A Problem Solving Approach To Mathematics For Elementary School Teachers (13th Edition)
13th Edition
ISBN: 9780135183885
Author: Rick Billstein, Shlomo Libeskind, Johnny Lott, Barbara Boschmans
Publisher: PEARSON
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Chapter 11.3A, Problem 6A
To determine
To find:
The measurement of supplementary angles of given angle.
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Let G
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(V,E) be a 3-connected graph. We prove that for every x, y, z Є V, there is a
cycle in G on which x, y, and z all lie.
(a) First prove that there are two internally disjoint xy-paths Po and P₁.
(b) If z is on either Po or P₁, then combining Po and P₁ produces a cycle on which
x, y, and z all lie. So assume that z is not on Po and not on P₁. Now prove that
there are three paths Qo, Q1, and Q2 such that:
⚫each Qi starts at z;
• each Qi ends at a vertex w; that is on Po or on P₁, where wo, w₁, and w₂ are
distinct;
the paths Qo, Q1, Q2 are disjoint from each other (except at the start vertex
2) and are disjoint from the paths Po and P₁ (except at the end vertices wo,
W1, and w₂).
(c) Use paths Po, P₁, Qo, Q1, and Q2 to prove that there is a cycle on which x, y, and
z all lie. (To do this, notice that two of the w; must be on the same Pj.)
Chapter 11 Solutions
A Problem Solving Approach To Mathematics For Elementary School Teachers (13th Edition)
Ch. 11.1 - MATHEMATICAL CONNECTIONS In the drawings below, we...Ch. 11.1 - If two lines, a and b, are perpendicular to the...Ch. 11.1 - Straight railroad tracks are sometime considered a...Ch. 11.1 - MATHEMATICAL CONNECTIONS Circles with the same...Ch. 11.1 - MATHEMATICAL CONNECTIONS The drawing below shows a...Ch. 11.1 - Answer each of the following and explain your...Ch. 11.1 - Given a line and a point not on the line, explain...Ch. 11.1 - How are pairs of parallel lines and skew lines a....Ch. 11.1 - Is it possible for ABC to have vertex C? Why?Ch. 11.1 - If planes and are distinct planes having points...
Ch. 11.1 - MATHEMATICAL CONNECTIONS A line n is perpendicular...Ch. 11.1 - Prob. 13MCCh. 11.1 - MATHEMATICAL CONNECTIONS a. Sketch three rays that...Ch. 11.1 - Identify a physical object within the classroom to...Ch. 11.1 - Prob. 17MCCh. 11.1 - Prob. 18MCCh. 11.1 - Bonn named the line below as ABC. How do you...Ch. 11.1 - A student says that ABBA because AB starts at A...Ch. 11.1 - Lu claims that a line segment has a finite number...Ch. 11.1 - A student claims that if any two planes that do...Ch. 11.1 - A student says that it is actually impossible to...Ch. 11.1 - Cade claims that to make the measure of an angle...Ch. 11.1 - A student says there can be only 360 different...Ch. 11.1 - MATHEMATICAL CONNECTIONS On the School Book Page...Ch. 11.1 - On the School Book Page on page 663, compare the...Ch. 11.1 - A formal definition of a line segment is as a...Ch. 11.1 - Prob. 1NAEPCh. 11.1 - Lines a and b are parallel to each other. Lines c...Ch. 11.1 - Prob. 3NAEPCh. 11.1 - In the map above, which two streets appear to be...Ch. 11.1A - Label each of the following as true or false. If...Ch. 11.1A - Prob. 2ACh. 11.1A - Name two rays that contain a. BC. b. EG.Ch. 11.1A - Given three collinear points A,B,C with B between...Ch. 11.1A - Prob. 5ACh. 11.1A - a. Points A,B,C and D are collinear. In how many...Ch. 11.1A - The following figure is a rectangular box in which...Ch. 11.1A - Assessment Use the following drawing of one of the...Ch. 11.1A - ASSESSMENT Trace each of the following drawings....Ch. 11.1A - ASSESSMENT Determine