Title: Reasoning and Proof
Course: quiz duplicate error

Q: Determine whether the conjecture is true or false. Give a counterexample for any false conjecture.

Given: [Image] are coplanar.
Conjecture: They are vertical angles.
- False; the angles may be supplementary. (Just because two angles share a common point does not mean they are vertical. They could be nearly adjacent or one could be in the interior of the other one.)
- True (Just because two angles share a common point does not mean they are vertical. They could be nearly adjacent or one could be in the interior of the other one.)
- False; one angle may be in the interior of the other. (Just because two angles share a common point does not mean they are vertical. They could be nearly adjacent or one could be in the interior of the other one.)
- False; the angles may be adjacent. (Just because two angles share a common point does not mean they are vertical. They could be nearly adjacent or one could be in the interior of the other one.)
A: False; one angle may be in the interior of the other.

Q: Determine whether the conjecture is true or false. Give a counterexample for any false conjecture.

Given: [Image]
Conjecture: [Image]
- False; [Image]. (Because m is squared in the example, m could be positive or negative.)
- True (Because m is squared in the example, m could be positive or negative.)
- False; [Image]. (Because m is squared in the example, m could be positive or negative.)
- False; [Image]. (Because m is squared in the example, m could be positive or negative.)
A: False; [Image].

Q: Use the following statements to write a compound statement for the conjunction or disjunction. Then find its truth value.

p: An isosceles triangle has two congruent sides.
q: A right angle measures 90º
r: Four points are always coplanar.
s: A decagon has 12 sides.

[Image]
- An isosceles triangle has two congruent sides and a right angle measures 90º. True (Two or more statements can be joined to form a compound statement. A disjunction is a compound statement formed by joining two or more statements with the word or. The symbol for logical or is [Image].)
- An isosceles triangle has two congruent sides or a right angle measures 90º. True (Two or more statements can be joined to form a compound statement. A disjunction is a compound statement formed by joining two or more statements with the word or. The symbol for logical or is [Image].)
- An isosceles triangle has two congruent sides or four points are always coplanar. True (Two or more statements can be joined to form a compound statement. A disjunction is a compound statement formed by joining two or more statements with the word or. The symbol for logical or is [Image].)
- An isosceles triangle has two congruent sides or a right angle measures 90º. False (Two or more statements can be joined to form a compound statement. A disjunction is a compound statement formed by joining two or more statements with the word or. The symbol for logical or is [Image].)
A: An isosceles triangle has two congruent sides or a right angle measures 90º. True

Q: Determine whether the conjecture is true or false. Give a counterexample for any false conjecture.

Given: points A, B, C, and D
Conjecture: A, B, C, and D are coplanar.
- False; the four points do not have to be in a straight line. (Coplanar points always lie in the same plane. Three points are always coplanar but four are not.)
- True (Coplanar points always lie in the same plane. Three points are always coplanar but four are not.)
- False; two points are always coplanar but four are not. (Coplanar points always lie in the same plane. Three points are always coplanar but four are not.)
- False; three points are always coplanar but four are not. (Coplanar points always lie in the same plane. Three points are always coplanar but four are not.)
A: False; three points are always coplanar but four are not.

Q: Make a conjecture about the next item in the sequence.

[Image]
- 1024 (Start with 1. Add, subtract, or multiply the same number to each number to get the next one.)
- 1025 (Start with 1. Add, subtract, or multiply the same number to each number to get the next one.)
- 4096 (Start with 1. Add, subtract, or multiply the same number to each number to get the next one.)
- 1022 (Start with 1. Add, subtract, or multiply the same number to each number to get the next one.)
A: 1024

Q: Determine whether the conjecture is true or false. Give a counterexample for any false conjecture.

Given: segments RT and ST; twice the measure of [Image].
Conjecture: S is the midpoint of segment RT.
- True (Even though they have a common point, the two segments do not have to be on the same line.)
- False; point S may not be on [Image]. (Even though they have a common point, the two segments do not have to be on the same line.)
- False; lines do not have midpoints. (Even though they have a common point, the two segments do not have to be on the same line.)
- False; [Image] could be the segment bisector of [Image]. (Even though they have a common point, the two segments do not have to be on the same line.)
A: False; point S may not be on [Image].

Q: Determine whether the conjecture is true or false. Give a counterexample for any false conjecture.

Given: a concave polygon
Conjecture: It can be regular or irregular.
- False; to be concave the angles cannot be congruent. (Concave polygons must be irregular. This means all sides and angles are not congruent.)
- True (Concave polygons must be irregular. This means all sides and angles are not congruent.)
- False; all concave polygons are regular. (Concave polygons must be irregular. This means all sides and angles are not congruent.)
- False; a concave polygon has an odd number of sides. (Concave polygons must be irregular. This means all sides and angles are not congruent.)
A: False; to be concave the angles cannot be congruent.

Q: Determine whether the conjecture is true or false. Give a counterexample for any false conjecture.

Given: [Image]
Conjecture: [Image]
- False; the angles are not vertical. (Unless there are specific angle measures mentioned, even though the angles in the picture may look congruent you cannot assume that they are congruent.)
- True (Unless there are specific angle measures mentioned, even though the angles in the picture may look congruent you cannot assume that they are congruent.)
- False; the angles are not complementary. (Unless there are specific angle measures mentioned, even though the angles in the picture may look congruent you cannot assume that they are congruent.)
- False; there is no indication of the measures of the angles. (Unless there are specific angle measures mentioned, even though the angles in the picture may look congruent you cannot assume that they are congruent.)
A: False; there is no indication of the measures of the angles.

