Elementary Geometry For College Students, 7e
7th Edition
ISBN: 9781337614085
Author: Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher: Cengage,
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Chapter 1.2, Problem 18E
Exercises
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Qll consider the problem -abu+bou+cu=f., u=0 ondor
I prove atu, ul conts.
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if Blu,v) = (b. 14, U) + ((4,0) prove that
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③if c±vbo prove that acuius v. elliptic.
Q3: Define the linear functional J: H₁(2)
R by
¡(v) =
a(v, v) - L(v)
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Let u be the unique weak solution to a(u,v) = L(v) in H(2) and suppose that
a(...) is a symmetric bilinear form on H(2) prove that
1- u is minimizer. 2- u is unique. 3- The minimizer J(u) can be rewritten under
1(u) = u Au-ub,
algebraic form
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Where A, b are repictively the stiffence matrix and the load vector
Q4: A) Answer
1- show that the solution to -Au = f in A, u = 0 on a satisfies the
stability Vullfll and show that ||V(u u)||||||2 - ||vu||2
2- Prove that
Where
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B) Consider the bilinea forta
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a(u, v) = (Au, Av) (Vu, Vv + (Vu, v) + (u,v)
Show that a(u, v) continues and V- elliptic on H(2)
7) In the diagram below of quadrilateral ABCD, E and F are points on AB and CD respectively,
BE=DF, and AE = CF. Which conclusion can be proven?
A
1) ED = FB
2) AB CD
3) ZA = ZC
4) ZAED/CFB
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B
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Chapter 1 Solutions
Elementary Geometry For College Students, 7e
Ch. 1.1 - In Exercises 1 and 2, complete the statement....Ch. 1.1 - In Exercises 1 and 2, complete the statement....Ch. 1.1 - In Exercises 3 and 4, use the fact that...Ch. 1.1 - Prob. 4ECh. 1.1 - Prob. 5ECh. 1.1 - In Exercises 5 and 6, use the fact that...Ch. 1.1 - Eleanor finds the 15-mile road from A to C is...Ch. 1.1 - A cross-country runner jogs at a rate of 15 feet...Ch. 1.1 - In Exercises 9 to 28, use the drawings as needed...Ch. 1.1 - In Exercises 9 to 28, use the drawings as needed...
Ch. 1.1 - In Exercises 9 to 28, use the drawings as needed...Ch. 1.1 - In Exercises 9 to 28, use the drawings as needed...Ch. 1.1 - In Exercises 9 to 28, use the drawings as needed...Ch. 1.1 - In Exercises 9 to 28, use the drawings as needed...Ch. 1.1 - In Exercises 9 to 28, use the drawings as needed...Ch. 1.1 - In Exercises 9 to 28, use the drawings as needed...Ch. 1.1 - In Exercises 9 to 28, use the drawings as needed...Ch. 1.1 - In Exercises 9 to 28, use the drawings as needed...Ch. 1.1 - In Exercises 9 to 28, use the drawings as needed...Ch. 1.1 - Prob. 20ECh. 1.1 - In Exercises 9 to 28, use the drawings as needed...Ch. 1.1 - Prob. 22ECh. 1.1 - Prob. 23ECh. 1.1 - In Exercises 9 to 28, use the drawings as needed...Ch. 1.1 - In Exercises 9 to 28, use the drawings as needed...Ch. 1.1 - In Exercises 9 to 28, use the drawings as needed...Ch. 1.1 - Prob. 27ECh. 1.1 - In Exercises 9 to 28, use the drawings as needed...Ch. 1.1 - Prob. 29ECh. 1.1 - In Exercises 29 to 32, use only a compass and a...Ch. 1.1 - In Exercises 29 to 32, use only a compass and a...Ch. 1.1 - Prob. 32ECh. 1.1 - Can you use the construction for the midpoint of a...Ch. 1.1 - Generalize your findings in Exercise 33. 