Which biconditional is not a good definition?
(1 point)
Responses
A whole number is odd if and only if the number is not divisible by 2.
A whole number is odd if and only if the number is not divisible by 2.
An angle is straight if and only if its measure is 180°.
An angle is straight if and only if its measure is 180°.
A whole number is even if and only if it is divisible by 2.
A whole number is even if and only if it is divisible by 2.
A ray is a bisector of an angle if and only if it splits the angle into two angles.
Name the Property of Equality that justifies the statement:
If RS = ST and ST = TU, then RS = TU
(1 point)
Responses
Reflexive Property
Reflexive Property
Symmetric Property
Symmetric Property
Transitive Property
Transitive Property
Addition Property
The correct answer is: Transitive Property
The correct answer is: A ray is a bisector of an angle if and only if it splits the angle into two angles.
If a conditional and its converse are always true, then the statement is a
(1 point)
Responses
converse.
converse.
conditional.
conditional.
biconditional.
biconditional.
counterexample.
The correct answer is: biconditional.
Which shows a true conditional with a correctly identified hypothesis and conclusion?
(1 point)
Responses
If it is raining, then the humidity level is 100% .
Hypothesis: The humidity level is 100%.
Conclusion: It is raining.
If it is raining, then the humidity level is 100% . Hypothesis: The humidity level is 100%. Conclusion: It is raining.
I hate humidity, except when it rains.
Hypothesis: I hate humidity.
Conclusion: It rains.
I hate humidity, except when it rains. Hypothesis: I hate humidity. Conclusion: It rains.
I hate humidity, except when it rains.
Hypothesis: It rains.
Conclusion: I hate humidity.
I hate humidity, except when it rains. Hypothesis: It rains. Conclusion: I hate humidity.
If it is raining, then the humidity level is 100%.
Hypothesis: It is raining.
Conclusion: The humidity level is 100%.
The correct answer is: If it is raining, then the humidity level is 100%.
Hypothesis: It is raining.
Conclusion: The humidity level is 100%.
Is the following definition of complementary reversible? If yes, write it as a true biconditional.
Complementary angles are two angles whose sum measures to 90°.
(1 point)
Responses
The statement is not reversible.
The statement is not reversible.
Yes, if angles are complementary, then their sum measures to 90°.
Yes, if angles are complementary, then their sum measures to 90°.
Yes, angles are complementary if (and only if) their sum measures to 90°.
Yes, angles are complementary if (and only if) their sum measures to 90°.
Yes, angles are complementary if their sum measures to 90°.
The correct answer is: Yes, angles are complementary if (and only if) their sum measures to 90°.
Name the Property of Congruence that justifies the statement: If AB is congruent to CD then CD is congruent to AB.
(1 point)
Responses
Transitive Property
Transitive Property
Symmetric Property
Symmetric Property
Reflexive Property
Reflexive Property
none of these
The correct answer is: Symmetric Property
Write the converse of the following true conditional statement. If the converse is false, write a counterexample.
If x less-than 20, then x less-than 30.
(1 point)
Responses
If x less-than 30, then x less-than 20; true.
If Image with alt text: x less-than 30 , then Image with alt text: x less-than 20 ; true.
If x less-than 30, then x less-than 20; false.
Counterexample: x = 27 and x greater-than 20.
If Image with alt text: x less-than 30 , then Image with alt text: x less-than 20 ; false. Counterexample: x = 27 and Image with alt text: x greater-than 20 .
If x greater-than 20, then x greater-than 30; false.
Counterexample: x = 25 and x less-than 30.
If Image with alt text: x greater-than 20 , then Image with alt text: x greater-than 30 ; false. Counterexample: x = 25 and Image with alt text: x less-than 30 .
If x greater-than 30, then x greater-than 20; true.
The correct answer is: If x greater-than 30, then x greater-than 20; false.
Counterexample: x = 25 and x less-than 30.
Use the Division Property of Equality to complete the following statement.
If 5x = 2y, then x = ?
