 27.1: Vocabulary Apply the vocabulary from this lesson to answer each que...
 27.2: Vocabulary Apply the vocabulary from this lesson to answer each que...
 27.3: Use the given flowchart proof to writea twocolumn proof.Given: 1 2...
 27.4: Use the given twocolumn proof to writea flowchart proof.Given: 2 a...
 27.9: Use the given paragraph proof to write a twocolumn proof.Given: 1 ...
 27.10: Use the given twocolumn proof to write a paragraph proof.Given: 1 ...
 27.11: Find each measure and name the theorem that justifies your answer.AB
 27.12: Find each measure and name the theorem that justifies your answer.m2
 27.13: Find each measure and name the theorem that justifies your answer.m3
 27.14: Algebra Find the value of each variable.14
 27.15: Algebra Find the value of each variable.15
 27.16: Algebra Find the value of each variable.16
 27.17: /////ERROR ANALYSIS///// Below are two drawings for the given proof...
 27.18: This problem will prepare you for the Concept Connection on page 12...
 27.19: Critical Thinking Two lines intersect, and one of the angles formed...
 27.20: Write About It Which style of proof do you find easiest to write? t...
 27.21: Which pair of angles in the diagram must be congruent?1 and 5 5 and...
 27.22: What is the measure of 2? 38 128 52 142
 27.23: Which statement is NOT true if 2 and 6 are supplementary? m2 + m6 =...
 27.24: Textiles Use the woven pattern to write a flowchart proof.Given: 1 ...
 27.25: Write a twocolumn proof.Given: AOC BODProve: AOB COD
 27.26: Write a paragraph proof.Given: 2 and 5 are right angles. m1 + m2 + ...
 27.27: MultiStep Find the value of each variable and the measures of all ...
 27.28: Solve each system of equations. Check your solution. (Previous cour...
 27.29: Solve each system of equations. Check your solution. (Previous cour...
 27.30: Solve each system of equations. Check your solution. (Previous cour...
 27.31: Use a protractor to draw an angle with each of the following measur...
 27.32: Use a protractor to draw an angle with each of the following measur...
 27.33: Use a protractor to draw an angle with each of the following measur...
 27.34: Use a protractor to draw an angle with each of the following measur...
 27.35: For each conditional, write the converse and a biconditional statem...
 27.36: For each conditional, write the converse and a biconditional statem...
Solutions for Chapter 27: Flowchart and Paragraph Proofs
Full solutions for Geometry  1st Edition
ISBN: 9780030923456
Solutions for Chapter 27: Flowchart and Paragraph Proofs
Get Full SolutionsChapter 27: Flowchart and Paragraph Proofs includes 32 full stepbystep solutions. This textbook survival guide was created for the textbook: Geometry, edition: 1. This expansive textbook survival guide covers the following chapters and their solutions. Geometry was written by and is associated to the ISBN: 9780030923456. Since 32 problems in chapter 27: Flowchart and Paragraph Proofs have been answered, more than 42072 students have viewed full stepbystep solutions from this chapter.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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).

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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!).

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.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.