Geometry

Return Exam to: Douglas Poole Fax: (308) 432-6434 e-mail: [email protected]

MATH 334, College Geometry Name Exam 3, Spring, 2015

Instructions: Credit will be given for correct steps leading to an incorrect solution or a partially correct proof; little credit will be given for unsupported answers or unsupported statements in proofs. Correct use of notation and symbols as adopted in class is required. Accurately drawn figures must accompany arguments. All work must be completed in the space provided on the exam paper.

Section 1: Problems 1 to 3 are restricted to axioms, definitions, and theorems for neutral geometry. No parallel postulate is stated for this section.

1. (6 points) In the figure to the right, B, D, F , and H lie on ! AC,

! CE,

! EG, and

! AG, respectively.

K is the intersection of ! BF , and

! DH. All right angles and all pairs of congruent sides are marked.

(a) Identify all Saccheri quadrilaterals in the figure.

(b) Identify all Lambert quadrilaterals in the figure.

A B C

D

EFG

H K

2. (12 points) Prove that the summit angles of a Saccheri quadri- lateral are congruent. Assume no other properties for Saccheri quadrilaterals.

3. (12 points) In �ABC, the foot of the perpendicular from A to ! BC

lies between B and C. Prove that \ABC and \ACB are acute.

Section 2: For problems 4 to 6 the Euclidean Parallel Postulate is valid as are all theorems proved using the Euclidean Parallel Postulate.

4. (6 points) Three sides of a triangle measure 9 cm, 40 cm, and 41 cm. Determine if the triangle is a right triangle. Justify your answer.

5. (20 points) Let �ABC be a right triangle with the right angle at vertex C. Let E be the point at which the bisector of \ABC (E/U of angle bisectors) intersects

! AC (Crossbar theorem). Drop

a perpendicular from E to ! AB and call the foot F (E/U of per-

pendiculars).

A

BC

(a) Draw a diagram in the box to the right showing the construc- tions. Add details as needed to complete (b) through (e).

(b) What theorem supports A⇤F ⇤B?

(c) Prove that �BCE ⇠= �BFE.

(d) Prove that �AFE ⇠ �ACB.

(e) Let a = BC, b = AC, c = AB, and x = EF . Determine lengths of other sides in terms of a, b, c, and x. Use the Fundamental Similar Triangles Theorem and algebra to prove that a2 + b2 = c2.

6. (10 points) Provide a response that completes each statement.

(a) The orthocenter of a triangle is the point of concurrency of three .

(b) The circumcenter of a triangle is the point of concurrency of three .

(c) The centroid of a triangle is the point of concurrency of three .

(d) The three triangle centers that lie on the Euler line are .

(e) Which triangle center never lies outside the triangle? .

Section 3: For problems 7 to 9 the Hyperbolic Parallel Postulate is valid as are all theorems proved using the Hyperbolic Parallel Postulate.

7. (8 points) In �ABC and �DEF , \ABC ⇠= \DEF and \BAC ⇠= \EDF . Explain why �ABC ⇠= �DEF .

8. (14 points) In the figure below, A ⇤ B ⇤ E and ⇤ABCD and ⇤BEFC are Saccheri quadrilaterals. Prove that

! CD and

! CF are distinct lines and that

! CD ||

! AE and

! CF ||

! AE.

A B E

FCD

9. (12 points) Let ` and m be parallel lines. Prove that if ` and m admit a common perpendicular, then ` and m are not asymptotically parallel (no pair of limiting rays can be found in the two lines).