Systems of Test File Spring 2010

Test 1

1.) Consider a geometry with undefined terms point, line, and contains (or belongs) and the following axioms.

Axiom 1. Given any two distinct points, there is exactly one line containing both of them Axiom 2. Each line contains at least two points. Axiom 3. There exist at least three distinct points. Axiom 4. Not all points are contained on the same line.

a.) Prove that the intersection of two distinct lines contains at most one point. b.) Prove that there exist three distinct lines that do not contain a common point.

2.) Discuss what we mean by the concepts of independence and consistency in the context of an axiomatic system.

3.) Discuss the contributions to geometry of two of the following men: Thales, Pythagoras, Hippocrates, Plato, Eudoxus, Euclid, Clint Eastwood.

4.) Suppose point, line, contains (belongs to), end point and congruence (of segments and ) are taken as undefined terms. Define the following terms.

a.) A is between B and C . ____ b.) AB  c.) AB d.) BAC

5.) If a and b are the lengths of the legs of a right and c is the length of the hypotenuse, Babylonian geometers approximated the value of c by the formula  a 2  c  b    . Is this formula correct? If not, how does it compare with the  2b  correct value for c?

6.) The undefined terms for a geometry are point, line, and on. The axioms are:

Axiom 1. There exist exactly four points. Axiom 2. Any two distinct points have exactly one line on both of them. Axiom 3. Each line is on exactly two points. a.) Construct a model for this geometry. b.) Based on this geometry, define parallel lines. c.) Prove that each distinct line has exactly one line parallel to it. (You may use the theorem that, in this geometry, each point has exactly three lines on it.)

Test #2

1.) (Assume the Euclidean .) Prove: The opposite sides of a are congruent (definition of parallelogram - with both pairs of opposite sides parallel).

2.) Prove that if two lines are intersected by a transversal such that a pair of alternate interior angles is congruent, then the lines have a common perpendicular.

3.) Vertical angles are congruent.

4.) (SASAS) If the vertices of two convex are in one-to-one correspondence such that three sides and the two included interior angles of one quadrilateral are congruent to the corresponding parts of a second quadrilateral, then the quadrilaterals are congruent.

Hilbert

Exercises 1.) Every line contains an infinite number of points.

2.) Prove that two lines that are parallel to the same line are parallel to each other.

These all use SMSG and Euclid's first four postulates. 1.) Prove that the Euclidean parallel postulate is equivalent to the following statement: If two parallel lines are intersected by a transversal, then the pairs of interior angles on the same side of the transversal are supplementary.

2.) Prove that the Euclidean parallel postulate is equivalent to the following: The diagonals of a parallelogram bisect each other.

3.) If we define the sum of triangle ABC as m∠A + m∠B + m∠C, prove that the Euclidean parallel postulate implies that every triangle has an angle sum of 180.

4.) An has three congruent sides. Prove that an equilateral triangle is also equiangular.

Test #3

Euclidean Parallel Postulate - Given a line and a point not on the line, there exists exactly one line on the point parallel to the given line.

SMSG - (also assume Euclid's first four postulates) Note: If a problem is labeled "Chapter 3," then you may only assume Euclid's first four postulates and SMSG postulates 1-15.

1.) Define the following: a.) b.) c.) Similar

2.) Which of the following statements are true in neutral geometry: _____ The Alternate Interior Angle Theorem. _____ The Converse of the Alternate Interior Angle Theorem. _____ A Saccheri quadrilateral exists. _____ A exists. _____ The angle sum of any triangle is 180.

3.) Prove that the diagonals of a parallelogram bisect each other.

4.) Prove that the altitude to the hypotenuse of a divides the triangle into two smaller , each of which is similar to the original right triangle.

5.) State and prove the Pythagorean Theorem. (You may assume without proof that the altitude to the hypotenuse of a right triangle divides the triangle into two smaller triangles, each of which is similar to the original right triangle.)

6.) State and prove the Alternate Interior Angle Theorem.

7.) (Chapter 3) Prove: If ABCD is a quadrilateral, where A and B are right angles and C  D, then ABCD is a Saccheri quadrilateral.

Test #4

1.) Prove: If a diameter of a circle is perpendicular to a chord of the circle, then the diameter bisects the chord. 2.) (Assume the Euclidean Parallel Postulate.) Prove: The opposite sides of a parallelogram are congruent. 3.) Prove that in neutral geometry, the angle sum of any convex quadrilateral is less than or equal to 3600 . 4.) Prove that if a convex quadrilateral ABCD is such that opposite sides are congruent, then it is a parallelogram. (You may assume the Euclidean Parallel Postulate.) 5.) The following picture shows a circle with center, A. For each of the following, identify an example from the picture.

a.) chord ______

D b.) tangent ______E c.) diameter ______

d.) secant ______F C A e.) central angle ______B f.) semicircle ______

G g.) minor arc ______I h.) major arc ______H i.) inscribed angle ______J

j.) radius ______

6.) Define the angle of parallelism. (Feel free to use the picture on #7.)

7.) Prove: If T-R-S and RPQ and RPQ' are the angles of parallelism for P and and P and , respectively, then mRPQ = mRPQ'. (Assume PR is perpendicular to TS)

P

Q' Q

T R S

Final Exam

1.) Define the following : a.) A is between B and C ____ b.) AB  c.) AB d.) BAC e.) Lambert quadrilateral f.) ABC g.) linear pair h.) Congruent polygons i.) Saccheri quadrilateral j.) Similar polygons k.) chord l.) circumcenter of a triangle m.) median of a triangle n.) secant of a circle o.) tangent to a circle 2.) Consider a geometry with undefined terms point, line, and contains (or belongs) and the following axioms. Axiom 1. Given any two distinct points, there is exactly one line containing both of them Axiom 2. Each line contains at least two points. Axiom 3. There exist at least three distinct points. Axiom 4. Not all points are contained on the same line. a.) Prove that the intersection of two distinct lines contains at most one point. b.) Prove that there exist three distinct lines that do not contain a common point. 3.) (Assume the Euclidean Parallel Postulate.) Prove: The opposite sides of a parallelogram are congruent (definition of parallelogram - quadrilateral with both pairs of opposite sides parallel). 4.) Prove that vertical angles are congruent. 5.) Prove that the Euclidean parallel postulate is equivalent to the following statement: If two parallel lines are intersected by a transversal, then the pairs of interior angles on the same side of the transversal are supplementary. (Remember, "A is equivalent to B" means you have to prove "A implies B" and "B implies A.") 6.) If we define the angle sum of triangle ABC as m∠A + m∠B + m∠C, prove that the Euclidean parallel postulate implies that every triangle has an angle sum of 180. 7.) Prove: If a diameter of a circle is perpendicular to a chord of the circle, then the diameter bisects the chord. 8.) Given AB=BC and AD= DC. Prove BD  AC 9.) (Assume the Euclidean Parallel Postulate) Straight Edge/Compass construction. Consider the following figure. It began with line segment . I then constructed segment . Next I constructed and such that D, F, E and G are collinear and = = . I then constructed . Lastly I constructed and such that || || . (Note: a number of segments and circles used in the construction were hidden.)

G

E

F

D C B A

Do the following. a.) Explain how we can construct , and such that D, F, E and G are collinear and = = . b.) Explain how can be constructed such that || . c.) This construction produces a trisection of . Explain why. 10.) Prove: If T-R-S and RPQ and RPQ' are the angles of parallelism for P and and P and , respectively, then mRPQ = mRPQ'. (Assume PR is perpendicular to TS)

P

Q' Q

T R S