Q: Determine whether the conjecture is true or false. Give a counterexample for any false conjecture.

Given: point B is in the interior of [Image].
Conjecture: [Image]
- False; [Image] may be obtuse. (Angles are congruent only if their measures are equal. Point B may be closer to line AD or line DC so the measures would not be equal.)
- True (Angles are congruent only if their measures are equal. Point B may be closer to line AD or line DC so the measures would not be equal.)
- False; just because it is in the interior does not mean it is on the bisecting line. (Angles are congruent only if their measures are equal. Point B may be closer to line AD or line DC so the measures would not be equal.)
- False; [Image]. (Angles are congruent only if their measures are equal. Point B may be closer to line AD or line DC so the measures would not be equal.)
A: False; just because it is in the interior does not mean it is on the bisecting line.

Q: Determine whether the conjecture is true or false. Give a counterexample for any false conjecture.

Given: Two angles are supplementary.
Conjecture: They are both acute angles.
- False; either both are right or they are adjacent. (If two angles are supplementary their measures total 180. Either both are right or one is obtuse and the other acute.)
- True (If two angles are supplementary their measures total 180. Either both are right or one is obtuse and the other acute.)
- False; either both are right or one is obtuse. (If two angles are supplementary their measures total 180. Either both are right or one is obtuse and the other acute.)
- False; they must be vertical angles. (If two angles are supplementary their measures total 180. Either both are right or one is obtuse and the other acute.)
A: False; either both are right or one is obtuse.

Q: Use the following statements to write a compound statement for the conjunction or disjunction. Then find its truth value.

p: An isosceles triangle has two congruent sides.
q: A right angle measures 90º
r: Four points are always coplanar.
s: A decagon has 12 sides.

p and r
- An isosceles triangle has two congruent sides and four points are always coplanar. False (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and.)
- An isosceles triangle has two congruent sides and a decagon has 12 sides. False (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and.)
- An isosceles triangle has two congruent sides and four points are always coplanar. True (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and.)
- An isosceles triangle has two congruent sides and a decagon has 12 sides. True (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and.)
A: An isosceles triangle has two congruent sides and four points are always coplanar. False

Q: Determine whether the conjecture is true or false. Give a counterexample for any false conjecture.

Given: [Image] is supplementary to [Image] and [Image] is supplementary to [Image].
Conjecture: [Image] is supplementary to [Image].
- False; they could be right angles. (If two angles are supplementary their measures total 180. [Image] could only be supplementary to [Image] if they are both right angles.)
- False; they could be congruent angles. (If two angles are supplementary their measures total 180. [Image] could only be supplementary to [Image] if they are both right angles.)
- True (If two angles are supplementary their measures total 180. [Image] could only be supplementary to [Image] if they are both right angles.)
- False; they could be vertical angles. (If two angles are supplementary their measures total 180. [Image] could only be supplementary to [Image] if they are both right angles.)
A: False; they could be congruent angles.

Q: Make a conjecture about the next item in the sequence.

[Image]
- [Image] (Start with 6. Do one operation to get the next number. Do a different operation to get the next number. Repeat.)
- [Image] (Start with 6. Do one operation to get the next number. Do a different operation to get the next number. Repeat.)
- [Image] (Start with 6. Do one operation to get the next number. Do a different operation to get the next number. Repeat.)
- [Image] (Start with 6. Do one operation to get the next number. Do a different operation to get the next number. Repeat.)
A: [Image]

Q: Determine whether the conjecture is true or false. Give a counterexample for any false conjecture.

Given: points R, S, and T
Conjecture: R, S, and T are coplanar.
- False; the points do not have to be in a straight line. (Coplanar points always lie in the same plane. Three points are always coplanar but four are not.)
- True (Coplanar points always lie in the same plane. Three points are always coplanar but four are not.)
- False; the points to not have to form right angles. (Coplanar points always lie in the same plane. Three points are always coplanar but four are not.)
- False; one point may not be between the other two. (Coplanar points always lie in the same plane. Three points are always coplanar but four are not.)
A: True

Q: Write the contrapositive of the conditional statement. Determine whether the contrapositive is true or false. If it is false, find a counterexample.

If you are 16 years old, then you are a teenager.
- If you are not a teenager, then you are not 16 years old. True (In the contrapositive you negate both the hypothesis and conclusion of the converse statement. [Image])
- If you are not 16 years old, then you are not a teenager. False; you could be 17 years old. (In the contrapositive you negate both the hypothesis and conclusion of the converse statement. [Image])
- If you are not a teenager, then you are 16 years old. True (In the contrapositive you negate both the hypothesis and conclusion of the converse statement. [Image])
- If you are a teenager, then you are 16 years old. False; you could be 17 years old. (In the contrapositive you negate both the hypothesis and conclusion of the converse statement. [Image])
A: If you are not a teenager, then you are not 16 years old. True

Q: Complete the truth table.

