Assignments for Week 1

Reading for Week 1

B&B, Chapter 1.

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. You should 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.

B&B, Chapter 2.

This chapter starts with the basic Ruler and Protractor Axioms that give the assumed properties of lines and angles. It also states SAS for similar triangles. The first week we will be working woth SAS and also ASA and SSS for congruent triangles (see below).

Heilbron, pp.54-57 and information in class

Read about SSS, SAS, ASA congruence criteria

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.
• Perpendicular bisector as locus
• A way to prove concurrence

Exercises 1 (Due Wed 9/29)

Exercises are counted as part of the participation grade. They may be practice problems, problems to focus your reading, or a warmup for a class discussion or activity. They will typically be inspected rather than grades. Since they are designed to help at a particular time of the course, to count they must be ready at the beginning of class on the date due.

E1.1.

Write a proof of Theorem B (B&B Chapter 1) using one of the 3 criteria for triangle congruence.

E1.2 .

Write a proof of Theorem C (B&B Chapter 1) using one of the 3 criteria for triangle congruence.

E1.3. If-Then

Exercises from B&B, pp. 24-25, #5, 6. Converses on p. 28, #1-4

Exercises 2 (Due Fri 10/1)

Exercises are counted as part of the participation grade. They may be practice problems, problems to focus your reading, or a warmup for a class discussion or activity. They will typically be inspected rather than grades. Since they are designed to help at a particular time of the course, to count they must be ready at the beginning of class on the date due.

E2.1.

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.

E2.2 .

Draw a segment AB. Then construct with straightedge and compass points C and D so that ABCD is a square.

E2.3.

Draw an obtuse triangle ABC. Construct the 3 altitudes with straightedge and compass (construct the whole altitude line, not just the segements).

Assignment 1 (55 points, Due Mon 10/4)

In these proofs you can use the triangle congruence criteria, SAS, SSS, ASA and the basic properties of isosceles triangles stated in Theorems A, B, C of B&B, p. 20-23.

1.1 Kites and Rhombi (15 points)

If a quadrilateral ABCD has AB = DA and BC = CD, we call this quadrilateral a kite. (Compare B&B, page 24, problems #2 and #3, but notice the difference in labeling the vertices.)

(a) State and prove a proposition about a relationship between angles of such a kite.

(b) State and prove a proposition about a relationship about angles between the diagonal lines, line AC and line BD, of such a kite. Can you say something stronger about the relationship between AC and BD than just the angle?

Caution: Do your proofs include all cases?

If a quadrilateral ABCD has a four sides equal, we call this quadrilateral a rhombus.

(c) State and prove the strongest proposition you can about the diagonals of a rhombus.

1.2 Construction of perpendicular bisector (10 points)

Prove this statement: Given a line segment AB, construct two circles of the same radius, with centers A and B. If the two circles intersect at point R and R', then line RR' is the perpendicular bisector of segment AB.

1.3 Trisecting a right angle. (10 points)

(a) Carry out an example of the construction on B&B, page 189.

(b) Explain why this works. What is the angle measure of the angle you get?

1.4 Concurrence of perpendicular bisectors (10 points)

Given any triangle ABC, prove that the perpendicular bisectors of the sides are concurrent.

Hint: Be careful not to assume your conclusion (Begging the Question). Try intersecting two of the lines and proving the point of intersection is on the third line.

1.5 Dirichlet domains (10 points)

Given two points A and B, the perpendicular bisector of AB divides the plane into two half-planes. Prove that the half-plane containing A is the set of points in the plane closer to A than to B. (Do you have to prove separately that the half-plane containing B is the set of points in the plane closer to B than to A?)

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