DUE THURSDAY, AUGUST 28:  In order to be able to do geometry, we need a common set of definitions.  Definitions are important because all results depend on the definition used.  It is often possible that more than one definition is acceptable.

  1. Write definitions for square, rectangle, quadrilateral, parallelogram, rhombus, kite, and trapezoid.  Make your definition as formal and clear as possible.  Be prepared to discuss why you chose your definition, and what you think makes a good definition.
  2. A goat is tied to a post on the outside of a barn using a chain.  The barn has a square base.  All around the barn is a field of grass.  How much grazing area does the goat have?  Notice that this problem is very open-ended.  Many details are missing.  Begin by making up some special numbers or cases, but try to expand your solution to cover as many possibilities as you can; try to make a completely general solution.


  1. Write definitions for scalene, isosceles, and equilateral triangle, and for circle.
  2. Euclidean geometry is named for the Greek mathematician Euclid.  Euclidean geometry is the type of geometry studied in high school, and the type of geometry we will focus on for the first 8 weeks or so of this course.  One important feature of Euclidean geometry is known as Euclid’s Fifth Postulate, or the Parallel Postulate.  A postulate is not something that can be proven; it is something that is accepted as true.  All mathematics is built on certain definitions and postulates.  They are the basic building blocks of every system of mathematics.  One way to state the Parallel Postulate 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. Explain why this is called the Parallel Postulate.
    2. Draw a diagram to illustrate the postulate as it is stated.  Then rewrite the postulate in your own words.
    3. It is known that the Parallel Postulate is equivalent to the following statement:  The angle sum of the three angles of a triangle is 180 degrees.  Prove the equivalence.  (Note that this requires doing two proofs.)


  1. Define vertical angles, and prove that vertical angles are congruent.
  2. 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 all your assertions.
  3. What does the word “is” mean?  Look at the following examples:
    1. A square is a rectangle.
    2. A scalene triangle is a triangle with no two sides having a common length.
    3. An equilateral triangle is isosceles.
    4. A square is a rhombus.

Describe how the word “is” is used in each example above.  Does “is” always have the same meaning?  How could some or all of the examples be reworded to be clearer?


  1. Define congruence.  You may start by writing a definition that is specific to the item being considered, for instance, “Two triangles are congruent if …” but you should then try to find a definition that covers congruence of all kinds.
  2. Define transformation, and the three kinds of transformation:  reflection, rotation, and translation.  How are these related to congruence?
  3. In order to study translations, we will want to consider the plane as having the coordinates (x, y).  Suppose that T is a function which is also a translation.  Describe T in terms of coordinates.  Do the same for the case where T is a reflection across the x-axis, and the case where T is a rotation around the origin by an angle α.
  4. Define ellipse, hyperbola, and parabola.
    1. Write the distance formula for two points in the plane.
    2. Define circle.
    3. Write the usual (x, y) equation-description of a circle.
    4. How does your answer to (c) fit with your answer to (b)?


  1. This is really a continuation of problem 10.  We want to consider more general types of transformations.
    1. Build upon your solution to number 10 and describe, in terms of coordinates, a function T which is a rotation, but not necessarily around the origin.
    2. Describe, in terms of coordinates, a function T which is a reflection across a line other than the x-axis.
    1. A congruence transformation will be any transformation that preserves congruence; such a transformation is also called an isometry.  Define what it means for a transformation to be an isometry.
    2. Prove that any translation is an isometry.
    3. Prove that the transformation T(x, y)=(x+5, 2y) is an isometry or show that it is not an isometry.
  3. What happens to the plane (or objects in the plane, if you prefer) if you perform two reflections?  Make your answer as complete as possible, and include proofs.


  1. Can the following definition of line be extended to three dimensions?  Explain clearly.  A line is a set of points of the form {(x, y)| ax + by = c, where a, b, and c are real numbers, and a^2 + b^2 ≠ 0}.
  2. What happens to the plane if you perform three reflections?  Give a complete answer with proofs.
  3. Ellipses, hyperbolae, parabolas, and circles are all examples of conics.  In general, a conic in the plane can be written in the form (*) ax^2+bxy+cy^2+dx+ey+f=0, with a, b, c, d, e, ,f, real numbers.
    1. What are the “usual” forms that you see for equations of parabolas, ellipses, and hyperbolae?
    2. Find a criterion for determining when an equation of the form (*) is a circle, a parabola, an ellipse, or a hyperbola? [Hint:  Use transformations to help you.]


(Problem 18 continued)  We saw that if, in the equation for 18, b = 0, then we have criteria for determining when the equation describes an ellipse, a circle, a hyperbola, or a parabola.  We want to figure out what to do when b ≠ 0, but we will take our time developing an answer to this question.  Part (c) is the beginning of an answer.

