(a) To Explain: Consider a circle or congruent circle, the statement congruent arcs have congruent central angles is true.

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Answer 1

Question

Polygon with three sides, three angles, and three vertices. Based on the properties of each side, the types of triangles are scalene (triangle with three three different lengths and three different angles), isosceles (angle with two equal sides and two equal angles), and equilateral (three equal sides and three angles of 60°). The types of angles are acute (less than 90°); obtuse (greater than 90°); and right (90°).

Chapter 6.1, Problem 23E

To determine

(a)

To Explain:

Consider a circle or congruent circle, the statement congruent arcs have congruent central angles is true.

To determine

(b)

To explain:

Consider a circle or congruent circle, the statement congruent central angles have congruent arcs is true.

To determine

(c)

To explain:

Consider a circle or congruent circle, the statement congruent chords have congruent arcs is true.

To determine

(d)

To explain:

Consider a circle or congruent circle, the statement congruent arcs have congruent chords is true.

To determine

(e)

To explain:

Consider a circle or congruent circle, the statement congruent central angles have congruent chords is true.

To determine

(f)

To explain:

Consider a circle or congruent circle, the statement congruent chords have congruent central angles is true.

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A tournament is a complete directed graph, for each pair of vertices x, y either (x, y) is an arc or (y, x) is an arc. One can think of this as a round robin tournament, where the vertices represent teams, each pair plays exactly once, with the direction of the arc indicating which team wins. (a) Prove that every tournament has a direct Hamiltonian path. That is a labeling of the teams V1, V2,..., Un so that vi beats Vi+1. That is a labeling so that team 1 beats team 2, team 2 beats team 3, etc. (b) A digraph is strongly connected if there is a directed path from any vertex to any other vertex. Equivalently, there is no partition of the teams into groups A, B so that every team in A beats every team in B. Prove that every strongly connected tournament has a directed Hamiltonian cycle. Use this to show that for any team there is an ordering as in part (a) for which the given team is first. (c) A king in a tournament is a vertex such that there is a direct path of length at most 2 to any…

The following is known. The complete graph K2t on an even number of vertices has a 1- factorization (equivalently, its edges can be colored with 2t - 1 colors so that the edges incident to each vertex are distinct). This implies that the complete graph K2t+1 on an odd number of vertices has a factorization into copies of tK2 + K₁ (a matching plus an isolated vertex). A group of 10 people wants to set up a 45 week tennis schedule playing doubles, each week, the players will form 5 pairs. One of the pairs will not play, the other 4 pairs will each play one doubles match, two of the pairs playing each other and the other two pairs playing each other. Set up a schedule with the following constraints: Each pair of players is a doubles team exactly 4 times; during those 4 matches they see each other player exactly once; no two doubles teams play each other more than once. (a) Find a schedule. Hint - think about breaking the 45 weeks into 9 blocks of 5 weeks. Use factorizations of complete…

. The two person game of slither is played on a graph. Players 1 and 2 take turns, building a path in the graph. To start, Player 1 picks a vertex. Player 2 then picks an edge incident to the vertex. Then, starting with Player 1, players alternate turns, picking a vertex not already selected that is adjacent to one of the ends of the path created so far. The first player who cannot select a vertex loses. (This happens when all neighbors of the end vertices of the path are on the path.) Prove that Player 2 has a winning strategy if the graph has a perfect matching and Player 1 has a winning strategy if the graph does not have a perfect matching. In each case describe a strategy for the winning player that guarantees that they will always be able to select a vertex. The strategy will be based on using a maximum matching to decide the next choice, and will, for one of the cases involve using the fact that maximality means no augmenting paths. Warning, the game slither is often described…

Answer 2

Question

Polygon with three sides, three angles, and three vertices. Based on the properties of each side, the types of triangles are scalene (triangle with three three different lengths and three different angles), isosceles (angle with two equal sides and two equal angles), and equilateral (three equal sides and three angles of 60°). The types of angles are acute (less than 90°); obtuse (greater than 90°); and right (90°).