how many acute angles are...Ch. 11.1A - Identify a physical model for each of the...Ch. 11.1A - Find the measure of each of the following angles....Ch. 11.1A - Assessment a. Perform each of the following...Ch. 11.1A - Assessment Consider a correctly set clock that...Ch. 11.1A - In parts a and b of the following figures,...Ch. 11.1A - Prob. 18ACh. 11.1A - Prob. 19ACh. 11.1A - ASSESSMENT An angle of elevation on an object is...Ch. 11.1A - ASSESSMENT Describe the size of the angles of the...Ch. 11.1B - Label each of the following as true or false. If...Ch. 11.1B - Let C be the set of coplanar lines with a given...Ch. 11.1B - ASSESSMENT Name three points that are not...Ch. 11.1B - ASSESSMENT In each of the following pairs,...Ch. 11.1B - ASSESSMENT On the dot paper below, draw all...Ch. 11.1B - Prob. 7ACh. 11.1B - ASSESSMENT The following figure is a box in which...Ch. 11.1B - Prob. 9ACh. 11.1B - ASSESSMENT Describe each of the following sets of...Ch. 11.1B - ASSESSMENT Determine the number of obtuse angles...Ch. 11.1B - Identify a physical model for each of the...Ch. 11.1B - ASSESSMENT Find the measures of each of the...Ch. 11.1B - a. Perform a following operations. Leave your...Ch. 11.1B - Prob. 15ACh. 11.1B - Assessment In each of the following figures,...Ch. 11.1B - Prob. 17ACh. 11.1B - ASSESSMENT In the figure below, O is the center of...Ch. 11.1B - If the measure of A is 30 more than twice the...Ch. 11.1B - The figure below shows different types of cuts:...Ch. 11.1B - ASSESSMENT An angle of depression on an object is...Ch. 11.1B - Describe the size of the angles of the pie-shaped...Ch. 11.2 - Prob. 1MCCh. 11.2 - Classify each of the following as true or false....Ch. 11.2 - MATHEMATICAL CONNECTIONS Copy and complete...Ch. 11.2 - MATHEMATICAL CONNECTIONS On a geoboard or dot...Ch. 11.2 - Prob. 6MCCh. 11.2 - Prob. 7MCCh. 11.2 - Milli claims that a rhombus is regular because all...Ch. 11.2 - Jano heard a student say that all squares are...Ch. 11.2 - A student asks if ABCD is a valid name for the...Ch. 11.2 - MATHEMATICAL CONNECTIONS A student claims that all...Ch. 11.2 - MATHEMATICAL CONNECTIONS Two students are thinking...Ch. 11.2 - MATHEMATICAL CONNECTIONS On the School Book Page...Ch. 11.2 - Prob. 15MCCh. 11.2 - Prob. 16MCCh. 11.2 - Prob. 17MCCh. 11.2 - Prob. 18MCCh. 11.2 - Prob. 19MCCh. 11.2 - Which of the following figures can be formed by...Ch. 11.2 - NATIONAL ASSESSMENT OF EDUCATIONAL PROGRESS NAEP...Ch. 11.2 - NATIONAL ASSESSMENT OF EDUCATIONAL PROGRESS NAEP...Ch. 11.2A - Can a regular polygon be concave? Explain.Ch. 11.2A - Determine which of the following figures labeled 1...Ch. 11.2A - What is the maximum number of intersection points...Ch. 11.2A - What type of polygon must have a diagonal such...Ch. 11.2A - Describe the shaded region as simply as possible.Ch. 11.2A - If possible, draw the following triangles. If it...Ch. 11.2A - Identify each of the following triangles as...Ch. 11.2A - Determine how many diagonals each of the following...Ch. 11.2A - Draw all lines of symmetry if any exist for each...Ch. 11.2A - Various international signs have symmetries....Ch. 11.2A - Prob. 12ACh. 11.2A - Design symbols that have each of the following...Ch. 11.2A - In each of the following figures, complete the...Ch. 11.2A - Prob. 15ACh. 11.2A - Prob. 16ACh. 11.2A - Complete the following figures so that they have...Ch. 11.2A - Prob. 18ACh. 11.2A - The photograph below is of a Taj Mahal tiling....Ch. 11.2A - Prob. 20ACh. 11.2B - ASSESSMENT Which of the following figures are...Ch. 11.2B - ASSESSMENT Determine which of the following...Ch. 