Assignment Sheet 1

Please note:  Read carefully for the difference between lines, rays, and segments.  Not all notation will appear correctly.

 

Historically, there have been several different approaches to doing geometry, not all of them axiomatic.  In order to be able to do geometry, we need a common set of definitions and axioms.  Definitions are important because all results depend on the definition used.  It is often possible that more than one definition is acceptable.  The same can be said of axioms.  There are many possible sets of axioms that result in what is typically referred to as Euclidean geometry.  I have chosen one particular set of axioms for this class, similar to the ones used by the mathematician Birkhoff, but this is certainly not the only choice.  However, it is important that whenever you do any proofs in this class, you do not rely on results we have not assumed or proven.  Therefore, you should carefully read all your proofs to be sure that you state the justification for each step.

 

As we have discussed, it is impossible to define every term in mathematics.  The terms point, line, and plane will be undefined for us.  Although we can discuss what we mean by these terms, these are the basic objects of study for us and we cannot define them in terms of other things.

 

Assumptions and postulates:

We assume the properties of the real numbers, of sets and set operations, and of algebra.

Postulate 1:  Given any two different points, there is exactly one line that contains both of them.  We often restate this as, “Two points determine a line.”

 

For the next postulate, we assume that we have picked a system of measurement.

Postulate 2 (Distance Postulate)/ Definition of Distance:  To each pair of points there is a unique number.  This number is called the distance between the two points.  For two points P and Q, the distance between them will be written PQ.

Postulate 3 (Number Line Postulate):  The points of a given line can be made to correspond to the real numbers in such a way that:

i.                     Every point of the line corresponds to exactly one real number

ii.                   Every real number corresponds to exactly one point on the line

iii.                  The distance between any two points is the absolute value of the difference between the corresponding numbers

Definitions:  A coordinate system is a choice of correspondence between points and numbers as described in Postulate 3.  The coordinate of a point is the number assigned via this correspondence.

Postulate 4 (Number Line Placement):  Given two points on a line, say P and Q, the coordinate system can be chosen so that P is at 0 and the coordinate of Q has a positive value.

Definition:  For three points, P, Q, and R, Q is said to be between P and R if PQ + QR = PR.

Any collection of points is said to be collinear if they all lie on the same line.

The absolute value of a real number x, written |x|, is given by:  x when x > 0 or x = 0, and –x when x < 0.

The term space or 3-space will be undefined.  Informally, we are using the term to mean three-dimensional space.

Definitions:  Objects are said to be coplanar if they lie in the same plane.

A segment is a set of two points together with all the points between them.  For two points P and Q, the segment will be denoted PQ, and P and Q are called the endpoints of the segment.  The length of this segment is the distance PQ.

A ray PQ, is the set of points P and Q together with all points R such that either R is between P and Q, or Q is between P and R.  P is called the endpoint of the ray.

A point R is the midpoint of a segment PQ, if R is between P and Q and PR = QR.  R, or any object which intersects PQ at R, is said to bisect PQ.

Postulate 5.1:  Every plane contains at least 3 non-collinear points.

Postulate 5.2:  Space contains at least 4 non-coplanar points.

Postulate 6:  If a plane contains two given points, it contains the line through the two points.

DUE THURSDAY, JANUARY 27:

These problems will be discussed over a period of several classes, but will be collected next class.

FOR DISCUSSION:

  1. Write definitions for square, rectangle, quadrilateral, parallelogram, rhombus, kite, and trapezoid.  Make your definitions as formal and clear as possible.  Be prepared to discuss why you chose your definition, and what you think makes a good definition.
  2. What does the word “is” mean?  Look at the following examples and describe how the word “is” is used in each.  Does “is” always have the same meaning?  How could some or all of the examples be reworded to be clearer?

                                                               i.      A square is a rectangle.

                                                             ii.      A scalene triangle is a triangle with no two sides having a common length.

                                                            iii.      An equilateral triangle is isosceles.

                                                           iv.      A square is a rhombus.

  1. Let P, Q, and R be three points of a line with coordinates p, q, and r.  Prove that if p<q<r, then Q is between P and R.
  2. Let P, Q, and R be three points on a line.  Prove that one of the points must be between the other two.
  3. Let P, Q, and R be three points on a line.  Prove that at most one of the points is between the other two.

