 26.1: Vocabulary Apply the vocabulary from this lesson to answer each que...
 26.2: Vocabulary Apply the vocabulary from this lesson to answer each que...
 26.3: Write a justification for each step, given that mA = 60 and mB = 2m...
 26.4: Fill in the blanks to complete the twocolumn proof. Given: 2 3 Pro...
 26.5: Use the given plan to write a twocolumn proof.Given: X is the midp...
 26.6: 6. Write a justification for each step, given that BX bisects ABC a...
 26.7: Fill in the blanks to complete each twocolumn proof.Given: 1 and2 ...
 26.8: Fill in the blanks to complete each twocolumn proof.Given: BAC is ...
 26.9: Use the given plan to write a twocolumn proof.Given:BE CE , DE AE ...
 26.10: Use the given plan to write a twocolumn proof.Given: 1 and 3 are c...
 26.11: Find each angle measure.m1
 26.12: Find each angle measure. m2
 26.13: Find each angle measure.m3
 26.14: Engineering The Oresund Bridge, whichconnects the countries of Denm...
 26.15: Critical Thinking Explain why there aretwo cases to consider when p...
 26.16: Tell whether each statement is sometimes, always,or never true.An a...
 26.17: Tell whether each statement is sometimes, always,or never true.A pa...
 26.18: Tell whether each statement is sometimes, always,or never true.An a...
 26.19: Tell whether each statement is sometimes, always,or never true.A li...
 26.20: Algebra Find the value of each variable.20
 26.21: Algebra Find the value of each variable.21
 26.22: Algebra Find the value of each variable.22
 26.23: Write About It How are a theorem and a postulate alike? How are the...
 26.24: This problem will prepare you for the Concept Connection on page 12...
 26.25: Which theorem justifies the conclusion that 1 4? Linear Pair Theore...
 26.26: What can be concluded from the statement m1 + m2 = 180?1 and 2 are ...
 26.27: Given: Two angles are complementary. The measure of one angle is 10...
 26.28: Write a twocolumn proof. Given: mLAN = 30, m1 = 15Prove: AM bisect...
 26.29: MultiStep Find the value of the variable and the measure of each a...
 26.30: MultiStep Find the value of the variable and the measure of each a...
 26.31: The table shows the number of tires replaced by a repair company du...
 26.32: The table shows the number of tires replaced by a repair company du...
 26.33: Sketch a figure that shows each of the following. (Lesson 11)Throu...
 26.34: Sketch a figure that shows each of the following. (Lesson 11)A pai...
 26.35: Identify the property that justifies each statement. (Lesson 25)JK...
 26.36: Identify the property that justifies each statement. (Lesson 25)If...
Solutions for Chapter 26: Geometric Proof
Full solutions for Geometry  1st Edition
ISBN: 9780030923456
Solutions for Chapter 26: Geometric Proof
Get Full SolutionsSince 36 problems in chapter 26: Geometric Proof have been answered, more than 43708 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Geometry, edition: 1. Chapter 26: Geometric Proof includes 36 full stepbystep solutions. Geometry was written by and is associated to the ISBN: 9780030923456. This expansive textbook survival guide covers the following chapters and their solutions.

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).

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Column space C (A) =
space of all combinations of the columns of A.

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

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

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Outer product uv T
= column times row = rank one matrix.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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

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).

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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