Chapter 6.1, Problem 23E

To determine

(a)

To Explain:

Consider a circle or congruent circle, the statement congruent arcs have congruent central angles is true.

To determine

(b)

To explain:

Consider a circle or congruent circle, the statement congruent central angles have congruent arcs is true.

To determine

(c)

To explain:

Consider a circle or congruent circle, the statement congruent chords have congruent arcs is true.

To determine

(d)

To explain:

Consider a circle or congruent circle, the statement congruent arcs have congruent chords is true.

To determine

(e)

To explain:

Consider a circle or congruent circle, the statement congruent central angles have congruent chords is true.

To determine

(f)

To explain:

Consider a circle or congruent circle, the statement congruent chords have congruent central angles is true.

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Answer 3

Question

Polygon with three sides, three angles, and three vertices. Based on the properties of each side, the types of triangles are scalene (triangle with three three different lengths and three different angles), isosceles (angle with two equal sides and two equal angles), and equilateral (three equal sides and three angles of 60°). The types of angles are acute (less than 90°); obtuse (greater than 90°); and right (90°).

Chapter 6.1, Problem 23E

To determine

(a)

To Explain:

Consider a circle or congruent circle, the statement congruent arcs have congruent central angles is true.

To determine

(b)

To explain:

Consider a circle or congruent circle, the statement congruent central angles have congruent arcs is true.

To determine

(c)

To explain:

Consider a circle or congruent circle, the statement congruent chords have congruent arcs is true.

To determine

(d)

To explain:

Consider a circle or congruent circle, the statement congruent arcs have congruent chords is true.

To determine

(e)

To explain:

Consider a circle or congruent circle, the statement congruent central angles have congruent chords is true.

To determine

(f)

To explain:

Consider a circle or congruent circle, the statement congruent chords have congruent central angles is true.

Expert Solution & Answer

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Answer 4

Question

Polygon with three sides, three angles, and three vertices. Based on the properties of each side, the types of triangles are scalene (triangle with three three different lengths and three different angles), isosceles (angle with two equal sides and two equal angles), and equilateral (three equal sides and three angles of 60°). The types of angles are acute (less than 90°); obtuse (greater than 90°); and right (90°).

Chapter 6.1, Problem 23E

To determine

(a)

To Explain:

Consider a circle or congruent circle, the statement congruent arcs have congruent central angles is true.

To determine

(b)

To explain:

Consider a circle or congruent circle, the statement congruent central angles have congruent arcs is true.

To determine

(c)

To explain:

Consider a circle or congruent circle, the statement congruent chords have congruent arcs is true.

To determine

(d)

To explain:

Consider a circle or congruent circle, the statement congruent arcs have congruent chords is true.

To determine

(e)

To explain:

Consider a circle or congruent circle, the statement congruent central angles have congruent chords is true.

To determine

(f)

To explain:

Consider a circle or congruent circle, the statement congruent chords have congruent central angles is true.

Expert Solution & Answer

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See solution

Answer 5

Chapter 6.1, Problem 23E

To determine

(a)

To Explain:

Consider a circle or congruent circle, the statement congruent arcs have congruent central angles is true.

To determine

(b)

To explain:

Consider a circle or congruent circle, the statement congruent central angles have congruent arcs is true.

To determine

(c)

To explain:

Consider a circle or congruent circle, the statement congruent chords have congruent arcs is true.

To determine

(d)

To explain:

Consider a circle or congruent circle, the statement congruent arcs have congruent chords is true.

To determine

(e)

To explain:

Consider a circle or congruent circle, the statement congruent central angles have congruent chords is true.

To determine

(f)

To explain:

Consider a circle or congruent circle, the statement congruent chords have congruent central angles is true.

Answer 6

Chapter 6.1, Problem 23E

To determine

(a)

To Explain:

Consider a circle or congruent circle, the statement congruent arcs have congruent central angles is true.