[Image]
- [Image] (The first statement column in a truth table contains half Ts, half Fs, grouped together. The second statement column in a truth table contains the same, but they are grouped by half the number that the first column was. The third statement column contains the same but they are grouped by half the number that the second column was. Use the truth values of the first three columns to determine the truth values for the last two columns. The symbol for not is [Image]. The symbol for logical and is [Image].)
- [Image] (The first statement column in a truth table contains half Ts, half Fs, grouped together. The second statement column in a truth table contains the same, but they are grouped by half the number that the first column was. The third statement column contains the same but they are grouped by half the number that the second column was. Use the truth values of the first three columns to determine the truth values for the last two columns. The symbol for not is [Image]. The symbol for logical and is [Image].)
- [Image] (The first statement column in a truth table contains half Ts, half Fs, grouped together. The second statement column in a truth table contains the same, but they are grouped by half the number that the first column was. The third statement column contains the same but they are grouped by half the number that the second column was. Use the truth values of the first three columns to determine the truth values for the last two columns. The symbol for not is [Image]. The symbol for logical and is [Image].)
- [Image] (The first statement column in a truth table contains half Ts, half Fs, grouped together. The second statement column in a truth table contains the same, but they are grouped by half the number that the first column was. The third statement column contains the same but they are grouped by half the number that the second column was. Use the truth values of the first three columns to determine the truth values for the last two columns. The symbol for not is [Image]. The symbol for logical and is [Image].)
A: [Image]

Q: Write the statement in if-then form.

Spiders can walk on walls.
- If it can walk on walls, then it can walk on walls. (The format of if-then form is “If hypothesis, then conclusion.”)
- If it is a spider, then it can walk on walls. (The format of if-then form is “If hypothesis, then conclusion.”)
- If it is a spider, then it is a spider. (The format of if-then form is “If hypothesis, then conclusion.”)
- If it can walk on walls, then it is a spider. (The format of if-then form is “If hypothesis, then conclusion.”)
A: If it is a spider, then it can walk on walls.

Q: Use the following statements to write a compound statement for the conjunction or disjunction. Then find its truth value.

p: An isosceles triangle has two congruent sides.
q: A right angle measures 90º
r: Four points are always coplanar.
s: A decagon has 12 sides.

[Image]
- An isosceles triangle has two congruent sides and a decagon has 12 sides. True (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. A disjunction is a compound statement formed by joining two or more statements with the word or. The symbol for logical and is [Image]. The symbol for logical or is [Image].)
- An isosceles triangle has two congruent sides or a decagon has 12 sides. False (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. A disjunction is a compound statement formed by joining two or more statements with the word or. The symbol for logical and is [Image]. The symbol for logical or is [Image].)
- An isosceles triangle has two congruent sides or a decagon has 12 sides. True (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. A disjunction is a compound statement formed by joining two or more statements with the word or. The symbol for logical and is [Image]. The symbol for logical or is [Image].)
- An isosceles triangle has two congruent sides and a decagon has 12 sides. False (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. A disjunction is a compound statement formed by joining two or more statements with the word or. The symbol for logical and is [Image]. The symbol for logical or is [Image].)
A: An isosceles triangle has two congruent sides and a decagon has 12 sides. False

Q: Write the inverse of the conditional statement. Determine whether the inverse is true or false. If it is false, find a counterexample.

Independence Day in the United States is July 4.
- July 4 is not Independence Day in the United States. False; it is Independence Day. (The inverse is negating both the hypothesis and conclusion of the conditional. [Image])
- Non-Independence Day in the United States is not July 4. True (The inverse is negating both the hypothesis and conclusion of the conditional. [Image])
- Non-Independence Day in the United States is July 4. False; July 4 is Independence Day in the United States. (The inverse is negating both the hypothesis and conclusion of the conditional. [Image])
- Non-July 4 is not Independence Day in the United States. True (The inverse is negating both the hypothesis and conclusion of the conditional. [Image])
A: Non-Independence Day in the United States is not July 4. True

Q: Write the inverse of the conditional statement. Determine whether the inverse is true or false. If it is false, find a counterexample.

An equilateral triangle has three congruent sides.
- A non-equilateral triangle has three congruent sides. False; an isosceles triangle has two congruent sides. (The inverse is negating both the hypothesis and conclusion of the conditional. [Image])
- A figure that has three non-congruent sides is not an equilateral triangle. True (The inverse is negating both the hypothesis and conclusion of the conditional. [Image])
- A non-equilateral triangle does not have three congruent sides. True (The inverse is negating both the hypothesis and conclusion of the conditional. [Image])
- A figure with three congruent sides is an equilateral triangle. True (The inverse is negating both the hypothesis and conclusion of the conditional. [Image])
A: A non-equilateral triangle does not have three congruent sides. True

Write the inverse of the conditional statement. Determine whether the inverse is true or false. If it is false, find a counterexample.

The segment bisector is the midpoint.

Q: Write the contrapositive of the conditional statement. Determine whether the contrapositive is true or false. If it is false, find a counterexample.

A converse statement is formed by exchanging the hypothesis and conclusion of the conditional.
- A non-converse statement is not formed by exchanging the hypothesis and conclusion of the conditional. True (In the contrapositive you negate both the hypothesis and conclusion of the converse statement. [Image])
- A statement not formed by exchanging the hypothesis and conclusion of the conditional is a converse statement. False; an inverse statement is not formed by exchanging the hypothesis and conclusion of the conditional. (In the contrapositive you negate both the hypothesis and conclusion of the converse statement. [Image])
- A non-converse statement is formed by exchanging the hypothesis and conclusion of the conditional. False; an inverse statement is formed by negating both the hypothesis and conclusion of the conditional. (In the contrapositive you negate both the hypothesis and conclusion of the converse statement. [Image])
- A statement not formed by exchanging the hypothesis and conclusion of the conditional is not a converse statement. True (In the contrapositive you negate both the hypothesis and conclusion of the converse statement. [Image])
A: A statement not formed by exchanging the hypothesis and conclusion of the conditional is not a converse statement. True

Write the inverse of the conditional statement. Determine whether the inverse is true or false. If it is false, find a counterexample.