11.2B - ASSESSMENT What is the maximum number of...Ch. 11.2B - ASSESSMENT A pentagon has only two diagonals that...Ch. 11.2B - ASSESSMENT Describe the shaded region as simply as...Ch. 11.2B - Prob. 6ACh. 11.2B - ASSESSMENT Identify each of the following...Ch. 11.2B - Prob. 8ACh. 11.2B - Prob. 9ACh. 11.2B - Prob. 10ACh. 11.2B - Prob. 11ACh. 11.2B - Prob. 12ACh. 11.2B - Prob. 13ACh. 11.2B - Prob. 14ACh. 11.2B - Prob. 15ACh. 11.2B - Prob. 16ACh. 11.2B - Prob. 17ACh. 11.2B - Prob. 18ACh. 11.2B - Prob. 19ACh. 11.2B - Prob. 20ACh. 11.3 - Prob. 1MCCh. 11.3 - Prob. 2MCCh. 11.3 - Prob. 3MCCh. 11.3 - Prob. 4MCCh. 11.3 - Prob. 5MCCh. 11.3 - Prob. 6MCCh. 11.3 - Prob. 7MCCh. 11.3 - Prob. 8MCCh. 11.3 - Regular hexagons have been used to tile floors....Ch. 11.3 - Lines a and b are cut by transversals c and d. If...Ch. 11.3 - A beam of light from A hits the surface of a...Ch. 11.3 - Find a possible set of measures of the interior...Ch. 11.3 - Draw three different concave polygons. When you...Ch. 11.3 - Draw three concave polygons. Measure all the...Ch. 11.3 - Jan wants to make tiles in the shape of a convex...Ch. 11.3 - A student wonders if there exists a convex decagon...Ch. 11.3 - A student wants to know if a triangle can have two...Ch. 11.3 - In each of the following, name the required...Ch. 11.3 - Prob. 21MCCh. 11.3 - Prob. 22MCCh. 11.3 - Design symbols that have each of the following...Ch. 11.3 - Prob. 24MCCh. 11.3 - Explain whether the following quilt patterns have...Ch. 11.3 - Find the lines of symmetry, if any, for each of...Ch. 11.3 - Prob. 1NAEPCh. 11.3 - In the given figure, what is the measure of angle...Ch. 11.3 - Prob. 3NAEPCh. 11.3 - Prob. 4NAEPCh. 11.3A - If three lines all meet in a single point, how...Ch. 11.3A - Prob. 2ACh. 11.3A - Find the measure of a complementary angle, a...Ch. 11.3A - Assessment In ABC, the measure of B is three times...Ch. 11.3A - Prob. 5ACh. 11.3A - Prob. 6ACh. 11.3A - Prob. 7ACh. 11.3A - Prob. 8ACh. 11.3A - Prob. 9ACh. 11.3A - Prob. 10ACh. 11.3A - Find the sum of the measures of the numbered...Ch. 11.3A - Prob. 12ACh. 11.3A - Prob. 13ACh. 11.3A - Prob. 14ACh. 11.3A - Prob. 15ACh. 11.3A - Prob. 16ACh. 11.3A - Prob. 17ACh. 11.3A - Find the measures of ange 1, 2, and 3 given that...Ch. 11.3A - Prob. 19ACh. 11.3A - Prob. 20ACh. 11.3A - Prob. 21ACh. 11.3A - Assessment 11-3A If m(2)=m(3), prove m(1)=m(4).Ch. 11.3A - a. Find the measure of an interior angle of a...Ch. 11.3A - Prob. 24ACh. 11.3A - Suppose a circle circumscribes contains every...Ch. 11.3A - Assessment 11-3A Suppose an isosceles trapezoid...Ch. 11.3B - If two planes intersect in a single line forming...Ch. 11.3B - Assessment 11-3B Find the measures of angle marked...Ch. 11.3B - Prob. 3ACh. 11.3B - Assessment 11-3B An angle measures 15 more than 4...Ch. 11.3B - Prob. 5ACh. 11.3B - Prob. 6ACh. 11.3B - Prob. 7ACh. 11.3B - In part (a), prove that kl .In parts (b)and(c),...Ch. 11.3B - Prob. 9ACh. 11.3B - Prob. 10ACh. 11.3B - Prob. 11ACh. 11.3B - Calculate the measure of each angle of a pentagon,...Ch. 11.3B - The sides of DEF are parallel to the sides of BCA....Ch. 11.3B - In each of the following figures, find the...Ch. 11.3B - Prob. 15ACh. 11.3B - Prob. 16ACh. 11.3B - Prob. 17ACh. 11.3B - Find the measure of x in the figure below.Ch. 11.3B - Prob. 19ACh. 11.3B - Prob. 20ACh. 11.3B - Given the figure shown with AXDY, find the...Ch. 11.3B - Prob. 22ACh. 11.3B - a. Find the measure of an interior angle of a...Ch. 11.3B - Prob. 24ACh. 11.3B - Suppose a circle circumscribes contains every...Ch. 11.3B - Prob. 26ACh. 11.4 - How many possible pairs of