33. Can...Ch. 1.1 - Consider points A, B, C, and D, no three of which...Ch. 1.1 - Consider noncoplanar points A, B, C, and D. Using...Ch. 1.1 - Line l is parallel to plane P that is, it will not...Ch. 1.1 - AB and EF are said to be skew lines because they...Ch. 1.1 - Exercises 3840 In the box shown for Exercise 38,...Ch. 1.1 - Exercises 3940 In the box shown for Exercise 38,...Ch. 1.1 - Let AB=a and BC=b. Point M is the midpoint of BC....Ch. 1.2 - What type of angle has the given measure? a 47 b...Ch. 1.2 - What type of angle has the given measure? a 115 b...Ch. 1.2 - What relationship, if any, exists between two...Ch. 1.2 - What relationship, if any, exists between two...Ch. 1.2 - In Exercises 5 to 8, describe in one word the...Ch. 1.2 - In Exercises 5 to 8, describe in one word the...Ch. 1.2 - In Exercises 5 to 8, describe in one word the...Ch. 1.2 - In Exercises 5 to 8, describe in one word the...Ch. 1.2 - Use drawings as needed to answer each of the...Ch. 1.2 - Suppose that AB,AC,AD,AE, and AF are coplanar,...Ch. 1.2 - Exercises 10-13 Without using a protractor, name...Ch. 1.2 - Exercises 10-13 What, if anything, is wrong with...Ch. 1.2 - Exercises 10-13 FAC and CAD are adjacent and AF...Ch. 1.2 - For Exercises 14 and 15, let m1=x and m2=y. Using...Ch. 1.2 - For Exercises 14 and 15, let m1=x and m2=y. Using...Ch. 1.2 - Given: mRST=39 mTSV=23 Find: mRSV Exercises 1624Ch. 1.2 - Exercises 1624 Given: mRSV=59 mTSV=17 Find: mRSTCh. 1.2 - Exercises 1624 Given: mRST=2x+9 mTSV=3x2 mRSV=67...Ch. 1.2 - Exercises 1624 Given: mRST=2x10 mTSV=x+6...Ch. 1.2 - Exercises 1624 Given: mRST=5(x+1)3 mTSV=4(x2)+3...Ch. 1.2 - Exercises 1624 Given: mRST=x2 mTSV=x4 mRSV=45...Ch. 1.2 - Exercises 1624 Given: mRST=2x3 mTSV=x2 mRSV=49...Ch. 1.2 - Exercises 1624 Given: STbisectsRSV mRST=x+y...Ch. 1.2 - Exercises 1624 Given: STbisectsRSV mRST=2x+3y...Ch. 1.2 - Given: AB and AC in plane P as shown AD intersects...Ch. 1.2 - Two angles are complementary. One angle is 12...Ch. 1.2 - Two angles are supplementary. One angle is 24 more...Ch. 1.2 - For two complementary angles, find an expression...Ch. 1.2 - Suppose that the two angles are supplementary....Ch. 1.2 - On the protractor shown, NP bisects MNQ. Find x....Ch. 1.2 - Exercises 30,31 On the protractor shown for...Ch. 1.2 - Classify as true or false: a If points P and Q lie...Ch. 1.2 - Prob. 33ECh. 1.2 - Prob. 34ECh. 1.2 - In Exercises 33 to 40, use only a compass and a...Ch. 1.2 - Prob. 36ECh. 1.2 - Draw a triangle with three acute angles. Construct...Ch. 1.2 - Prob. 38ECh. 1.2 - Prob. 39ECh. 1.2 - Prob. 40ECh. 1.2 - Given: 1 and 2 as shown in, sketch the bisectors...Ch. 1.2 - Prob. 42ECh. 1.2 - Prob. 43ECh. 1.2 - If mTSV=38,mUSW=40, and mTSW=61, find mUSV....Ch. 1.2 - Exercises 44, 45 If mTSU=x+2z,mUSV=xz, and...Ch. 1.2 - Prob. 46ECh. 1.2 - On the hanging sign, the three angles...Ch. 1.2 - With 0x90, an acute angle has a measure x. Find...Ch. 1.3 - In Exercises 1 to 6, which property justifies the...Ch. 1.3 - In Exercises 1 to 6, which property