Two angles measuring 90° are complementary.

Q: Use the following statements to write a compound statement for the conjunction or disjunction. Then find its truth value.

p: An isosceles triangle has two congruent sides.
q: A right angle measures 90º
r: Four points are always coplanar.
s: A decagon has 12 sides.

[Image]
- Four points are always coplanar and a right angle measures 90º. False (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. A disjunction is a compound statement formed by joining two or more statements with the word or. The symbol for logical and is [Image]. The symbol for logical or is [Image].)
- Four points are always coplanar and a right angle measures 90º. True (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. A disjunction is a compound statement formed by joining two or more statements with the word or. The symbol for logical and is [Image]. The symbol for logical or is [Image].)
- Four points are always coplanar or a right angle measures 90º. False (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. A disjunction is a compound statement formed by joining two or more statements with the word or. The symbol for logical and is [Image]. The symbol for logical or is [Image].)
- Four points are always coplanar or a right angle measures 90º. True (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. A disjunction is a compound statement formed by joining two or more statements with the word or. The symbol for logical and is [Image]. The symbol for logical or is [Image].)
A: Four points are always coplanar and a right angle measures 90º. False

Q: Write the inverse of the conditional statement. Determine whether the inverse is true or false. If it is false, find a counterexample.

People who live in Texas live in the United States.
- People who do not live in the United States do not live in Texas. True (The inverse is negating both the hypothesis and conclusion of the conditional. [Image])
- People who do not live in Texas do not live in the United States. False; they could live in Oklahoma. (The inverse is negating both the hypothesis and conclusion of the conditional. [Image])
- People who live in the United States live in Texas. False; they could live in Oklahoma. (The inverse is negating both the hypothesis and conclusion of the conditional. [Image])
- People who do not live in Texas live in the United States. True (The inverse is negating both the hypothesis and conclusion of the conditional. [Image])
A: People who do not live in Texas do not live in the United States. False; they could live in Oklahoma.

Q: Use the following statements to write a compound statement for the conjunction or disjunction. Then find its truth value.

p: An isosceles triangle has two congruent sides.
q: A right angle measures 90º
r: Four points are always coplanar.
s: A decagon has 12 sides.

[Image]
- Four points are always coplanar, or a right angle measures 90º and a decagon has 12 sides. False (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. A disjunction is a compound statement formed by joining two or more statements with the word or. The symbol for logical and is [Image]. The symbol for logical or is [Image].)
- Four points are always coplanar, and a right angle measures 90º or a decagon has 12 sides. True (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. A disjunction is a compound statement formed by joining two or more statements with the word or. The symbol for logical and is [Image]. The symbol for logical or is [Image].)
- Four points are always coplanar, or a right angle measures 90º and a decagon has 12 sides. True (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. A disjunction is a compound statement formed by joining two or more statements with the word or. The symbol for logical and is [Image]. The symbol for logical or is [Image].)
- Four points are always coplanar, and a right angle measures 90º or a decagon has 12 sides. False (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. A disjunction is a compound statement formed by joining two or more statements with the word or. The symbol for logical and is [Image]. The symbol for logical or is [Image].)
A: Four points are always coplanar, and a right angle measures 90º or a decagon has 12 sides. False

Write the inverse of the conditional statement. Determine whether the inverse is true or false. If it is false, find a counterexample.

A conditional statement is formed by a given hypothesis and conclusion.

Q: Determine whether statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. If it does not, write invalid.

(1) She is a girl.
(2) Her name is Chris.
(3) Chris is a girls name.
- invalid (The Law of Syllogism states, “If [Image] and [Image] are true, then [Image] is also true.” The Law of Detachment states, “If [Image] is true and p is true, then q is also true.”)
- yes; Law of Syllogism (The Law of Syllogism states, “If [Image] and [Image] are true, then [Image] is also true.” The Law of Detachment states, “If [Image] is true and p is true, then q is also true.”)
- yes; Law of Detachment (The Law of Syllogism states, “If [Image] and [Image] are true, then [Image] is also true.” The Law of Detachment states, “If [Image] is true and p is true, then q is also true.”)
A: invalid

Q: Use the following statements to write a compound statement for the conjunction or disjunction. Then find its truth value.

p: An isosceles triangle has two congruent sides.
q: A right angle measures 90º
r: Four points are always coplanar.
s: A decagon has 12 sides.

[Image]
- A right angle measures 90º or an isosceles triangle has two congruent sides. True (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. The symbol for logical and is [Image].)
- A right angle measures 90º and an isosceles triangle has two congruent sides. True (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. The symbol for logical and is [Image].)
- A right angle measures 90º and an isosceles triangle does not have two congruent sides. False (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. The symbol for logical and is [Image].)
- A right angle measures 90º or an isosceles triangle does not have two congruent sides. True (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. The symbol for logical and is [Image].)
A: A right angle measures 90º and an isosceles triangle has two congruent sides. True

Q: Use the following statements to write a compound statement for the conjunction or disjunction. Then find its truth value.

p: An isosceles triangle has two congruent sides.
q: A right angle measures 90º
r: Four points are always coplanar.
s: A decagon has 12 sides.

