Assignment 2 (45 points, Due Monday, October 2)

Reading

Look over the whole chapter. Skip the "real-world applications" to family disputes but do pay attention to the sections on geometry, pp. 12-36. Your goal should be to understand the role of undefined terms, axioms, "if-then" statements and their converses, "if and only if" statements, "begging the question" and the role of diagrams, good and bad. You may not take all this in on the first reading; it will be useful to refer back to this section from time to time.

Study Principles 1 through 5. These will be our axioms, but at the beginning we will only focus on the case of Principle 5 where the constant of proportionality is 1 (the case of congruent triangles, not similar triangles).

Main ideas

• Basic properties of isosceles triangles, including Theorems A, B, C in the reading.
• Use of triangle congruence criteria in arguments. Also, which "criterion" is not true?
• Converses and "if and only if" statements.

Problems to Turn In

In these proofs you can use Principles 1 — 5 and what was proved in Assignment 1 and also in Chapter 1 of B&B..

2.1 Construction of perpendicular bisector (10 points)

1. Draw a segment AB and construct the perpendicular bisector as explained on pp. 172-3 of B&B.
2. Prove why this construction works, based on the axioms and what was proved in Assignment 1.

2.2 Construction of angle bisector (10 points)

1. Draw angle ABC and construct the angle bisector as explained on pp. 174 of B&B.
2. Prove why this construction works, based on the axioms and what was proved in Assignment 1.

2.3 Perpendicular Bisector as locus (10 points)

1. Prove problem 16 on page 63 of B&B.
2. Combine (a) with what has been proved before to prove the following statement: Given a segment AB, the perpendicular bisector of AB is the set of points equidistant from A and B. (Be careful, this is an if and only if statement.)

2.4 Constructions (15 points)

Carry out these constructions with straightedge and compass. You may wish to read more of Chapter 6 in B&B for ideas. Make each construction large enough to take up one side of a page. Do not make tiny constructions.

1. Draw 3 triangles. One should be an acute triangle, one an obtuse triangle, and one a right triangle. They should be general, or random (i.e., not isosceles or special angles). Construct with straightedge and compass the perpendicular bisectors of the 3 sides of each triangle. What do you observe?
2. Draw a segment AB. Then construct with straightedge and compass points C and D so that ABCD is a square.
3. Draw an obtuse triangle ABC. Construct the 3 altitudes with straightedge and compass (construct the whole altitude line, not just the segments