Consider the axiomatic system and theorem below:
Axiom 1: For any two points, there exists a line so that each of the two points is on the line.
Axiom 2: There exist at least two points on any line.
Axiom 3: There exist at least three distinct points.
Axiom 4: Not all points are on the same line.

Theorem 1: Each point is on at least two distinct lines.

A. List the undefined terms involved in the given axiomatic system.

1. Prove Theorem 1 using the axioms as needed.

Note: You do not need to use all four axioms. Be sure to indicate the axioms you use in your proof, but do not mention any axioms you do not use.

B. Acknowledge sources, using in-text citations and references, for content that is quoted, paraphrased, or summarized.

 

Consider the theorems below:

The following is a theorem of Euclidean geometry:
Euclidean angle sum theorem: The sum of the measures of the angles of a triangle is 180°.

Theorem 1: An exterior angle of a triangle is greater than either of the nonadjacent interior angles of the triangle.

A. Using the Euclidean angle sum theorem, prove Theorem 1. Your proof must refer to the definitions provided below.

Definitions:

• adjacent: Two angles are adjacent if they share a common vertex and common side, and they do not overlap. Otherwise, the two angles are nonadjacent.

• supplementary:Two angles are supplementary if their measures sum to 180°.

• exterior:An angle that is both adjacent and supplementary to an angle of a triangle is an exterior angle of the triangle.

1. Explain why this theorem is also true in hyperbolic geometry.

2. Explain why this theorem is not true in spherical geometry.

B. Acknowledge sources, using in-text citations and references, for content that is quoted, paraphrased, or summarized.

 

A. Use analytic methods to compute the coordinates for point B, showing all work.

B. Use analytic methods to demonstrate that one of the following triangles is an isosceles right triangle, showing all work:

• ∆AFG

• ∆ADG

• ∆EHJ

• ∆CJK

C. Use analytic methods to demonstrate one of the following, showing all work:

• ACKF is an isosceles trapezoid

• DEHG is a parallelogram

D. Use analytic methods to demonstrate that one of the following is a square, showing all work:

• CEDB

• CJGA

E. Acknowledge sources, using in-text citations and references, for content that is quoted, paraphrased, or summarized.

 

A. Use Geometer’s Sketchpad to graph a triangle and two transformations as described below. You should submit a single image showing the original triangle specified in part A1 and the two transformations specified in parts A2 and A3 with all points labeled.

1. Graph the triangle with vertices A(4,4), B(3,6), and C(–2,2).

2. Graph the image ∆A'B'C' of ∆ABC under the translation t2,–4.

3. Graph the image ∆A″B″C″ of ∆A’B’C’ under a reflection about the line y = –2x – 3.

B. Use Geometer’s Sketchpad to graph a triangle and two transformations as described below. You should submit a single image showing the original triangle specified in part B1 and the two transformations specified in parts B2 and B3 with all points labeled.

1. Graph the triangle with vertices D(4,0), E(6,1), and F(3,2).

2. Graph the image ∆D'E'F' of ∆DEF under a reflection about the line y = x.

3. Graph the image ∆D″E″F″ of ∆D’E’F' under a counterclockwise rotation about (1,1) by 180°.

C. Acknowledge sources, using in-text citations and references, for content that is quoted, paraphrased, or summarized.

 

A. Demonstrate, using a counter example, that the product of two isometries P and Q is not always commutative (i.e., PQ does not always equal QP).

B. Acknowledge sources, using in-text citations and references, for content that is quoted, paraphrased, or summarized.