[Image]
- A decagon has 12 sides and an isosceles triangle has two congruent sides. False (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. A disjunction is a compound statement formed by joining two or more statements with the word or. The symbol for logical and is [Image]. The symbol for logical or is [Image].)
- A decagon has 12 sides and an isosceles triangle has two congruent sides. True (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. A disjunction is a compound statement formed by joining two or more statements with the word or. The symbol for logical and is [Image]. The symbol for logical or is [Image].)
- A decagon has 12 sides or an isosceles triangle has two congruent sides. True (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. A disjunction is a compound statement formed by joining two or more statements with the word or. The symbol for logical and is [Image]. The symbol for logical or is [Image].)
- A decagon has 12 sides or an isosceles triangle has two congruent sides. False (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. A disjunction is a compound statement formed by joining two or more statements with the word or. The symbol for logical and is [Image]. The symbol for logical or is [Image].)
A: A decagon has 12 sides or an isosceles triangle has two congruent sides. True

Q: In the figure below, points A, B, C, and F lie on plane P. State the postulate that can be used to show each statement is true.
[Image]

A and B are collinear.
- If two points lie in a plane¸ then the entire line containing those points lies in that plane. (Postulates: 1. Through any two points, there is exactly one line. 2. Through any three points not on the same line, there is exactly one plane. 3. A line contains at least two points. 4. A plane contains at least three points not on the same line. 5. If two points lie in a plane, then the entire line containing those points lies in that plane. 6. If two lines intersect, then their intersection is exactly one point. 7. If two planes intersect, then their intersection is a line.)
- Through any two points there is exactly one line. (Postulates: 1. Through any two points, there is exactly one line. 2. Through any three points not on the same line, there is exactly one plane. 3. A line contains at least two points. 4. A plane contains at least three points not on the same line. 5. If two points lie in a plane, then the entire line containing those points lies in that plane. 6. If two lines intersect, then their intersection is exactly one point. 7. If two planes intersect, then their intersection is a line.)
- If two lines intersect¸ then their intersection is exactly one point. (Postulates: 1. Through any two points, there is exactly one line. 2. Through any three points not on the same line, there is exactly one plane. 3. A line contains at least two points. 4. A plane contains at least three points not on the same line. 5. If two points lie in a plane, then the entire line containing those points lies in that plane. 6. If two lines intersect, then their intersection is exactly one point. 7. If two planes intersect, then their intersection is a line.)
- A line contains at least two points. (Postulates: 1. Through any two points, there is exactly one line. 2. Through any three points not on the same line, there is exactly one plane. 3. A line contains at least two points. 4. A plane contains at least three points not on the same line. 5. If two points lie in a plane, then the entire line containing those points lies in that plane. 6. If two lines intersect, then their intersection is exactly one point. 7. If two planes intersect, then their intersection is a line.)
A: Through any two points there is exactly one line.

Q: Write the converse of the conditional statement. Determine whether the converse is true or false. If it is false, find a counterexample.

All Jack Russells are terriers.
- If a dog is a terrier, then it is a Jack Russell. False; it could be a Scottish terrier. (The converse of a conditional statement [Image] exchanges the hypothesis and conclusion of the conditional. It is also known as [Image].)
- If it is a Jack Russell, then a dog is a terrier. True (The converse of a conditional statement [Image] exchanges the hypothesis and conclusion of the conditional. It is also known as [Image].)
- If a dog is a terrier, then a dog is a terrier. True (The converse of a conditional statement [Image] exchanges the hypothesis and conclusion of the conditional. It is also known as [Image].)
- All Jack Russells are terriers. True (The converse of a conditional statement [Image] exchanges the hypothesis and conclusion of the conditional. It is also known as [Image].)
A: If a dog is a terrier, then it is a Jack Russell. False; it could be a Scottish terrier.

Write the inverse of the conditional statement. Determine whether the inverse is true or false. If it is false, find a counterexample.

If it is a spider, then it can walk on walls.

Q: Write the converse of the conditional statement. Determine whether the converse is true or false. If it is false, find a counterexample.

If you have a dog, then you are a pet owner.
- If you are a pet owner, then you have a dog. True (The converse of a conditional statement [Image] exchanges the hypothesis and conclusion of the conditional. It is also known as [Image].)
- A dog owner owns a pet. True (The converse of a conditional statement [Image] exchanges the hypothesis and conclusion of the conditional. It is also known as [Image].)
- If you are a pet owner, then you have a dog. False; you could own a hamster. (The converse of a conditional statement [Image] exchanges the hypothesis and conclusion of the conditional. It is also known as [Image].)
- If you have a dog, then you are a pet owner. True (The converse of a conditional statement [Image] exchanges the hypothesis and conclusion of the conditional. It is also known as [Image].)
A: If you are a pet owner, then you have a dog. False; you could own a hamster.

Q: Determine whether statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. If it does not, write invalid.