FOR CAREFUL WRITE-UP:

  1. Let there be a ray with endpoint A, and let x be a positive number.  Prove that there is exactly one point X on the ray with AX = p.
  2. Prove that two different lines can intersect in at most one point.
  3. Write definitions for scalene, isosceles, and equilateral triangle (assume we know what a triangle is), and circle.
  4. If a, b, and c are coordinates of collinear points, and |a – c|  + |c – b|= |a – b|, what can be said as to the relationship among the three points?  Must one of the three points be between the other two?  Justify your answer.
  5. Prove that if a line intersects a plane, then either:  the line and the plane intersect in one point, or the plane contains the entire line.

 

 

Postulate 7:  There is at least one plane containing any three given points.  If the points are non-collinear, then there is only one such plane.

Postulate 8:  If two different planes intersect, then their intersection is a line.

Definition:  A set of points is called convex if, for any two points in the set, every point on the segment joining the points is contained in the set.

Postulate 9 (Plane Separation):  Given a line and a plane containing it.  The points of the plane that do not lie on the line form two sets such that each set is convex and, given two points, one in each set, the segment joining the points intersects the line.

Definitions:  The two sets described in Postulate 9 are called half-planes, and the line is called the edge of each half-plane.  The line is also said to separate the plane into two half-planes.  Two points that lie in the same half-plane are said to lie on the same side of the line; if they are in different half-planes, they lie on opposite sides of the line.

Postulate 10 (Space Separation): [You will state this one yourself.]

Definitions:  The two sets described in Postulate 10 are called half-spaces, and the plane is called a face of each half-space.

An angle is the union of two rays that have the same endpoint.  The two rays are each called the sides of the angle, and their common endpoint is called the vertex of the angle.

For any three points, the union of the segments joining them is called a triangle, the segments are called the sides, and the three points are called the vertices of the triangle.

Postulate 11 (Angle Measurement):  To every angle there corresponds a number greater than or equal to 0 and less than or equal to 180.

Definition:  The measure of an angle is the number assigned through the correspondence in Postulate 11.

Postulate 12 (Angle Construction):  For any ray AB, such that the ray lies on the edge of a half-plane, and for any number r, 0 < r < 180, there is exactly one ray AC with the same endpoint A and with C in the half-plane, with m<BAC=r.  For r=0 or 180, C will lie on the line AB.

Postulate 13 (Angle Addition):  Angle measure is additive for angles which share a common ray:  For angles <BAC and <CAD, m<BAC + m<CAD = m<BAD.

Definitions:  If two angles, <BAC and <CAD, share a common ray and B, A, and D are collinear, then the two angles form a linear pair.   Two angles are supplementary if the sum of their measures is 180, and each angle is said to be a supplement of the other.  If two angles form a linear pair and have the same measure, then each angle is a right angle.  Two intersecting sets, each of which is a line, a segment, or a ray, are perpendicular if the angles formed by the intersection are right angles.

Two angles are complementary if the sum of their measures is 90, and each angle is said to be a complement of the other. 

An angle with a measure less than 90 is called acute, while an angle with measure greater than 90 is called obtuse.

Two angles are congruent if their measures are equal.  Two segments are congruent if they have the same length.  Two triangles are congruent if there is a correspondence between the angles and segments of each triangle such that the corresponding angles and segments are congruent.

A ray AC is a bisector of angle <BAD if <BAC @ <CAD and the measures of these congruent angles are each not greater than 90.

A median of a triangle is a segment that has one endpoint at a vertex of the triangle and the other endpoint at the midpoint of the opposite side.

Postulate 14 (Supplement Postulate):  If two angles form a linear pair, then they are supplementary.

Postulate 15 (SAS Postulate):  If there is a correspondence between two triangles such that there are two sides and the angle included between the sides of the first triangle congruent to the corresponding parts of the second triangle, then the triangles are congruent.

Note:  We may later take a different look at congruence using isometries of the plane in which we will consider alternative postulates for congruence.

DUE TUESDAY, FEBRUARY 1:

FOR DISCUSSION:

  1. The concatenation issue:  Concatenation is a word used to refer to the placing of numbers or symbols together in mathematics as a short notation for some operation which is taken to be understood.  This can be very confusing, as there are many meanings, depending on the context.  Take a look at the examples below and explain what meaning is suggested by the different symbols.

