Introduction

It occurs to me that there are some basic questions that you may have, but it's been such a long time since geometry that you may not know how to answer them, or they may not even occur to you. For instance, after class I was asked about using SAS, SSS properties to prove the bisection results. The underlying question being, in the order in which we learn geometry did we learn about triangles first or bisection first?

Additionally, I said in class that classically a "straightedge" is NOT a ruler, it is not marked, and that the compass "collapses". This leads to some fundamental questions. If we don't have a ruler and our compass collapses, then how do we duplicate angles or lengths?
Okay, some basics on the order in which we learned things, and then some more discussion on "duplicating" lengths and angles. Then a bit more, too.

Similarity and Congruence

  1. First thing that is developed is the idea of similarity. Two triangles are similar if
    1. the corresponding sides all have the same ratio,
      OR
    2. the corresponding angles are all equal.
      Notice that this means that if TWO corresponding angles are the same then the triangles are similar - we will call this the AA property.
    3. similarity for figures with more than three sides requires an AND instead of an OR (can you think of an example proving this? how about a rhombus and a square? They have the sides property, but not necessarily the angles property. How about non-similar with the angles property, but not the sides property?).
  2. Next we developed the idea of congruence for triangles. Two triangles are congruent if
    1. (definition) the ratio of corresponding sides is 1.
    2. Note that this gives us the SSS (side-side-side) property for congruence - IT'S the definition!
    3. Now how can we extend the AA similarity to congruence? by requiring AA and one pair of corresponding sides are equal - we'll call this the AAS property for congruence.
    4. We also have a third congruence property. Two triangles are congruent if two pairs of corresponding sides are equal and the angle between the two pairs of sides is equal - we'll call this SAS.
      (Can you see why this is true? Take a triangle and erase one side. Is there any way to put a side back in and have it a different length without changing the angle? That's not the proof, but it is an insight. Note that the angle must be the one formed by the two sides considered corresponding in the two triangles. Can you find two different triangles that would satisfy an ASS property?)
Note that intuitively, congruence means that we can slide on of the triangles so that it lies exactly on top of the other one. BUT, we'll also take the view that if we can flip the triangle and then make it align that it will be congruent too. (This is called a reflection property).

Basic Rules About Intersecting lines.

Recall the basic rules about lines intersecting and the angles involved. They are illustrated by the pictures here
Basic Rules about angles and intersecting lines.

Basic Constructions

  1. Bisection of a line segment was done in class on Tuesday too. Note that the bisector constructed is perpendicular to the line segment, and is sometimes called a perpendicular bisector.
    Construction of perpendicular bisectors
  2. Bisection of an angle - covered in class Tuesday.
    Construction of angle bisectors can be found here
  3. Constructing a line perpendicular to given line through a given point.
    Construction of perpendiculars through a given point.
  4. Constructing a integer number line. In geometry we use constructions based on the use of the straightedge and compass. The straightedge is not marked like a ruler. Effectively, we get to define a unit length, just by choosing a special line segment.
    Construction of a line with equal length segments.

More involved constructions

  1. We can use the above number line to divide any segment into an arbitrary number of equal length subintervals. We have to understand how to construct perpendiculars, and properties of similar triangles to accomplish this.
    Dividing a line into equal length segments.
    Dividing a line into equal length segments(continued).
  2. Duplicating a length. Remember that if we want to duplicate a length we can't just use the compass or straightedge to measure it. The straightedge is not marked and the compass will collapse. (Note that actually the construction technique will be more accurate than your eye anyway, if we do it right.) Note that we actually did duplicate a length in the previous construction, but that was a very simple case.
    We will consider two additional cases: duplicating a length on the same line and duplicating it on a different line.
    Duplicating a length.
    THEOREM: We can use a rigid compass to duplicate lengths.
    Proof: Well, we just showed how to copy a length anywhere we wanted to by using a collapsing compass and a straightedge. Thus if our compass is rigid (does not collapse when we lift it of the paper) then we might as well take the short cut of using the compass to measure. Again this is actually more acurate than using a ruler in real life.
  3. Duplicating an angle. How do we reproduce an angle in a different place? We are going to use the shortcut of allowing ourselves to use the compass to measure now. (Challange: how would you do this if you had to use a collapsing compass?)
    We did this in class on Thursday. Can you still do it?