bases does a right...Ch. 11.4 - Prob. 2MCCh. 11.4 - Prob. 3MCCh. 11.4 - What is the name of a pyramid that has 9 vertices?Ch. 11.4 - Prob. 5MCCh. 11.4 - What is the minimum number of faces that intersect...Ch. 11.4 - Sketch at least 2 nets for the prism shown below.Ch. 11.4 - Jon asks why a sphere is not a polyhedron. How do...Ch. 11.4 - Prob. 13MCCh. 11.4 - Joay asks if his soccer ball is a regular...Ch. 11.4 - Jed has a model of a tetrahedron shown below and...Ch. 11.4 - A net for a right cylinder consists of two figures...Ch. 11.4 - Sketch a net for the lateral surface of a right...Ch. 11.4 - Find the supplement of 181342.Ch. 11.4 - Prob. 20MCCh. 11.4 - Can two adjacent angles be vertical angles?...Ch. 11.4 - How many diagonals are there in a decagon?Ch. 11.4 - In a regular 20-gon, what is the measure of each...Ch. 11.4 - If mn find m(1).Ch. 11.4 - The solids shown have bases that are regular...Ch. 11.4 - Kyle makes a 3-dimensional shape using 3...Ch. 11.4 - Prob. 3NAEPCh. 11.4 - The figure above shown a pyramid with a square...Ch. 11.4 - Which of the following geometric solids has both...Ch. 11.4A - Identify each of the following polyhedral. If a...Ch. 11.4A - Prob. 2ACh. 11.4A - Given the tetrahedron shown, name the following....Ch. 11.4A - Prob. 4ACh. 11.4A - Prob. 5ACh. 11.4A - Prob. 6ACh. 11.4A - Classify each of the following statements as true...Ch. 11.4A - Prob. 8ACh. 11.4A - Prob. 9ACh. 11.4A - Sketch a pyramid that is hexahedron six facesCh. 11.4A - Prob. 11ACh. 11.4A - Prob. 12ACh. 11.4A - Prob. 13ACh. 11.4A - Name the intersection of each of the following...Ch. 11.4A - Prob. 15ACh. 11.4A - Prob. 16ACh. 11.4A - Prob. 17ACh. 11.4A - Prob. 18ACh. 11.4A - Prob. 19ACh. 11.4A - On the left of each of the following figures is a...Ch. 11.4B - Identify each of the following three-dimensional...Ch. 11.4B - The following are pictures of solid cubes lying on...Ch. 11.4B - Prob. 4ACh. 11.4B - Prob. 5ACh. 11.4B - Determine the minimum number of edges possible for...Ch. 11.4B - Prob. 8ACh. 11.4B - Prob. 9ACh. 11.4B - Prob. 10ACh. 11.4B - Prob. 11ACh. 11.4B - Prob. 12ACh. 11.4B - Prob. 13ACh. 11.4B - Prob. 14ACh. 11.4B - Prob. 15ACh. 11.4B - A diagonal of a prism is any segment determined by...Ch. 11.4B - Name the polyhedron that can be constructed using...Ch. 11.4B - On the left of each of the following figure is a...Ch. 11.4B - Answer each of the following questions about a...Ch. 11.4B - A soccer ball resembles a polyhedron with 32 faces...Ch. 11.CR - Prob. 1CRCh. 11.CR - Prob. 3CRCh. 11.CR - Prob. 4CRCh. 11.CR - Prob. 5CRCh. 11.CR - a. Can a triangle have two obtuse angles? Justify...Ch. 11.CR - In a certain triangle, the measure of one angle is...Ch. 11.CR - If ABC is a right angles triangle and m(A)=42,...Ch. 11.CR - Prob. 9CRCh. 11.CR - Prob. 10CRCh. 11.CR - Prob. 11CRCh. 11.CR - Prob. 12CRCh. 11.CR - Prob. 13CRCh. 11.CR - Prob. 14CRCh. 11.CR - Prob. 15CRCh. 11.CR - Prob. 16CRCh. 11.CR - Prob. 18CRCh. 11.CR - Prob. 19CRCh. 11.CR - Prob. 20CRCh. 11.CR - Prob. 21CRCh. 11.CR - Prob. 22CRCh. 11.CR - Sketch three planes that intersect in a point. a....Ch. 11.CR - In each of the following figures, determine the...Ch. 11.CR - Prob. 25CRCh. 11.CR - Prob. 27CRCh. 11.CR - Verify Eulers formula (VE+F=2) for figures d, e,...Ch. 11.CR - Prob. 29CRCh. 11.CR - Prob. 30CRCh. 11.CR - Carefully draw nets that can be folded into each...Ch. 11 - Now try this a. Can skew lines have a point in...Ch. 11 - Now try this Convert 8.42 to degrees, minutes and...Ch. 11 - Prob. 5NTCh. 11 - Prob. 8NT
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