                                                               i.      3a

                                                             ii.      xy

                                                            iii.      PQ

                                                           iv.      3 1/2

                                                             v.      31

  1. Given a line and a point not on a line, prove that there is exactly one plane containing both of them.
  2. Sometimes definitions are given without the use of variables, other times the description is given only in terms of variables, and sometimes both descriptions are given.
    1. The definition of convex is one example of a definition given without variables.  Rewrite the definition by giving variable names to the various objects.  Then give three examples of sets that are convex and two examples of sets that are not convex.
    2. The definition of segment is given without variables and then with variables.  Find a definition given only using variables.  Rewrite it without using variables.
  3. Prove that the intersection of any two convex sets is also convex.
  4. Prove that supplements of congruent angles are congruent.  To do this:
    1. Restate the above in terms of the four angles suggested by the statement.
    2. Prove your statement from (a).
  5.  
    1. Define vertical angles.
    2. Prove that vertical angles are congruent.
  6.  
    1. Prove that if two of the sides of a triangle are congruent, then the angles opposite these sides are also congruent.
    2. Prove that every equilateral triangle is also equiangular.
  7. Prove that every angle has exactly one angle bisector.

 

FOR CAREFUL WRITE-UP

  1. Carefully state Postulate 10.  It should be analogous to Postulate 9.
  2. Prove that if two angles are both congruent and supplementary, then they are each right angles.
  3. Prove that if two intersecting lines form one right angle, then they form four right angles.

 

Definition:  In a given plane, the perpendicular bisector of a segment is the line that is perpendicular to the segment and intersects the segment at its midpoint.

 

DUE TUESDAY, FEBRUARY 8:

FOR DISCUSSION:

  1. State and prove the angle-side-angle theorem.
  2. Prove that if two angles of a triangle are congruent, the sides opposite these angles are congruent.
  3. An equiangular triangle is equilateral.
  4. State and prove the theorem of side-side-side triangle congruence.
  5. State and prove the theorem of side-angle-angle triangle congruence.
  6.  
    1. Is it possible for two lines to intersect in such a way that three of the angles formed measure 30, 80, and 30?
    2. Carefully state conditions under which it is impossible for two lines to intersect in four angles of measures a, b, c, and d.
  7. Prove that all points on the perpendicular bisector of a segment are equidistant from the endpoints of the segment.
  8. Prove that given a point on a line in a plane, there is exactly one line perpendicular to the given line through that point.

 

FOR CAREFUL WRITE-UP:

  1. If two medians of a triangle are perpendicular to their respective sides, then the triangle is equilateral.
  2. Prove that given a point not on a given line in a plane, there is exactly one line perpendicular to the given line.

 

Definition:  Two lines are parallel if they are coplanar and do not intersect, and are skew if they are not coplanar and do not intersect.

A transversal of two lines in a plane is a third line that intersects the two lines in two different points.

DUE TUESDAY, FEBRUARY 15:

FOR DISCUSSION:

  1. Prove that if two sides of a triangle are not congruent, then the larger angle is opposite the larger side.
  2. Prove that if two angles of a triangle are not congruent, then the larger side is opposite the larger angle.
  3. Prove the triangle inequality:  in any triangle ABC, AB +BC is greater than or equal to AC.
  4. Prove that if two lines in a plane are both perpendicular to the same line, then they are parallel.
  5. Prove that there is at least one line through a given point not on a given line that is parallel to the given line.
  6.  
    1. Define alternate interior angles.
    2. Prove that if two lines are cut by a transversal, and if one pair of alternate interior angles is congruent, then the other pair of alternate interior angles is also congruent.

 

FOR CAREFUL WRITE-UP

  1.  
    1. Define exterior angle and remote interior angle for a triangle.
    2. Prove that the measure of any exterior angle of a triangle is always larger than the measure of either of its remote interior angles.
  2. Prove that if two lines are cut by a transversal, and if one pair of alternate interior angles is congruent, then the lines are parallel.

 

Euclidean geometry is named for the Greek mathematician Euclid, and it is what we have focused on so far in this course.  One important feature of Euclidean geometry is known as Euclid’s Fifth Postulate, or the Parallel Postulate.  We state it here:

Postulate 16 (Parallel Postulate):  Through a given point not on a given line, there is at most one line parallel to the given line.  [We already proved that there is at least one such line.]

In this week’s problems, we will look at some alternative choices for Postulate 16.

Amazingly, it is not possible to prove the Parallel Postulate from the other 15 postulates and their consequences.  This confounded mathematicians for a very long time.  We will see later that in non-Euclidean geometry, it is possible to have the other postulates hold true, but to have more than one parallel through a point not on a given line, or to alter things so that there are no parallels, that is, so that lines always intersect.