    1. Rather than try to solve this problem by looking at what to do with b, let us work from the other direction.  We will produce some equations which have b ≠ 0.  One form for an ellipse is (1) x^2/a^2 + y^2/b^2=1 (do not confuse the b here with the one above).  This describes any ellipse with foci aligned either horizontally or vertically and centered at the origin.  What if we want to move the ellipse away from the origin?  The solution is to translate the ellipse.  If we want the ellipse to be centered at (h, k), then we are translating by h in the x-direction and k in the y-direction.  If the new coordinates after translation are (x’, y’), then (x’, y’) = (x + h, y + k), or, equivalently, (x’ – h, y’ – k) = (x, y).  The original coordinates satisfied equation (1) above, so what equation do the new coordinates satisfy?
    2. Now, rather than translate the ellipse, we are interested in rotating the ellipse.  We will use Alex’s form of the rotation equations.  Start from equation (1).  If we rotate the ellipse which is described in (x, y) coordinates by equation (1) around the origin by an angle α, and call the new coordinates (x’’, y’’), then (x’’, y’’) = (x cos α – y sin α, x sin α + y cos α).  As in (c), we need to solve for x and y in terms of x’’ and y’’.  Remember that since the new coordinates were obtained from the old ones by a rotation by α, the old ones can be obtained from the new ones by rotation by –α. Therefore, (x’’ cos (-α) – y’’ sin (-α), x’’ sin (-α) + y’’ cos (-α)) = (x, y).  Substitute x and y in equation (1) with the new coordinates to obtain the equation for the rotated ellipse.  Simplify the equation by multiplying out and getting the equation in the form of (18).What are the values of the coefficients of x^2, xy, y^2, x, y, and the constant term in terms of a, b, and sin(α) and cos(α)?  Notice that you need to simplify the sines and cosines using sin(-α) = -sin(α) and cos(-α) = cos(α).  What relation(s), if any, do the coefficients of x^2, xy, and y^2 satisfy?  This is our first example of a conic with a nonzero coefficient of xy, i.e., with b ≠ 0.
    3. Conics received their name because they are the intersections of cones (in 3 dimensions) with planes.  The equation of a cone in 3 dimensions is z^2=x^2+y^2.  We know from a recent discussion that the equation of a plane in three dimensions is ax + by + cz + d = 0.  If a cone and a plane intersect, then any point (x, y, z) on the intersection must satisfy both the plane equation and the cone equation.  Solve for one of the variables x, y, or z in the plane equation and plug it into the cone equation.  This is the general form of the intersection.  Now try some special cases:  look at planes cz + d = 0 with some particular c and d values, if you like; look at planes where one or more of the constants a, b, c, or d are 0 (there are many different cases here); then look at some planes with all of the constants nonzero.  In each case, discuss what shape the intersection gives you.


    1. Prove that reflections preserve distance, that is, if r is a reflection map, d denotes distance, and P and Q are two points in the plane, then d(P,Q)=d(r(P), r(Q)).  We will use this later.
    2. 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.  In the first case, reflecting twice across the same line puts all the points back where they started.  If the lines are parallel, the result is a translation.  Try at least two examples to convince yourself that this works.  Then determine exactly what translation results from reflecting across l1: ax + by = c and then l2:  ax + by = c’, where a, b, c, and c’ are fixed real numbers.  You may need to use one of our formulas for the coordinate change caused by a reflection, or you can find a way around this. 
    3. If the lines are transverse, that is, if they intersect, then the result of doing the two reflections is equivalent to a rotation.  Try a couple of examples and see if you believe this is true (I might not be as trustworthy as I look).  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 (we will need to do several different cases for A), 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, by 19(a), reflections preserve distance, then d(X,A) = d(X, A’) = d(X, A’’).  Since X is fixed and A and A’’ are two points at the same distance from X, they are on a circle centered at X.  This proves that the two reflections result in a rotation (you may want to spend a moment considering how that proves it).  It remains to show that the angle of rotation is the angle I claimed above.  This is your job.  Try all possible cases for the location of A:  A may be on either line, or it may be between the lines in one of a couple of different ways.  Draw a diagram and find the angle.
    4. Now we are ready for three reflections.  There are many cases, but there are only two different kinds of end results.  Depending on the original three lines’ position relative to each other, three reflections result in either a reflection or what’s known as a glide reflection—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.  Your job is to determine which positions give you a reflection and which positions give you a glide reflection.  There are many cases.  Be systematic and try to analyze each one.


  1. The last case to consider is four reflections.  The composite of four reflections is either a translation or a rotation.  I will not ask you to work through this case.  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.

