Determine whether the conjecture is true or false. If false, give a counterexample.
Given: ∠A is supplementary to ∠B and ∠B is supplementary to ∠C.
Conjecture: ∠A is supplementary to ∠C.
Select one:
a. False; they could be right angles.
b. False; they could be congruent angles.
c. True
d. False; they could be vertical angles.
Determine whether the conjecture is true or false. If false, give a counterexample.
Given: x2+4=8
Conjecture: x=−2
Select one:
a. False; x = 4
b. True
c. False; x = 3
d. False; x = 2
Make a conjecture about the next item in the sequence. 2, -20, 200, -2000, ?.
Determine whether the following conjecture is true or false. Give a counterexample if the answer is false.
AB¯¯¯¯¯¯¯¯⊥BC¯¯¯¯¯¯¯¯, then ∠ABC is a right angle
Make a conjecture about the next item in the sequence. 9, 14, 19, 24, 29, ?.
Make a conjecture about the next item in the sequence. 11, 19, 27, 35, 43, ?.
Make a conjecture about the next item in the sequence.
1, -4, 9, -16, ?
Determine whether the conjecture is true or false. If false, give a counterexample.
Given: ∠LMN and ∠XMZ are coplanar.
Conjecture: The angles are vertical angles.
Select one:
a. False; the angles may be supplementary.
b. True
c. False; one angle may be in the interior of the other.
d. False; the angles may be adjacent.
Write a conjecture about the following relationship.
∠A and ∠B are both right angles
To determine whether a conjecture is true or false, we need to carefully analyze the given information and apply our knowledge of math concepts. Let's break down each question and explain how to get the answers:
1. Given: ∠A is supplementary to ∠B and ∠B is supplementary to ∠C.
Conjecture: ∠A is supplementary to ∠C.
To check the conjecture, we need to understand the definition of supplementary angles. Supplementary angles add up to 180 degrees. If ∠A and ∠B are supplementary, their sum is 180 degrees. Similarly, if ∠B and ∠C are supplementary, their sum is also 180 degrees.
To determine if the conjecture is true or false, we evaluate if the sum of ∠A and ∠C is 180 degrees. If it is, then the conjecture is true; otherwise, it is false.
Answer: The correct answer is (c) True. Since both ∠A and ∠C are supplementary to ∠B, it follows that they are supplementary to each other.
2. Given: x^2 + 4 = 8
Conjecture: x = -2
To validate the conjecture, we solve the equation to find the value of x. By subtracting 4 from both sides of the equation: x^2 = 4. Taking the square root of both sides: x = ±2.
To determine if the conjecture x = -2 is true or false, we substitute -2 into the original equation. If it satisfies the equation, then the conjecture is true.
Answer: The correct answer is (b) True. When we substitute x = -2 into the equation, we get (-2)^2 + 4 = 8, which is true.
3. Make a conjecture about the next item in the sequence: 2, -20, 200, -2000, ?.
To determine the pattern in the sequence, we can observe that each term alternates between being positive and negative, while the magnitude increases by a power of 10 (10, 100, 1000, ...).
Based on this pattern, we can conjecture that the next term in the sequence will be a negative number whose magnitude is 10,000 (10^4).
Answer: The conjecture is that the next item in the sequence is -10,000.
4. Determine whether the following conjecture is true or false: AB¯¯¯¯¯¯¯¯ ⊥ BC¯¯¯¯¯¯¯¯, then ∠ABC is a right angle.
To determine the truth of the conjecture, we need to recall the definition of perpendicular lines. When two lines are perpendicular, their corresponding angles are right angles (90 degrees).
To confirm the conjecture, we need to verify if ∠ABC is a right angle given that AB¯¯¯¯¯¯¯¯ is perpendicular to BC¯¯¯¯¯¯¯¯.
Answer: The correct answer is (b) True. When AB¯¯¯¯¯¯¯¯ is perpendicular to BC¯¯¯¯¯¯¯¯, then ∠ABC is indeed a right angle.
5. Make a conjecture about the next item in the sequence: 9, 14, 19, 24, 29, ?.
To identify the pattern in the sequence, we can observe that each term increases by 5.
Based on this pattern, we can conjecture that the next item in the sequence will be obtained by adding 5 to the previous term.
Answer: The conjecture is that the next item in the sequence is 34.
6. Make a conjecture about the next item in the sequence: 11, 19, 27, 35, 43, ?.
To identify the pattern in the sequence, we can observe that each term increases by 8.
Based on this pattern, we can conjecture that the next item in the sequence will be obtained by adding 8 to the previous term.
Answer: The conjecture is that the next item in the sequence is 51.
7. Make a conjecture about the next item in the sequence: 1, -4, 9, -16, ?.
To identify the pattern in the sequence, we can observe that each term is obtained by taking the square of the next odd number and alternating signs.
Based on this pattern, we can conjecture that the next item in the sequence will be obtained by taking the square of the next odd number, which is 5 in this case, and assigning it a negative sign.
Answer: The conjecture is that the next item in the sequence is -25.
8. Determine whether the conjecture is true or false: Given ∠LMN and ∠XMZ are coplanar. Conjecture: The angles are vertical angles.
To determine the truth of the conjecture, we need to review the definition of vertical angles. Vertical angles are formed by two intersecting lines, and they are congruent (equal in measure).
Given the information that ∠LMN and ∠XMZ are coplanar, we can evaluate if they meet the conditions of being vertical angles.
Answer: The correct answer is (d) False. Coplanar angles do not necessarily have to be vertical angles. Coplanar angles could be adjacent angles, supplementary angles, or any other type of angles.
9. Write a conjecture about the following relationship: ∠A and ∠B are both right angles.
To write a conjecture, we need to analyze the given relationship. If ∠A and ∠B are both right angles, it implies that both angles measure 90 degrees.
Based on this information, we can conjecture that ∠A and ∠B are congruent angles.
Answer: The conjecture is that ∠A and ∠B are congruent angles.
Given: ∠A is supplementary to ∠B and ∠B is supplementary to ∠C.
Conjecture: ∠A is supplementary to ∠C.
Answer: c. True
Given: x^2 + 4 = 8
Conjecture: x = -2
Answer: b. True
Next item in the sequence: 2, -20, 200, -2000, ?:
Conjecture: -20000
Reason: The sequence is multiplied by -10 each time.
Given: AB is perpendicular to BC
Conjecture: ∠ABC is a right angle
Answer: b. True
Next item in the sequence: 9, 14, 19, 24, 29, ?:
Conjecture: 34
Reason: The sequence increases by 5 each time.
Next item in the sequence: 11, 19, 27, 35, 43, ?:
Conjecture: 51
Reason: The sequence increases by 8 each time.
Next item in the sequence: 1, -4, 9, -16, ?:
Conjecture: 25
Reason: The sequence alternates between squares of odd numbers and their negatives.
Given: ∠LMN and ∠XMZ are coplanar.
Conjecture: The angles are vertical angles.
Answer: a. False; the angles may be supplementary.
Conjecture about the relationship between ∠A and ∠B:
The angles are congruent.
What do you think the answers are?
We can check your work once you have posted some.
For #1 consider that supplementary angles are two or more angles that add up to 180 degrees. I would suggest sketching a diagram to aid you.
THat should give you some ideas for your first answer : )