Geometry

MATH 334, College Geometry Spring, 2015 Assignment No. 11 Sections 6.1 - 6.5

Instructions: Complete the following problems on separate sheets of paper. Your work must be organized and must follow a logical process. Correct use of notation and symbols as adopted in class is required. Proofs must be accompanied by figures.

1. Consider the four triangle congruences, SAS, ASA, AAS, and SSS.

(a) Do all four congruences hold in Euclidean geometry? Explain why or why not.

(b) Do all four congruences hold in hyperbolic geometry? Explain why or why not.

(c) Are there any other congruence theorems for triangles that hold in Euclidean geometry? If so, point out where the proof is.

(d) Are there any other congruence theorems for triangles that hold in hyperbolic geometry? If so, point out where the proof is.

2. Given distinct lines `, m, and n, let ` || m and ` || n.

(a) In neutral geometry, what can we conclude about m and n with respect to parallelism? Support your answer with an appropriate theorem or counter-example.

(b) In Euclidean geometry, what can we conclude about m and n with respect to parallelism? Support your answer with an appropriate theorem or counter-example.

(c) In hyperbolic geometry, what can we conclude about m and n with respect to parallelism? Support your answer with an appropriate theorem or counter-example.

3. Let ! AB and

! CD admit a common perpendicular. Let

! PQ be the common perpendicular with

A⇤P ⇤B, C ⇤Q⇤D and A and C lie on the same side of ! CD.

(a) In Euclidean geometry, is the common perpendicular unique? Explain why or why not.

(b) In Euclidean geometry, is ! AB ||

! CD? Explain why or why not.

(c) In hyperbolic geometry, is ! AB ||

! CD? Explain why or why not.

(d) In hyperbolic geometry, are the rays �! PA and

�! QC limiting parallel rays. Explain why or

why not.

(e) In hyperbolic geometry, are the rays ��! PB and

��! QD limiting parallel rays. Explain why or

why not.

4. Let ��! PD and

�! AB be limiting rays. Let E be a point on the ray opposite to

��! PD and F be a

point on the ray opposite to �! AB. Compare the angle of parallelism for point E and

! AB with

that of point P and ! AB. Explain why they are the same or how they are di↵erent.