(1) You are in ninth grade.
(2) People who are in ninth grade floss their teeth regularly.
(3) You floss your teeth regularly.
- yes; Law of Syllogism (The Law of Syllogism states, “If [Image] and [Image] are true, then [Image] is also true.” The Law of Detachment states, “If [Image] is true and p is true, then q is also true.”)
- invalid (The Law of Syllogism states, “If [Image] and [Image] are true, then [Image] is also true.” The Law of Detachment states, “If [Image] is true and p is true, then q is also true.”)
- yes; Law of Detachment (The Law of Syllogism states, “If [Image] and [Image] are true, then [Image] is also true.” The Law of Detachment states, “If [Image] is true and p is true, then q is also true.”)
A: yes; Law of Detachment

Q: Write the inverse of the conditional statement. Determine whether the inverse is true or false. If it is false, find a counterexample.

All quadrilaterals are four-sided figures.
- All non-quadrilaterals are four-sided figures. False; a triangle is a non-quadrilateral. (The inverse is negating both the hypothesis and conclusion of the conditional. [Image])
- All four-sided figures are quadrilaterals. True (The inverse is negating both the hypothesis and conclusion of the conditional. [Image])
- No quadrilaterals are not four-sided figures. True (The inverse is negating both the hypothesis and conclusion of the conditional. [Image])
- No four-sided figures are not quadrilaterals. True (The inverse is negating both the hypothesis and conclusion of the conditional. [Image])
A: No quadrilaterals are not four-sided figures. True

Q: Use the following statements to write a compound statement for the conjunction or disjunction. Then find its truth value.

p: An isosceles triangle has two congruent sides.
q: A right angle measures 90º
r: Four points are always coplanar.
s: A decagon has 12 sides.

q or r
- A right angle measures 90º and four points are always coplanar. False (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. A disjunction is a compound statement formed by joining two or more statements with the word or.)
- A right angle measures 90º or a decagon has 12 sides. True (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. A disjunction is a compound statement formed by joining two or more statements with the word or.)
- A right angle measures 90º or four points are always coplanar. True (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. A disjunction is a compound statement formed by joining two or more statements with the word or.)
- A right angle measures 90º or four points are always coplanar. False (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. A disjunction is a compound statement formed by joining two or more statements with the word or.)
A: A right angle measures 90º or four points are always coplanar. True

Q: Write the contrapositive of the conditional statement. Determine whether the contrapositive is true or false. If it is false, find a counterexample.

If you have a gerbil, then you are a pet owner.
- If you are not a pet owner, then you do not have a gerbil. True (In the contrapositive you negate both the hypothesis and conclusion of the converse statement. [Image])
- If you do not have a gerbil, then you are not a pet owner. False; you could have a dog. (In the contrapositive you negate both the hypothesis and conclusion of the converse statement. [Image])
- If you are not a pet owner, then you have a gerbil. False; if you are not a pet owner then you have no pets. (In the contrapositive you negate both the hypothesis and conclusion of the converse statement. [Image])
- If you are not a gerbil, then you are not a pet owner. True (In the contrapositive you negate both the hypothesis and conclusion of the converse statement. [Image])
A: If you are not a pet owner, then you do not have a gerbil. True

Q: Write the contrapositive of the conditional statement. Determine whether the contrapositive is true or false. If it is false, find a counterexample.

Thanksgiving Day in the United States is November 25.
- If it is not November 25, it is Thanksgiving Day in the United States. True (Based on logic rather than general knowledge. In the contrapositive you negate both the hypothesis and conclusion of the converse statement. [Image])
- If it is not Thanksgiving Day in the United States, it is not November 25. False; Thanksgiving Day could be another date in a different year so November 25 could be not Thanksgiving Day. (Based on logic rather than general knowledge. In the contrapositive you negate both the hypothesis and conclusion of the converse statement. [Image])
- If it is not November 25, it is not Thanksgiving Day in the United States. True (Based on logic rather than general knowledge. In the contrapositive you negate both the hypothesis and conclusion of the converse statement. [Image])
- If it is not November 25, it is not Thanksgiving Day in the United States. False; Thanksgiving Day could be another date in a different year. (Based on logic rather than general knowledge. In the contrapositive you negate both the hypothesis and conclusion of the converse statement. [Image])
A: If it is not November 25, it is not Thanksgiving Day in the United States. True

Q: Write the statement in if-then form.

A counterexample invalidates a statement.
- If it invalidates the statement, then there is a counterexample. (The format of if-then form is “If hypothesis, then conclusion.”)
- If there is a counterexample, then it invalidates the statement. (The format of if-then form is “If hypothesis, then conclusion.”)
- If it is true, then there is a counterexample. (The format of if-then form is “If hypothesis, then conclusion.”)
- If there is a counterexample, then it is true. (The format of if-then form is “If hypothesis, then conclusion.”)
A: If there is a counterexample, then it invalidates the statement.

Q: Use the following statements to write a compound statement for the conjunction or disjunction. Then find its truth value.

p: An isosceles triangle has two congruent sides.
q: A right angle measures 90º
r: Four points are always coplanar.
s: A decagon has 12 sides.

s or q
- A decagon has 12 sides or a right angle measures 90º. True (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. A disjunction is a compound statement formed by joining two or more statements with the word or. The symbol for logical and is [Image]. The symbol for logical or is [Image].)
- A decagon has 12 sides or a right angle measures 90º. False (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. A disjunction is a compound statement formed by joining two or more statements with the word or. The symbol for logical and is [Image]. The symbol for logical or is [Image].)
- A decagon has 12 sides and a right angle measures 90º. True (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. A disjunction is a compound statement formed by joining two or more statements with the word or. The symbol for logical and is [Image]. The symbol for logical or is [Image].)
- A decagon has 12 sides and a right angle measures 90º. False (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. A disjunction is a compound statement formed by joining two or more statements with the word or. The symbol for logical and is [Image]. The symbol for logical or is [Image].)
A: A decagon has 12 sides or a right angle measures 90º. True

Q: Write the inverse of the conditional statement. Determine whether the inverse is true or false. If it is false, find a counterexample.