justifies the...Ch. 1.3 - In Exercises 1 to 6, which property justifies the...Ch. 1.3 - Prob. 4ECh. 1.3 - Prob. 5ECh. 1.3 - In Exercises 1 to 6, which property justifies the...Ch. 1.3 - In Exercises 7 10, state the property or...Ch. 1.3 - In Exercises 7 10, state the property or...Ch. 1.3 - Prob. 9ECh. 1.3 - In Exercises 7 10, state the property or...Ch. 1.3 - Prob. 11ECh. 1.3 - Prob. 12ECh. 1.3 - Prob. 13ECh. 1.3 - Prob. 14ECh. 1.3 - Prob. 15ECh. 1.3 - Prob. 16ECh. 1.3 - In Exercises 11 to 22, use the Given information...Ch. 1.3 - Prob. 18ECh. 1.3 - Prob. 19ECh. 1.3 - Prob. 20ECh. 1.3 - In Exercises 11 to 22, use the Given information...Ch. 1.3 - Prob. 22ECh. 1.3 - In Exercises 23 to 24, fill in the missing reasons...Ch. 1.3 - In Exercises 23 to 24, fill in the missing reasons...Ch. 1.3 - Prob. 25ECh. 1.3 - Prob. 26ECh. 1.3 - Prob. 27ECh. 1.3 - In Exercises 27 to 30, fill in the missing reasons...Ch. 1.3 - In Exercises 27 to 30, fill in the missing reasons...Ch. 1.3 - In Exercises 27 to 30, fill in the missing reasons...Ch. 1.3 - In Exercises 31 and 32, fill in the missing...Ch. 1.3 - In Exercises 31 and 32, fill in the missing...Ch. 1.3 - Prob. 33ECh. 1.3 - Prob. 34ECh. 1.3 - Prob. 35ECh. 1.3 - The Division Property of Inequality requires that...Ch. 1.3 - Prob. 37ECh. 1.3 - Prob. 38ECh. 1.3 - Prob. 39ECh. 1.3 - Write a proof for: If a=b and c=d, then ac=bd....Ch. 1.4 - In Exercise 1 and 2, supply reasons. Given: 13...Ch. 1.4 - In Exercise 1 and 2, supply reasons. Given: AB...Ch. 1.4 - In Exercise 3 and 4, supply statements. Given: 12...Ch. 1.4 - In Exercise 3 and 4, supply statements. Given:...Ch. 1.4 - In Exercise 5 to 9, use a compass and a...Ch. 1.4 - In Exercises 5 to 9, use a compass and a...Ch. 1.4 - In Exercise 5 to 9, use a compass and a...Ch. 1.4 - Prob. 8ECh. 1.4 - Prob. 9ECh. 1.4 - Prob. 10ECh. 1.4 - In Exercise 11 and 12, provide the missing...Ch. 1.4 - In Exercise 11 and 12, provide the missing...Ch. 1.4 - Does the relation is perpendicular to have a...Ch. 1.4 - Does the relation is greater than have a reflexive...Ch. 1.4 - Does the relation is complementary to for angles...Ch. 1.4 - Does the relation is less than for a numbers have...Ch. 1.4 - Does the relation is a brother of have a reflexive...Ch. 1.4 - Does the relation is a friend of have a reflexive...Ch. 1.4 - Prob. 19ECh. 1.4 - Sometimes symbols and abbreviations are used in...Ch. 1.4 - Prob. 21ECh. 1.4 - Prob. 22ECh. 1.4 - Prove the Extended Segment Addition Property by...Ch. 1.4 - The Segment-Addition Postulate can be generalized...Ch. 1.4 - Prove the Extended Angle Addition Property by...Ch. 1.4 - The Angle-Addition Postulate can be generalized as...Ch. 1.4 - Prob. 27ECh. 1.4 - In the proof below, provide the missing reasons....Ch. 1.4 - Prob. 29ECh. 1.4 - Prob. 30ECh. 1.5 - In Exercises 1 to 6, state the hypothesis H and...Ch. 1.5 - In Exercises 1 to 6, state the hypothesis H and...Ch. 1.5 - In Exercises 1 to 6, state the hypothesis H and...Ch. 1.5 - In Exercises 1 to 6, state the hypothesis H and...Ch. 1.5 - Prob. 5ECh. 1.5 - In Exercises 1 to 6, state the hypothesis H and...Ch. 1.5 - Name, in order, the five parts of the formal proof...Ch. 1.5 - Which part hypothesis or conclusion of a theorem...Ch. 1.5 - Which part Given or Prove