(1 point)
Responses
Start Fraction 2 over 5 lower y End Fraction
Image with alt text: Start Fraction 2 over 5 lower y End Fraction
10y
10 y
Start Fraction 2 lower y over 5 End Fraction
Image with alt text: Start Fraction 2 lower y over 5 End Fraction
2y — 5
To complete the statement using the Division Property of Equality, the answer would be:
Start Fraction 2 over 5 lower y End Fraction
Which statement is an example of the Addition Property of Equality?
(1 point)
Responses
If p = q, then p · s = q · s.
If p = q, then p · s = q · s .
If p = q, then p + s = q + s.
If p = q, then p + s = q + s.
If p = q, then p – s = q – s.
If p = q, then p – s = q – s.
p = q
The correct answer is: If p = q, then p + s = q + s.
The measure of two vertical angles are 6 x minus 22 and 4 x plus 2. Find x.
(1 point)
Responses
50
50
12
12
10
10
130
To find the value of x, we need to set the measures of the two vertical angles equal to each other:
6x - 22 = 4x + 2
Now, we will solve this equation for x:
6x - 4x = 2 + 22
2x = 24
x = 12
So, the value of x is 12.
m angle 2 equals 34 degrees. Find m angle 4 and explain how you know.
Two lines intersect in the shape of an x to form four angles. The angle at the top is labeled 1. The angle on the right is labeled 2. The angle at the bottom is labeled 3. The angle on the left is labeled 4.
To find m angle 4, we need to understand the relationship between angle 2 and angle 4.
In an X-shaped intersection of two lines, angle 2 and angle 4 are vertical angles. Vertical angles are always congruent. Therefore, if m angle 2 is given as 34 degrees, we can conclude that m angle 4 is also 34 degrees.
Complete the two-column proof.
DiagramTwo horizontal lines that appear to be parallel are cut by a transversal. Where the transversal intersects the top horizontal line, the angle in the top right corner is obtuse and labeled with a 2. Where the transversal intersects the bottom horizontal line, the angle in the top left corner is acute and labeled with a 1, the angle in the bottom left corner is obtuse and labeled with a 4, and the angle in the bottom right corner is acute and labeled with a 3.
Drawing not to scale.
Given: Angle 2 is congruent to angle 4, the measure of angle 2 is 110 degrees
Prove: m angle D equals 70 degrees
Statement Proof
Angle 2 is congruent to angle 4, the measure of angle 2 is 110 degrees Given
measure of angle 2 equal to measure of angle 4 Definition of congruent angles
m angle 4 equals 110 degrees a.
m angle 3 and m angle 4are a linear pair Definition of a linear pair (shown in diagram)
m angle 3 and m angle 4are supplementary Linear Pair Postulate
m angle 3 plus m angle 4 equals 180 degrees b.
m angle 3 plus 110 degrees equals 180 degrees Substitution Property
m angle D equals 70 degrees c.
Statement | Proof
----------------|-----------
Angle 2 is congruent to angle 4, the measure of angle 2 is 110 degrees | Given
Measure of angle 2 equals measure of angle 4 | Definition of congruent angles
Measure of angle 4 equals 110 degrees | a. (Using the Transitive Property)
Angles 3 and 4 are a linear pair | Definition of a linear pair (shown in diagram)
Angles 3 and 4 are supplementary | Linear Pair Postulate
Measure of angle 3 plus measure of angle 4 equals 180 degrees | b. (Definition of supplementary angles)
Measure of angle 3 plus 110 degrees equals 180 degrees | Substitution Property
Measure of angle D equals 70 degrees | c. (Subtraction Property of Equality)
Complete the paragraph proof.
Given: ∠1 and ∠2 are supplementary, and ∠2 and ∠3 are supplementary.
Prove: ∠1 ≅ ∠3
Angle 1 is acute, angle 2 is obtuse, and angle three is acute.
By definition of supplementary angles, m∠1 + m∠2 = _____ (a) and m∠2 + m∠3 = _____ (b). Then, m∠1 + m∠2 = m∠2 + m∠3 by the __________ (c). Subtract m∠2 from each side. You get m∠1 = ______ (d), or ∠1 ≅ _____ (e).
Given: ∠1 and ∠2 are supplementary, and ∠2 and ∠3 are supplementary.