All country names are capitalized words.
- All capitalized words are country names. False; the first word in the sentence is capitalized. (The inverse is negating both the hypothesis and conclusion of the conditional. [Image])
- All non-capitalized words are not country names. True (The inverse is negating both the hypothesis and conclusion of the conditional. [Image])
- All non-country names are capitalized words. False; most of the words in the sentence are non-capitalized words. (The inverse is negating both the hypothesis and conclusion of the conditional. [Image])
- All non-country names are non-capitalized words. False; the first word in the sentence is capitalized. (The inverse is negating both the hypothesis and conclusion of the conditional. [Image])
A: All non-country names are non-capitalized words. False; the first word in the sentence is capitalized.

Q: Complete the truth table.

[Image]
- [Image] (The first statement column in a truth table contains half Ts, half Fs, grouped together. The second statement column in a truth table contains the same, but they are grouped by half the number that the first column was. The third statement column contains the same but they are grouped by half the number that the second column was. Use the truth values of the first three columns to determine the truth values for the last three columns. The symbol for not is [Image]. The symbol for logical and is [Image].)
- [Image] (The first statement column in a truth table contains half Ts, half Fs, grouped together. The second statement column in a truth table contains the same, but they are grouped by half the number that the first column was. The third statement column contains the same but they are grouped by half the number that the second column was. Use the truth values of the first three columns to determine the truth values for the last three columns. The symbol for not is [Image]. The symbol for logical and is [Image].)
- [Image] (The first statement column in a truth table contains half Ts, half Fs, grouped together. The second statement column in a truth table contains the same, but they are grouped by half the number that the first column was. The third statement column contains the same but they are grouped by half the number that the second column was. Use the truth values of the first three columns to determine the truth values for the last three columns. The symbol for not is [Image]. The symbol for logical and is [Image].)
- [Image] (The first statement column in a truth table contains half Ts, half Fs, grouped together. The second statement column in a truth table contains the same, but they are grouped by half the number that the first column was. The third statement column contains the same but they are grouped by half the number that the second column was. Use the truth values of the first three columns to determine the truth values for the last three columns. The symbol for not is [Image]. The symbol for logical and is [Image].)
A: [Image]

Q: Write the contrapositive of the conditional statement. Determine whether the contrapositive is true or false. If it is false, find a counterexample.

Two angles measuring 180 are supplementary.
- Two angles not measuring 180 are not supplementary. True (In the contrapositive you negate both the hypothesis and conclusion of the converse statement. [Image])
- More than two angles measuring 180 are non-supplementary. True (In the contrapositive you negate both the hypothesis and conclusion of the converse statement. [Image])
- Non-supplementary angles are not two angles measuring 180. True (In the contrapositive you negate both the hypothesis and conclusion of the converse statement. [Image])
- Non-supplementary angles are two angles measuring 180. False; supplementary angles must measure 180. (In the contrapositive you negate both the hypothesis and conclusion of the converse statement. [Image])
A: Non-supplementary angles are not two angles measuring 180. True

Q: Use the following statements to write a compound statement for the conjunction or disjunction. Then find its truth value.

p: An isosceles triangle has two congruent sides.
q: A right angle measures 90º
r: Four points are always coplanar.
s: A decagon has 12 sides.

[Image]
- A right angle measures 90º or four points are always coplanar, and a decagon has 12 sides. False (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. A disjunction is a compound statement formed by joining two or more statements with the word or. The symbol for logical and is [Image]. The symbol for logical or is [Image].)
- A right angle measures 90º and four points are always coplanar, or a decagon has 12 sides. False (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. A disjunction is a compound statement formed by joining two or more statements with the word or. The symbol for logical and is [Image]. The symbol for logical or is [Image].)
- A right angle measures 90º and four points are always coplanar, or a decagon has 12 sides. True (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. A disjunction is a compound statement formed by joining two or more statements with the word or. The symbol for logical and is [Image]. The symbol for logical or is [Image].)
- A right angle measures 90º or four points are always coplanar, and a decagon has 12 sides. True (Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. A disjunction is a compound statement formed by joining two or more statements with the word or. The symbol for logical and is [Image]. The symbol for logical or is [Image].)
A: A right angle measures 90º and four points are always coplanar, or a decagon has 12 sides. False

Q: In the figure below, points A, B, C, and F lie on plane P. State the postulate that can be used to show each statement is true.
[Image]