of the proof depends...Ch. 1.5 - Which of the following can be cited as a reason in...Ch. 1.5 - When can a theorem be cited as a reason reason in...Ch. 1.5 - Based upon the hypothesis of a theorem, do the...Ch. 1.5 - Prob. 13ECh. 1.5 - Prob. 14ECh. 1.5 - For each theorem stated in Exercises 13 to 18,...Ch. 1.5 - Prob. 16ECh. 1.5 - For each theorem stated in Exercises 13 to 18,...Ch. 1.5 - Prob. 18ECh. 1.5 - Prob. 19ECh. 1.5 - Prob. 20ECh. 1.5 - Prob. 21ECh. 1.5 - Prob. 22ECh. 1.5 - Prob. 23ECh. 1.5 - Prob. 24ECh. 1.5 - In Exercises 19 to 26, use the drawing in which AC...Ch. 1.5 - In Exercises 19 to 26, use the drawing in which AC...Ch. 1.5 - Prob. 27ECh. 1.5 - Prob. 28ECh. 1.5 - In Exercises 27 to 35, complete the formal proof...Ch. 1.5 - Prob. 30ECh. 1.5 - Prob. 31ECh. 1.5 - Prob. 32ECh. 1.5 - In Exercises 27 to 35, complete the formal proof...Ch. 1.5 - In Exercises 27 to 35, complete the formal proof...Ch. 1.5 - In Exercises 27 to 35, complete the formal proof...Ch. 1.CR - Name the four components of a mathematical system.Ch. 1.CR - Name three types of reasoning.Ch. 1.CR - Name the four characteristics of a good...Ch. 1.CR - In Review Exercises 4 to 6, name the type of...Ch. 1.CR - In Review Exercises 4 to 6, name the type of...Ch. 1.CR - In Review Exercises 4 to 6, name the type of...Ch. 1.CR - In Review Exercises 7 and 8, state the hypothesis...Ch. 1.CR - In Review Exercises 7 and 8, state the hypothesis...Ch. 1.CR - Prob. 9CRCh. 1.CR - Prob. 10CRCh. 1.CR - Prob. 11CRCh. 1.CR - A, B and C are three points on a line. AC=8, BC=4,...Ch. 1.CR - Prob. 13CRCh. 1.CR - Figure MNPQ is a rhombus. Draw diagonals MP and QN...Ch. 1.CR - In Review Exercises 15 to 17, sketch and label the...Ch. 1.CR - In Review Exercises 15 to 17, sketch and label the...Ch. 1.CR - In Review Exercises 15 to 17, sketch and label the...Ch. 1.CR - On the basis of appearance, what type of angle is...Ch. 1.CR - On the basis of appearance, what type of angle is...Ch. 1.CR - Given: BD bisects ABC mABD=2x+15 mDBC=3x+5 Find:...Ch. 1.CR - Given: mABD=2x+5 mDBC=3x4 mABC=86 Find: mDBCCh. 1.CR - Given: AM=3x1 MB=4x5 M is the midpoint of AB Find:...Ch. 1.CR - Given: AM=4x4 MB=5x+2 AB=25 Find: MBCh. 1.CR - Given: D is the midpoint of AC ACBC CD=2x+5...Ch. 1.CR - Given: m3=7x21 m4=3x+7 Find: mFMHCh. 1.CR - Given: mFMH=4x+1 m4=x+4 Find: m4Ch. 1.CR - In the figure, find: a KHFJ b MJMH c KMJJMH d MKMHCh. 1.CR - Given: EFG is a right angle. mHFG=2x6 mEFH=3mHFG...Ch. 1.CR - Two angles are supplementary. One angle is 40 more...Ch. 1.CR - aWrite an expression for the perimeter of the...Ch. 1.CR - The sum of the measures of all three angles of the...Ch. 1.CR - Susan wants to have a 4-ft board with some pegs on...Ch. 1.CR - State whether the sentences in Review Exercises 33...Ch. 1.CR - State whether the sentences in Review Exercises 33...Ch. 1.CR - State whether the sentences in Review Exercises 33...Ch. 1.CR - State whether the sentences in Review Exercises 33...Ch. 1.CR - State whether the sentences in Review Exercises 33...Ch. 1.CR - Fill in the missing statements or reasons. Given:...Ch. 1.CR - Write two-column proofs for Review Exercises 39 to...Ch. 1.CR - Write two-column proofs for Review Exercises 39 to...Ch. 1.CR - Prob. 41CRCh. 1.CR - Prob. 