We have done much of the work of proving this theorem.  I am not going to ask you to complete the proof.  Instead, we will turn to the idea of congruence.  We need some definitions:  A function T which is 1-1 and has the plane as both its domain and codomain is called a transformation.  Two objects A and B in the plane will be said to be congruent (A is congruent to B) if there is a transformation T, which preserves distance, and with T(A) = B. Prove that congruence is an equivalence relation by going through the steps below:

    1. Reflexive:  Show that A is congruent to A.  This amounts to finding an appropriate transformation to satisfy the definition above.
    2. Symmetric:  Show that if A is congruent to B, then B is congruent to A.
    3. Transitive:  Show that if A is congruent to B, and B is congruent to C, then A is congruent to C.
  1. Compare these definitions for “line” in three dimensions.  Alex’s Calculus definition:  a line is the set of all points (x, y, z) = (x0 + ta, y0 + tb, z0 + tc) for some t є R, where x0, y0, z0, a, b, and c are all fixed real numbers.  Intersection of Planes definition:  a line is the set of all point (x, y, z) such that ax + by + cz + d = 0 and ex + fy + gz + h = 0, where a, b, …, h are all fixed real numbers, and the planes are not identical and not parallel. 
    1. Given x0, y0, z0, a, b, and c, find two planes (that is, find appropriate coefficients a through h) such that the intersection is the line in the AC definition. 
    2. Is your answer to (a) the only possible answer?
    3. Given two planes (and hence the coefficients a through h), find the six numbers x0, y0, z0, a, b, and c that fit the AC definition.  Hence the definitions are equivalent.


  1. Now that we have determined what congruence means, we will spend some time focusing on proving congruence is what we think it is.
    1. Prove that two line segments AB and CD having the same length are congruent.  (Find the appropriate transformation.)
    2. Prove that two angles with the same measure are congruent.
    3. Prove SAS triangle congruence.
  2. A farmer, standing at point A, wants to fill her bucket of water and deliver the water to the horse trough at point B.  The farm is along one side of a straight river, so the farmer needs to walk over to the river, fill the bucket, and take it to the trough.  Find the shortest path from point A, to any part of the river, to point B.
  3. Conics again:  Find planes that, when intersected with the cone z^2 = x^2 + y^2, will give you (1) a circle, (2) an ellipse, (3) a hyperbola, (4) a parabola, (5) intersecting straight lines, (6) a single straight line, and (7) a single point.  Discuss your approach to solving each of these cases.


  1. Prove ASA congruence for triangles.  A suggestion:  Aim to map the included side (the “S” in ASA) onto its corresponding side in the other triangle.
  2. Show that with SSA, there can be two different, incongruent triangles which both satisfy SSA.
  3. 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.


  1. We will repeat the steps of 18(c-d) for parabolas.  Follow the steps below.
    1. Take the equation y=ax^2 and find its image after translation by (h, k).
    2. Find its image after rotation by α. 
    3. As a new step, take the rotated parabola and translate it by (h,k).
    4. If the parabola is rotated by an angle of 3π/4 radians, what equation should you get? (This is a familiar algebra II equation.)  Verify your answer by using α = 3π/4 in your solution to (b).
  2. Right triangles
    1. 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.  (I will be disappointed if, as a class, we come up with less than 3 proofs of this theorem.  I will give you one diagram in class which suggests one possible proof.)
    2. Show with a diagram how any triangle (acute, obtuse) can be seen as a sum of two right triangles, and in the case of the obtuse triangle, can also be seen as the difference of two right triangles.
    3. What is the formula for the area of a triangle?
  3. In measuring length, the unit of length is 1 inch, 1 foot, etc, or 1cm, 1m, 1km, etc.  In what units is area measured?


  1. The unit of measurement for area is a square, with length 1 unit on each side, and having an area of 1 unit2, or 1 u2.  Explain how you can derive that the area of any rectangle is equal to bh, where b is the length of the base and h is the height of the rectangle.
    1. Build upon your answer to 31 to explain how the area of any right triangle is bh/2, where b and h are the lengths of the legs of the triangle.
    2. For any (not necessarily right) triangle, define base and height.
    3. Use 29(b) and 32(a) to show that the area of any triangle is bh/2, where b and h are the base and height of the triangle.
    4. Prove that the area of a trapezoid ABCD, with side AB parallel to side CD, is equal to ˝ (length of AB + length of CD(height of the trapezoid).  You may want to use the results of part (c).
    5. Define base and height in a parallelogram.  Prove that the area of a parallelogram is bh, where b is the base and h is the height.
  3. The law of cosines states that for any triangle ABC, if c=AB, b=AC, and a=BC and C=m<ABC, then c2 = a2 + b2 – 2ab cos C.  Use 29(b) to help you prove the law of cosines.

Study for Thursday’s exam.


More to come…