DUE TUESDAY, FEBRUARY 22:

FOR DISCUSSION:

  1.  
    1. Define corresponding angles for lines cut by a transversal.
    2. Prove that if two lines are cut by a transversal, and one pair of corresponding angles is congruent, then the other three pairs of corresponding angles are congruent.
  2. Prove that if two parallel lines are cut by a transversal, then alternate interior angles are congruent.
  3. If the Theorem 40 is taken as a Postulate instead of Postulate 16, it is possible to prove Postulate 16 as a theorem.  (In other words, we could have taken Theorem 40 as postulate 16.)  Assume Theorem 40 and prove Postulate 16 as a consequence.
  4. A close translation of the Parallel Postulate as Euclid gave it is:  If a third line crosses two other lines, and the interior angles on the same side of the third line have an angle sum less than 180 degrees, then the lines, if extended sufficiently, meet on the side in which the angle sum is less than 180 degrees.
    1. Restate the postulate in your own words.
    2. Prove Euclid’s version as a consequence of Postulate 16.
  5.  
    1. Prove Postulate 16 as a consequence of Theorem 42.
    2. Is Theorem 42 equivalent to the following statement:  If two parallel lines are cut by a transversal, interior angles on the same side of the transversal are supplementary.
  6. Using Postulate 16 or any of the equivalent theorems, prove that the angle sum of the three angles of a triangle is 180 degrees.

FOR CAREFUL WRITE-UP:

  1. Prove that if two lines are cut by a transversal, and if one pair of corresponding angles is congruent, then the lines are parallel.
  2. Prove that the measure of an exterior angle of a triangle is the sum of the measures of the two remote interior angles.
  3. As a consequence of the Parallel Postulate, what can you state about the angles formed by two parallel lines which are crossed by a third line?  Include a diagram.  Prove any assertions we have not yet made.

NOTE:

The quiz on March 1 will cover all the work up through the problems on this page.  It will be open notes, and you may quote from any work we have done on the quiz.

 

Assignment sheet 2

We have, up to now, done Euclidean geometry without use of coordinates.  We will now begin to look at geometry in the plane and in 3 dimensions with coordinates and we will think about congruence in a new way.

Definitions:  A function T which is 1-1 and has the plane as both its domain and codomain is called a transformation.  A transformation T is an isometry if it preserves distance.  Two objects A and B in the plane will be said to be congruent (A is congruent to B) if there is an isometry T with T(A) = B. This will be referred to as the “new definition.”

The three most commonly discussed transformations are:  reflection, rotation, and translation.  We define them here:

A reflection T across a line m is a transformation satisfying the condition that, for every point P in the plane, setting P’ = T(P), we have PP’┴m, and, letting Q be the point of intersection of the lines, PQ=P’Q.  A rotation T around a point R by an angle e, e in radians, -π < e < π is a transformation such that, for every point P in the plane, if P’=T(P), then m<P’RP = e and P’R = PR.  Notice that in the preceding definition angles have an orientation.  A translation T by a directed segment AB is a transformation such that, for any point C, if C’=T(C), then C’ completes a parallelogram C’CAB, unless C’ is on the line AB, in which case C’C = BA and C’B = CA.

 

With a new definition of congruence, we want to show that the same objects are congruent as before.  To do this, we must show objects which were congruent under the new definition are congruent under the old definition, and that objects congruent under the old definition are congruent under the new definition.  Even though we redefined congruence, as a result of 49-54, you may continue to use the old theorems about congruence to prove new results.

DUE TUESDAY, MARCH 1

FOR DISCUSSION:

  1. Show that the image of a segment under an isometry is also a segment.
  2. Show that the image of a ray under an isometry is also a ray.
  3. Show that the image of an angle under an isometry is also an angle.
  4. Show that segments congruent under the new definition are congruent under the old definition.
  5. Repeat (52) above for angles.
  6. Assume that two triangles are congruent under the new definition.  Then there is an isometry which maps one triangle onto the other.  Show that these triangles are congruent under the old definition.

As a result of the above, we know that objects congruent under the new definition are congruent under the old definition.  We are not yet able to show that objects congruent under the old definition are congruent under the new definition.  The reason for this is that, given two objects congruent under the old definition, we need to find an isometry that maps the first object onto the second.  In order for this to happen, we need to know what maps are isometries.

  1. Prove that reflections are isometries.
  2. Prove that rotations are isometries.
  3. Prove that translations are isometries.

 

FOR CAREFUL WRITE-UP:

We will begin to do some work using coordinates in the plane (and in three dimensions).  We are going to assume that coordinates (x, y) have been established for the plane, and coordinates (x, y, z) have been established for 3 dimensions.

