Do/Answer (2pts): I will write out my response to the question, explaining my answer and what it means. I will explain why my answer makes sense.
For question 1, use each of the reasons found in the Reasoning Bank (below), to complete the proof. (+2 pts)
Pro-tip: The reason explains how you got the statement, not what to do next.
Given: AB = 28; M lies on the line AB
Prove: M is a midpoint.
Statements
Reasons
1) M lies on AB
AB = 28
1)
2) AM + MB = AB
2)
3) 2x + 2 + 4x - 10 = 28
3) Substitution Property
4) 6x - 8 = 28
4)
5) 6x = 36
5)
6) x = 6
6)
7) AM = 2(6) + 2 and
MB = 4(6) - 10
7) Substitution Property
8) AM = 14 and MB = 14
8) Simplifying
9) M is a midpoint
9)
Reasoning Bank:
Addition Property of Equality, Combine like terms, Definition of Midpoint, Division Property of Equality, Given, and Segment Addition Postulate
1) Given
2) Definition of Midpoint (M is the point that divides AB into two equal segments, AM and MB)
3) Substitution Property (replace AB with 28)
4) Combine like terms (-10 + 2 = -8)
5) Addition Property of Equality (add -8 to both sides of the equation)
6) Division Property of Equality (divide both sides by 6)
7) Substitution Property (replace x with 6)
8) Simplifying (2(6) + 2 = 14 and 4(6) - 10 = 14)
9) Definition of Midpoint (AM and MB have equal lengths and M is the point that divides AB into two equal segments)
Based on the given information and the steps in the proof, we can conclude that M is a midpoint because it satisfies the definition of a midpoint - it divides the line AB into two equal segments. Additionally, the calculations in steps 7 and 8 show that the lengths of AM and MB are both 14, further supporting the fact that M is a midpoint. Therefore, my answer makes sense because it is supported by the given information and the logical progression of the proof.
Do/Answer (2pts): I will write out my response to the question, explaining my answer and what it means. I will explain why my answer makes sense.
Now that you have completed a proof by filling in the reasons for someone else’s statements (their work/steps), try to do a little bit of both on your own. For question 2, use your GeomA Reference sheet (posted on the Message Board/padlet), to complete the proof. (2pts)
Pro-tip: What do the two expressions make?
Given: Angles 1 and 2 are a linear pair.
Prove that x = 11
Statements
Reasons
1) Angles 1 and 2 are a linear pair.
1) Given
2) Angles 1 and 2 are supplementary.
2) Linear Pair Postulate
3) m∠1 + m∠2 = 180°
3)
4) 11x - 6 + 4x + 21 = 180
4)
5) 15x + 15 = 180
5)
6) 15x = 165
6)
7) x = 11
7)
To complete the proof, we can use the information given and the properties from the GeomA Reference sheet.
1) Given
2) Linear Pair Postulate (Angles that form a linear pair are supplementary)
3) Definition of supplementary angles (m∠1 + m∠2 = 180°)
4) Simplifying the expression (11x - 6 + 4x + 21 = 180)
5) Combine like terms (15x + 15 = 180)
6) Isolate the variable (15x = 165)
7) Solve for x (x = 11)
Based on the given information and the steps in the proof, we can conclude that x = 11. This means that the value of x that satisfies the given conditions is 11. This conclusion aligns with the calculations and properties used in the proof. Therefore, my answer makes sense because it is supported by the given information and the logical progression of the proof.
Do/Answer (3pts): I will write out my response to the question, explaining my answer and what it means. I will explain why my answer makes sense.
For question 3, use your GeomA Reference sheet (posted on the Message Board/padlet ), to complete the proof(3 pts).
Pro-tip: Solve the problem first
Given: -2(x - 4) = 2x + 12
Prove: x = -1
Statements
Reasons
1)
1)
2)
2) Distributive Property
3)
3)
4) -4x = 4
4)
5)
5) Division Property of Equality
To complete the proof, we need to solve the equation -2(x - 4) = 2x + 12 and show that it leads to the conclusion x = -1.
1) Given
2) Distributive Property (-2 * x = -2x and -2 * -4 = 8)
3) Simplifying the expression (-2x + 8 = 2x + 12)
4) Combining like terms (subtracting 2x from both sides, we get -4x + 8 = 12)
5) Subtracting 8 from both sides (we get -4x = 4)
6) Division Property of Equality (dividing both sides by -4, we get x = -1)
Based on the given equation and the steps in the proof, we can conclude that x = -1. This means that when we solve the equation -2(x - 4) = 2x + 12, the value of x that satisfies the equation is -1. This conclusion aligns with the calculations and properties used in the proof. Therefore, my answer makes sense because it is supported by the given information and the logical progression of the proof.
Here is the step-by-step solution to prove that M is a midpoint:
1) Given: M lies on AB, AB = 28
Reason: Given information
2) AM + MB = AB
Reason: Definition of Midpoint (A midpoint divides a segment into two congruent parts)
3) 2x + 2 + 4x - 10 = 28
Reason: Substitution Property (Substituting the values of AM and MB)
4) 6x - 8 = 28
Reason: Combining like terms
5) 6x = 36
Reason: Addition Property of Equality (Adding 8 to both sides)
6) x = 6
Reason: Division Property of Equality (Dividing both sides by 6)
7) AM = 2(6) + 2 and MB = 4(6) - 10
Reason: Substitution Property (Substituting the value of x)
8) AM = 14 and MB = 14
Reason: Simplifying the expressions
9) M is a midpoint
Reason: Definition of Midpoint (Since AM = MB, M is a midpoint)
Each step in this proof is supported by a valid reason, and the proof concludes by showing that M is indeed a midpoint.
To prove that M is a midpoint, we need to use the given information and the reasoning bank provided.
First, we know that M lies on AB, so we can use the reason "Given" in statement 1.
Next, we can use the reason "Definition of Midpoint" in statement 7. This reason states that if a point divides a line segment into two congruent segments, then it is the midpoint of the line segment. In this case, we need to show that AM is congruent to MB.
To do this, we use the reason "Segment Addition Postulate" in statement 2. The Segment Addition Postulate states that if three points A, M, and B are collinear and M is between A and B, then AM + MB = AB. Since AB is given as 28 in the problem, we substitute 28 for AB in statement 2.
Now we need to simplify the equation. In statement 3, we use the reason "Substitution Property" to substitute the equation AM + MB = 28 into the equation 2x + 2 + 4x - 10 = 28.
Then, in statement 4, we combine like terms using the reason "Combine like terms". This gives us the equation 6x - 8 = 28.
To isolate the variable x, we subtract 8 from both sides of the equation in statement 4. This follows the reason "Addition Property of Equality".
In statement 5, we have the equation 6x = 36 after simplifying. We can now use the reason "Division Property of Equality" to divide both sides of the equation by 6, which gives us x = 6 in statement 6.
Now that we have found the value of x, we can substitute it back into the equations for AM and MB. Using the reason "Substitution Property" in statement 7, we substitute x = 6 into the equations AM = 2(6) + 2 and MB = 4(6) - 10.
Simplifying these equations in statement 8, we get AM = 14 and MB = 14, showing that the segments AM and MB are congruent.
Finally, in statement 9, we use the reason "Definition of Midpoint" to conclude that M is a midpoint because AM is congruent to MB.
Therefore, statement 9, which states "M is a midpoint," is proven using the given information and the reasoning bank provided.