 2.1: For Exercises 1 to 4, with transversal v. If , find: a) b)
 2.2: For Exercises 1 to 4, with transversal v. If , find: a) b)
 2.3: For Exercises 1 to 4, with transversal v. If , find: a) b)
 2.4: For Exercises 1 to 4, with transversal v. If , find: a) b)
 2.5: Use drawings, as needed, to answer each question. Does the relation...
 2.6: Use drawings, as needed, to answer each question. In a plane, and ....
 2.7: Use drawings, as needed, to answer each question. Suppose that . Bo...
 2.8: Use drawings, as needed, to answer each question. Make a sketch to ...
 2.9: Use drawings, as needed, to answer each question. Suppose that r is...
 2.10: Use drawings, as needed, to answer each question. In Euclidean geom...
 2.11: Use drawings, as needed, to answer each question. Lines r and s are...
 2.12: Use drawings, as needed, to answer each question. , , and . Find: a...
 2.13: Use drawings, as needed, to answer each question. , with transversa...
 2.14: Use drawings, as needed, to answer each question. Given:Transversal...
 2.15: Use drawings, as needed, to answer each question. Given:Transversal...
 2.16: Use drawings, as needed, to answer each question. Given:Transversal...
 2.17: Use drawings, as needed, to answer each question. Given:Transversal...
 2.18: Use drawings, as needed, to answer each question. In the threedime...
 2.19: Use drawings, as needed, to answer each question. Given: and Prove:...
 2.20: Given: and (See figure for Exercise 19.) Prove:
 2.21: Given:Transversal bisects bisects Prove:
 2.22: Given:Transversal bisects Prove:
 2.23: Given:Transversal t is a right Prove:
 2.24: Given:Find: (HINT: There is a line through C parallel to both and .)
 2.25: Given:Find: (See Hint in Exercise 24.)
 2.26: Given: , (See figure for Exercise 23.) Prove:
 2.27: In triangle ABC, line t is drawn through vertex A in such a way tha...
 2.28: In Exercises 28 to 30, write a formal proof of each theorem. If two...
 2.29: In Exercises 28 to 30, write a formal proof of each theorem. If two...
 2.30: In Exercises 28 to 30, write a formal proof of each theorem. If a t...
 2.31: Suppose that two lines are cut by a transversal in such a way that ...
 2.32: Given: Line and point P not on Construct:
 2.33: Given: Triangle ABC with three acute angles Construct: , with D on .
 2.34: Given: Triangle MNQ with obtuse Construct: , with E on .
 2.35: Given: Triangle MNQ with obtuse Construct: (HINT: Extend .)
 2.36: Given: A line m and a point T not on mSuppose that you do the follo...
 2.37: A carpenter drops a plumb line from point A to . Assuming that is h...
 2.38: Given:bisects bisects Prove:
 2.39: A lamppost has a design such that and . Find and .
 2.40: For the lamppost of Exercise 39, suppose that and that . Find , , a...
 2.41: The triangular symbol on the PLAY button of a DVD has congruent ang...
 2.42: A polygon with four sides is called a quadrilateral. Consider the f...
 2.43: Explain why the following statement is true. Each interior angle of...
 2.44: Explain why the following statement is true. The acute angles of a ...
 2.45: In Exercises 45 to 47, write a formal proof for each corollary. The...
 2.46: In Exercises 45 to 47, write a formal proof for each corollary. If ...
 2.47: In Exercises 45 to 47, write a formal proof for each corollary. Use...
 2.48: Given: , , and bisects bisects Prove: is a right angle
 2.49: Given: bisects bisects Find:
 2.50: Given: In rt. , bisects and bisects . Find:
Solutions for Chapter 2: Parallel Lines
Full solutions for Elementary Geometry for College Students  6th Edition
ISBN: 9781285195698
Solutions for Chapter 2: Parallel Lines
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Geometry for College Students, edition: 6. Since 50 problems in chapter 2: Parallel Lines have been answered, more than 4221 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 2: Parallel Lines includes 50 full stepbystep solutions. Elementary Geometry for College Students was written by and is associated to the ISBN: 9781285195698.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.