Prove: ∠1 ≅ ∠3
Angle 1 is acute, angle 2 is obtuse, and angle three is acute.
By definition of supplementary angles, m∠1 + m∠2 = 180° (a) and m∠2 + m∠3 = 180° (b). Then, m∠1 + m∠2 = m∠2 + m∠3 by the Transitive Property of Equality (c). Subtracting m∠2 from each side, we get m∠1 = m∠3 (d), or ∠1 ≅ ∠3 (e).
Explanation:
In this proof, we start by stating that ∠1 and ∠2 are supplementary, which means they add up to 180°. We also state that ∠2 and ∠3 are supplementary, meaning they also add up to 180°. We use the Transitive Property of Equality to say that if ∠1 + ∠2 is equal to ∠2 + ∠3, then ∠1 must be equal to ∠3. This is because we can subtract ∠2 from both sides of the equation. Therefore, we have proven that ∠1 ≅ ∠3, meaning they are congruent.
What is the value of x? Justify each step.
AC = 32
A horizontal line is labeled with three points, A, B, and C. Segment AB is labeled 2 x and segment BC is labeled 6 x plus 8. The diagram is not to scale.
The first column contains steps and the second column provides blanks where justification for each step should be written.
Apologies, but I cannot provide justifications for each step without a complete list of steps and corresponding blanks. Could you please provide the full list of steps and the corresponding blanks where the justifications should be written?
Write the conditional statement that the Venn diagram illustrates.
A Venn diagram is shown. The larger circle is labeled 'Quadrilaterals,' and a smaller circle inside the larger one is labeled 'Squares.'
The conditional statement that the Venn diagram illustrates is:
"If a shape is a square, then it is a quadrilateral."
Write a two-column proof.
Given: 7y = 8x – 14; y = 6
Prove: x = 7
Statement | Reason
---------------------|------------------
y = 6 | Given
7y = 8x – 14 | Given
7(6) = 8x – 14 | Substitution using y = 6
42 = 8x – 14 | Simplification
42 + 14 = 8x | Addition Property of Equality
56 = 8x | Simplification
8x = 56 | Symmetric Property of Equality
(1/8)(8x) = (1/8)(56)| Multiplication Property of Equality
x = 7 | Simplification
ray EC bisects angle BED, The measure of angle AEB equals 11x minus 12.and The measure of angle CED equals 4x plus 1.. Find measure of angle AEC.Horizontal line AED shown with line segments EB and EC extending upward.
We are given that ray EC bisects angle BED. Therefore, angle AEC is congruent to angle CED.
Let the measure of angle AEC be y. Since ray EC bisects angle BED, angle AEB is twice the measure of angle AEC. Therefore, we have:
Measure of angle AEB = 2y (since it is twice angle AEC)
Given the measures of angle AEB and angle CED, we can set up an equation and solve for y:
11x - 12 = 4x + 1
11x - 4x = 1 + 12
7x = 13
x = 13/7
Substituting this value of x back into the equation for angle AEB:
Measure of angle AEB = 2y
11(13/7) - 12 = 2y
143/7 - 84/7 = 2y
59/7 = 2y
Multiplying both sides by 7/2, we get:
59/2 = y
Therefore, the measure of angle AEC is 59/2.
11x - 12 = 4x + 1
11x - 4x = 1 + 12
7x = 13
x = 13/7
Substituting this value of x back into the equation for angle AEB:
Measure of angle AEB = 2y
11(13/7) - 12 = 2y
143/7 - 84/7 = 2y
59/7 = 2y
Multiplying both sides by 7/2, we get:
59/2 = y
Apologies for the oversight. Let's correct the calculations:
11x - 12 = 4x + 1
11x - 4x = 1 + 12
7x = 13
x = 13/7
Substituting this value of x back into the equation for angle AEB:
Measure of angle AEB = 11x - 12
Measure of angle AEB = 11(13/7) - 12
Measure of angle AEB = 143/7 - 12
Measure of angle AEB = 143/7 - 84/7
Measure of angle AEB = 59/7
Therefore, the measure of angle AEB is 59/7.