  1. Suppose that T is a translation.  Describe T in terms of coordinates. 
  2. Describe all ways in which three lines can be arranged in a plane.

 

DUE TUESDAY, MARCH 8

 

FOR DISCUSSION:

  1. Prove that the composition of two isometries is also an isometry.
  2. Hint:  Showing congruence under the new definition means that one must describe an isometry, or a sequence of isometries (allowed because of 54), in which the image of the first object is the other object.
    1. Prove that two segments congruent under the old definition are congruent under the new definition.
    2. Prove that two angles congruent under the old definition are congruent under the new definition.
  3. Prove that two triangles congruent under the old definition are congruent under the new definition.
  4. The Pythagorean Theorem states that if ABC is a right triangle with legs of length a and b, and a hypotenuse of length c, then a2 + b2 = c2.  Prove the Pythagorean Theorem.  [There are many proofs of this theorem.  We will have at least 3 presentations of different proofs.]

FOR CAREFUL WRITE-UP:

  1. Show that with SSA, there can be two different, incongruent triangles which both satisfy SSA.

IN-CLASS EXPLORATION FOR MARCH 10:  What happens to the plane (or objects in the plane, if you prefer) if you perform two reflections?  Put another way, what is the result of performing 2 reflections across lines l1 and l2 in the plane?  There are three possible cases for the two lines of reflection:  l1 and l2 are identical, they are parallel to each other, or they intersect. 

  1. In the first case, reflecting twice across the same line puts all the points back where they started.  Prove that this is true.
  2. Prove that if the lines are parallel, the result is a translation. 
  3. If the lines are transverse, that is, if they intersect, then the result of doing the two reflections is equivalent to a rotation.  In fact, the rotation you get is a rotation around the intersection point of the two lines, and is a rotation by an angle equal to twice the angle from l1 to l2.  Here is how we will prove this assertion:  Let X be the point of intersection of our two lines.  Let A be an arbitrary point in the plane, let A’ be its image after reflection about l1, and let A’’ be the image of A’ after reflection about l2.  Notice that the image of X after each reflection is itself; X is a fixed point of the reflections.  Since we proved that reflections preserve distance, then XA = XA’ = XA’’.  Complete the proof, and be sure to cover all possible cases for the location of A:  A may be on either line, or it may be not on the lines in one of a couple of different ways.

 

DUE TUESDAY, MARCH 15

FOR DISCUSSION:

  1. Describe T in terms of coordinates, where T is a reflection across the x-axis.
  2. Describe T in terms of coordinates, where T is a rotation around the origin by an angle α.
  3. Describe, in terms of coordinates, a function T that is a rotation, but not necessarily around the origin.
  4. Describe, in terms of coordinates, a function T that is a reflection across a line other than the x-axis.

 

FOR CAREFUL WRITE-UP:

  1. Prove that the transformation T(x, y)=(x+5, 2y) is an isometry or show that it is not an isometry.
  2.  
    1. Write the distance formula for two points in the plane.
    2. Write the usual (x, y) equation-description of a circle.
    3. How does your answer to (a) fit with your answer to (b)?
  3. What is the coordinate description of a plane in 3 dimensions?
  4. Prove or disprove:  If in quadrilaterals ABCD and EFGH, angles A, C, E and G are right angles, AB = EF, and BC = FG, then the quadrilaterals are congruent.

 

NOTE:  The first exam covers problems 1-75.  It will be open notes.

IN-CLASS EXPLORATION ON MARCH 24:  We want to know what happens to the plane if you perform three reflections across 3 not necessarily distinct lines.  Based on problem 59, you know how three lines can be arranged.  For each of these possibilities, and considering different orders, you should come out with two different possible outcomes for the type of transformation you get:  a glide reflection, which is a reflection followed by a translation (the most commonly cited example of a glide reflection is an idealized set of footprints, with two symmetric feet leaving tracks equidistant from a fixed line), or a reflection.  Use the two problems below to summarize your answer.

  1. Describe which possible arrangements and orderings of reflections across 3 lines give you a reflection.  Prove your answer.
  2. Describe which possible arrangements and orderings of reflections across 3 lines give you a glide reflection.  Prove your answer.

The last case to consider is four reflections.       It can be proven that the composition of four reflections is either a translation or a rotation.  As a result of all this, it turns out that any number of reflections is equivalent to either a reflection, rotation, translation, or glide reflection.  From here, there’s a short leap to prove the Theorem:  Every distance-preserving transformation T is a composite of reflections.  You do not need to prove the theorem.  Thus the new definition of congruence is equivalent to the old definition.