- 4.1: Use the fact that ABCD is a squareto classify ABD by its sidelength...
- 4.2: DB bisects ABC and ADC. DEbisects ADB. Find the measuresof the angl...
- 4.3: Given that DB bisects ABC andEDF, BE BF , and DE DF ,prove that EDB...
- 4.4: Classify each triangle by its side lengths. PQR
- 4.5: Classify each triangle by its side lengths. PRS
- 4.6: Classify each triangle by its side lengths. PQS
- 4.7: Find each angle measure.mM
- 4.8: Find each angle measure.mABC
- 4.9: A carpenter built a triangular support structure for a roof. Two of...
- 4.10: Given: JKL DEF. Identify the congruent corresponding parts.KL ?
- 4.11: Given: JKL DEF. Identify the congruent corresponding parts.DF ?
- 4.12: Given: JKL DEF. Identify the congruent corresponding parts.K ?
- 4.13: Given: JKL DEF. Identify the congruent corresponding parts.F ?
- 4.14: Given: PQR STU. Find each value.PQ
- 4.15: Given: PQR STU. Find each value.y
- 4.16: Given: AB CD , AB CD ,AC BD , AC CD , DB AB Prove: ACD DBA Proof:
- 4.17: Find each value.mS
- 4.18: Given: Isosceles ABC has coordinates A (2a, 0) , B (0, 2b), and C (...
- 4.19: The measure of P is 3 __12 times the measure of Q.If P and Q are co...
- 4.20: Given m with transversal n, explain why 2and 3 are complementary.
- 4.21: G and H are supplementary angles.mG = (2x + 12), and mH = x. a. Wri...
- 4.22: A manager conjectures that for every 1000 partsa factory produces, ...
- 4.23: BD is the perpendicular bisector of AC . a. What are the conclusion...
- 4.24: ABC and DEF are isosceles triangles. BC EF ,and AC DF . mC = 42.5, ...
Solutions for Chapter 4: Triangle Congruence
Full solutions for Geometry | 1st Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.
Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)
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.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = 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 B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
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).
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Every v in V is orthogonal to every w in W.
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.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.