
8.)
9.)
c.)
AB
BAC
d.)
e.)
Lambert quadrilateral
ABC
f.)
g.)
linear pair
h.)
Congruent polygons
i.)
Saccheri quadrilateral
j.)
Similar polygons
k.)
chord
l.)
circumcenter of a triangle
m.)
median of a triangle
n.)
secant of a circle
o.)
tangent to a circle
Consider a geometry with undefined terms point, line, and contains (or belongs)
and the following axioms.
Axiom 1.
Given any two distinct points, there is exactly one line containing
both of them
Axiom 2.
Each line contains at least two points.
Axiom 3.
There exist at least three distinct points.
Axiom 4.
Not all points are contained on the same line.
a.)
Prove that the intersection of two distinct lines contains at most one point.
b.)
Prove that there exist three distinct lines that do not contain a common
point.
(Assume the Euclidean Parallel Postulate.) Prove: The opposite sides of a
parallelogram are congruent (definition of parallelogram - quadrilateral with both
pairs of opposite sides parallel).
Prove that vertical angles are congruent.
Prove that the Euclidean parallel postulate is equivalent to the following
statement: If two parallel lines are intersected by a transversal, then the pairs of
interior angles on the same side of the transversal are supplementary.
(Remember, "A is equivalent to B" means you have to prove "A implies B" and "B
implies A.")
If we define the angle sum of triangle ABC as m∠A + m∠B + m∠C, prove that the
Euclidean parallel postulate implies that every triangle has an angle sum of 180.
Prove: If a diameter of a circle is perpendicular to a chord of the circle, then the
diameter bisects the chord.
Given AB=BC and AD= DC. Prove BD  AC
(Assume the Euclidean Parallel Postulate) Straight Edge/Compass construction.
Consider the following figure. It began with line segment
. I then constructed
segment
. Next I constructed
and
such that D, F, E and G are collinear
and
=
=
. I then constructed
. Lastly I constructed
and
such
that
||
||
. (Note: a number of segments and circles used in the
construction were hidden.)
G
E
F
D
10.)
C
B
A
Do the following.
a.)
Explain how we can construct
,
and
such that D, F, E and G are
collinear and
=
=
.
b.)
Explain how
can be constructed such that
||
.
c.)
This construction produces a trisection of
. Explain why.
Prove: If T-R-S and RPQ and RPQ' are the angles of parallelism for P and
and P and
, respectively, then mRPQ = mRPQ'. (Assume PR is
perpendicular to TS)
P
Q'
T
Q
R
S