42CRCh. 1.CR - Prob. 43CRCh. 1.CR - Prob. 44CRCh. 1.CR - Prob. 45CRCh. 1.CR - Prob. 46CRCh. 1.CR - Given: VP Construct: VW such that VW=4VPCh. 1.CR - Prob. 48CRCh. 1.CR - Prob. 49CRCh. 1.CR - Prob. 50CRCh. 1.CR - Prob. 51CRCh. 1.CR - Prob. 52CRCh. 1.CT - Given ABC as shown, provide a second correct...Ch. 1.CT - Using the Segment-Addition Postulate, state a...Ch. 1.CT - Complete each postulate: a If two lines intersect,...Ch. 1.CT - Prob. 4CTCh. 1.CT - Prob. 5CTCh. 1.CT - Given that NP bisects MNQ, state a conclusion...Ch. 1.CT - Complete each theorem: a If two lines are...Ch. 1.CT - Given the line AC and ray BD, suppose that BE...Ch. 1.CT - Given that 2x3=7. What reason allows the...Ch. 1.CT - Given A-B-C-D, and M is the midpoint of AB. If...Ch. 1.CT - Prob. 11CTCh. 1.CT - In the figure, mEFG=68 and m3=33. Find m4.Ch. 1.CT - In the figure, m3=x and m4=2x3. If mEFG=69, find:...Ch. 1.CT - Prob. 14CTCh. 1.CT - Prob. 15CTCh. 1.CT - Prob. 16CTCh. 1.CT - Prob. 17CTCh. 1.CT - Construct the angle bisector of obtuse angle RST.Ch. 1.CT - Prob. 19CTCh. 1.CT - In exercises 20 to 22, complete the missing...Ch. 1.CT - Prob. 21CTCh. 1.CT - In exercises 20 to 22, complete the missing...Ch. 1.CT - Prob. 23CT
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- 1) In parallelogram EFGH, diagonals EG and FH intersect at point I such that EI = 2x - 2 and EG = 3x + 11. Which of the following is the length of GH? a) 15 b) 28 c) 32 d) 56arrow_forward5) Which of the following are properties of all squares: 1. Congruent diagonals 2. Perpendicular diagonals 3. Diagonals that bisect vertex angles a) 1 and 2 only b) 1 and 3 only c) 2 and 3 only d) 1, 2, and 3arrow_forward6) In an isosceles trapezoid HIJK it is known that IJ || KH. Which of the following must also be true? a) IJ = KH b) HIJK c) HIJK d) IJ KHarrow_forward
- 1) Given: MNPQ is a parallelogram with MP 1 NQ. Prove: MNPQ is a rhombus. Statement Reason M R Parrow_forward4) Find a proposition with three variables p, q, and r that is never true. 5) Determine whether this proposition is a tautology using propositional equivalence and laws of logic: ((p (bv (bL ← →¬p [1 6) Explain why the negation of "Some students in my class use e-mail” is not "Some students in my class do not use e-mail".arrow_forwardMilgram lemma B) Consider Show that -Au= f in a u=0 on on llu-ulls Chllullz 02 Prove that Where ||ul| = a(u, u) = vu. Vu dx + fonu.u ds Q3: Let V = H' (2), a(u,v) = CR, a(u,v) = (f,v) where Vu. Vv dx + Ja cuv dx and ||u|=|||| Show that a(u, v) is V-ellipiticly and continuity.arrow_forward
- 7) Is the following statement True or False: AU BUA = Ā. Justify your answer. 8) Suppose g: A → B and f: B → C where A = {2,3,6,8}, and g and f are defined by g = {a, b, c, d}, B = {1,2,3}, C = {(a,2), (b,1), (c, 3), (d, 2)} and f = {(1,8),(2,3), (3,2)}. Find fog, gof and f−¹. 9) Verify that a₁ = 7(3") -π is a solution to the recurrence relation an = an 4an-1 -3an-2arrow_forward1) Find the prime factorization of 111111. 2) Find (-88 mod 13) 5 mod 7. 3) Use the Euclidean algorithm to find gcd(144,233).arrow_forward3 10) Suppose B = 2 4 5) and C = (b 2 prove that no such matrix exists. (6 1) . Find a matrix A such that AB = C orarrow_forward
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