Answer 7

Chapter 6.1, Problem 23E

To determine

(a)

To Explain:

Consider a circle or congruent circle, the statement congruent arcs have congruent central angles is true.

Answer 8

To determine

(b)

To explain:

Consider a circle or congruent circle, the statement congruent central angles have congruent arcs is true.

Answer 9

To determine

(b)

To explain:

Consider a circle or congruent circle, the statement congruent central angles have congruent arcs is true.

Answer 10

To determine

(c)

To explain:

Consider a circle or congruent circle, the statement congruent chords have congruent arcs is true.

Answer 11

To determine

(c)

To explain:

Consider a circle or congruent circle, the statement congruent chords have congruent arcs is true.

Answer 12

To determine

(d)

To explain:

Consider a circle or congruent circle, the statement congruent arcs have congruent chords is true.

Answer 13

To determine

(d)

To explain:

Consider a circle or congruent circle, the statement congruent arcs have congruent chords is true.

Answer 14

To determine

(e)

To explain:

Consider a circle or congruent circle, the statement congruent central angles have congruent chords is true.

Answer 15

To determine

(e)

To explain:

Consider a circle or congruent circle, the statement congruent central angles have congruent chords is true.

Answer 16

To determine

(f)

To explain:

Consider a circle or congruent circle, the statement congruent chords have congruent central angles is true.

Answer 17

To determine

(f)

To explain:

Consider a circle or congruent circle, the statement congruent chords have congruent central angles is true.

Answer 18

Expert Solution & Answer

Answer 19

Expert Solution & Answer

Answer 20

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Answer 21

(a) To Explain: Consider a circle or congruent circle, the statement congruent arcs have congruent central angles is true.

Solution Summary: The author explains that if two arcs have equal measures, then their associated central angles have the same measures.

Concept explainers

Videos

(a)

To Explain:

Consider a circle or congruent circle, the statement congruent arcs have congruent central angles is true.

(b)

To explain:

Consider a circle or congruent circle, the statement congruent central angles have congruent arcs is true.

(c)

To explain:

Consider a circle or congruent circle, the statement congruent chords have congruent arcs is true.

(d)

To explain:

Consider a circle or congruent circle, the statement congruent arcs have congruent chords is true.

(e)

To explain:

Consider a circle or congruent circle, the statement congruent central angles have congruent chords is true.

(f)

To explain:

Consider a circle or congruent circle, the statement congruent chords have congruent central angles is true.

Want to see the full answer?

Chapter 6 Solutions

(a) To Explain: Consider a circle or congruent circle, the statement congruent arcs have congruent central angles is true.

Solution Summary: The author explains that if two arcs have equal measures, then their associated central angles have the same measures.

Concept explainers

Videos

(a) To Explain: Consider a circle or congruent circle, the statement congruent arcs have congruent central angles is true.

(b) To explain: Consider a circle or congruent circle, the statement congruent central angles have congruent arcs is true.

(c) To explain: Consider a circle or congruent circle, the statement congruent chords have congruent arcs is true.

(d) To explain: Consider a circle or congruent circle, the statement congruent arcs have congruent chords is true.

(e) To explain: Consider a circle or congruent circle, the statement congruent central angles have congruent chords is true.

(f) To explain: Consider a circle or congruent circle, the statement congruent chords have congruent central angles is true.

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Chapter 6 Solutions

(a)

To Explain:

Consider a circle or congruent circle, the statement congruent arcs have congruent central angles is true.

(b)

To explain:

Consider a circle or congruent circle, the statement congruent central angles have congruent arcs is true.

(c)

To explain:

Consider a circle or congruent circle, the statement congruent chords have congruent arcs is true.

(d)

To explain:

Consider a circle or congruent circle, the statement congruent arcs have congruent chords is true.

(e)

To explain:

Consider a circle or congruent circle, the statement congruent central angles have congruent chords is true.

(f)

To explain:

Consider a circle or congruent circle, the statement congruent chords have congruent central angles is true.