Line AD contains points A and D.
- If two lines intersect¸ then their intersection is exactly one point. (Postulates: 1. Through any two points, there is exactly one line. 2. Through any three points not on the same line, there is exactly one plane. 3. A line contains at least two points. 4. A plane contains at least three points not on the same line. 5. If two points lie in a plane, then the entire line containing those points lies in that plane. 6. If two lines intersect, then their intersection is exactly one point. 7. If two planes intersect, then their intersection is a line.)
- If two points lie in a plane¸ then the entire line containing those points lies in that plane. (Postulates: 1. Through any two points, there is exactly one line. 2. Through any three points not on the same line, there is exactly one plane. 3. A line contains at least two points. 4. A plane contains at least three points not on the same line. 5. If two points lie in a plane, then the entire line containing those points lies in that plane. 6. If two lines intersect, then their intersection is exactly one point. 7. If two planes intersect, then their intersection is a line.)
- A line contains at least two points. (Postulates: 1. Through any two points, there is exactly one line. 2. Through any three points not on the same line, there is exactly one plane. 3. A line contains at least two points. 4. A plane contains at least three points not on the same line. 5. If two points lie in a plane, then the entire line containing those points lies in that plane. 6. If two lines intersect, then their intersection is exactly one point. 7. If two planes intersect, then their intersection is a line.)
- Through any two points¸ there is exactly one line. (Postulates: 1. Through any two points, there is exactly one line. 2. Through any three points not on the same line, there is exactly one plane. 3. A line contains at least two points. 4. A plane contains at least three points not on the same line. 5. If two points lie in a plane, then the entire line containing those points lies in that plane. 6. If two lines intersect, then their intersection is exactly one point. 7. If two planes intersect, then their intersection is a line.)
A: A line contains at least two points.

Write a two-column proof.

If [Image], then [Image].
[Image]

Write a two-column proof.

[Image]. Prove [Image].
[Image] [Image]

Write a paragraph proof.

Parallel lines TU and VW. Line XY intersects line TU at point Z and intersects line VW at point S. [Image]. Prove [Image].

Write a two-column proof.

[Image] intersect at point X. [Image]. [Image]. Prove [Image].
[Image]

Write the contrapositive of the conditional statement. Determine whether the contrapositive is true or false. If it is false, find a counterexample.

A counterexample invalidates a statement.

Write a two-column proof.

[Image] intersect at point X. Point X is the midpoint of both [Image]. [Image] is 4 units less than [Image]. If [Image] and [Image], prove [Image].
[Image]

Write a paragraph proof.

Regular hexagon ABCDFG; [Image]; [Image];  [Image] bisects [Image] and [Image] bisects [Image]. Prove: [Image].

Write a two-column proof.

If [Image], then [Image].

Write a two-column proof.

Regular hexagon ABCDFG; [Image] each bisect the hexagon and meet at point H. If [Image] and [Image], then [Image].
[Image]

Write a two-column proof.

[Image] intersect at S. [Image]. If [Image], then [Image].
[Image]

Write a paragraph proof.

Isosceles triangle GHJ. [Image] is a segment between points K and L on line KL. HG = JG. Prove [Image].
[Image]

Write the contrapositive of the conditional statement. Determine whether the contrapositive is true or false. If it is false, find a counterexample.

Two segments having the same measure are congruent.

Write a two-column proof.

Segments MN, LT, PQ, and OR all intersect at point S. If [Image], prove [Image].
[Image]

Write the inverse of the conditional statement. Determine whether the inverse is true or false. If it is false, find a counterexample.

A line is made up of points.

Write a paragraph proof.

Equilateral triangle ABC with points H and D on line AC, I and F on line AB, and J and G on line BC. [Image]. Prove [Image].
[Image]

Write the contrapositive of the conditional statement. Determine whether the contrapositive is true or false. If it is false, find a counterexample.

A line is determined by two points.

Write a two-column proof.

If [Image] and [Image], then [Image].
[Image]

Write a two-column proof.

If [Image] and [Image], then [Image].
[Image]2

Write a paragraph proof.

[Image] and [Image] are a linear pair. Prove that [Image].

Write a paragraph proof.

[Image] and [Image] intersect at point T. [Image]. Prove [Image].

Write a paragraph proof.

Point C is on line [Image] and line [Image]. [Image] Prove [Image].
[Image]

Write a two-column proof.

[Image] are all parallel to each other. [Image], [Image]. Prove [Image].
[Image]

Write the contrapositive of the conditional statement. Determine whether the contrapositive is true or false. If it is false, find a counterexample.

If you buy one can of soup, then you get the second one free.

Write the contrapositive of the conditional statement. Determine whether the contrapositive is true or false. If it is false, find a counterexample.

Vertical angles are two nonadjacent angles formed by two intersecting lines.

Write a paragraph proof.

[Image] and [Image] are vertical. Prove that [Image].

Write a two-column proof.

Line KL is parallel to line NM. Line PQ is parallel to line TU. If [Image], then [Image].
[Image]

Write a paragraph proof.

Adjacent angles [Image] and [Image]. Prove that point Q is in the interior of [Image].

Write a two-column proof.

[Image]. [Image] [Image]. [Image]. If [Image], prove [Image].
[Image] [Image]

Write a paragraph proof.

[Image] all intersect at point P. [Image] and [Image]. Prove [Image].
[Image]

[Image]
Jan has a patio room behind her house. The 15 ft wall of the patio room is perpendicular to the house. Jan would like to create a triangular flower bed using the two walls as the legs of the triangle. If she has 25 ft of edging material to go along the hypotenuse of the triangle, how long is the other leg of the flower bed? Show your work.

Given: [Image] and [Image] are supplementary angles. [Image], [Image].
Prove: [Image]