 4.5.1: The picture statement below represents the ASA Triangle Congruence ...
 4.5.2: Create a picture statement to represent the SAA Triangle Congruence...
 4.5.3: In the third investigation you discovered that the AAA case is not ...
 4.5.4: For Exercises 49, determine whether the triangles are congruent, an...
 4.5.5: For Exercises 49, determine whether the triangles are congruent, an...
 4.5.6: For Exercises 49, determine whether the triangles are congruent, an...
 4.5.7: For Exercises 49, determine whether the triangles are congruent, an...
 4.5.8: For Exercises 49, determine whether the triangles are congruent, an...
 4.5.9: For Exercises 49, determine whether the triangles are congruent, an...
 4.5.10: In Exercises 1017, if possible, name a triangle congruent to the tr...
 4.5.11: In Exercises 1017, if possible, name a triangle congruent to the tr...
 4.5.12: In Exercises 1017, if possible, name a triangle congruent to the tr...
 4.5.13: In Exercises 1017, if possible, name a triangle congruent to the tr...
 4.5.14: In Exercises 1017, if possible, name a triangle congruent to the tr...
 4.5.15: In Exercises 1017, if possible, name a triangle congruent to the tr...
 4.5.16: In Exercises 1017, if possible, name a triangle congruent to the tr...
 4.5.17: In Exercises 1017, if possible, name a triangle congruent to the tr...
 4.5.18: The perimeter of ABC is 138 cm and BC  DE. Is ABC ADE ? Which con...
 4.5.19: Use slope properties to show AB BC, CD DA, and BC  DA. ABC . Why?
 4.5.20: In Exercises 2022, use a compass or patty paper, and a straightedge...
 4.5.21: In Exercises 2022, use a compass or patty paper, and a straightedge...
 4.5.22: In Exercises 2022, use a compass or patty paper, and a straightedge...
 4.5.23: Construction Using only a compass and a straightedge, construct an ...
 4.5.24: If n concurrent lines divide the plane into 250 parts then n = .
 4.5.25: If the two diagonals of a quadrilateral are perpendicular, then the...
 4.5.26: Construction Construct an isosceles right triangle with KM as one o...
 4.5.27: Sketch five lines in a plane that intersect in exactly five points....
 4.5.28: Application Scientists use seismograms and triangulation to pinpoin...
Solutions for Chapter 4.5: Are There Other Congruence Shortcuts?
Full solutions for Discovering Geometry: An Investigative Approach  4th Edition
ISBN: 9781559538824
Solutions for Chapter 4.5: Are There Other Congruence Shortcuts?
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Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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