Part A: MUST SHOW ALL WORK FOR CREDIT. (10 points Total)
Understand & Think (1pt): What is being asked in the problem below and what does that mean? What do I know and what does it mean? What plan am I going to try?
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:
Understand & Think (1pt): What is being asked in the problem below and what does that mean? What do I know and what does it mean? What plan am I going to try?
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)
For question 1, we are given that AB = 28 and M lies on the line AB. We need to prove that M is a midpoint.
To prove this, we can use the fact that a point is a midpoint if the sum of the distances from one endpoint to the midpoint and from the midpoint to the other endpoint is equal to the length of the line segment.
Based on this, we can say that AM + MB = AB.
Substituting the given values, we have 2x + 2 + 4x - 10 = 28. (Using the fact that AM = 2x + 2 and MB = 4x - 10)
Simplifying this equation, we have 6x - 8 = 28.
Solving for x, we get 6x = 36, which gives us x = 6.
Substituting x = 6 back into the expressions for AM and MB, we get AM = 14 and MB = 14.
Since AM + MB = 28, we can conclude that M is indeed the midpoint of AB.
This makes sense because if M is the midpoint, then the distances from A to M and from M to B should be equal, and that is what we have shown.
For question 2, we are given that angles 1 and 2 are a linear pair. We need to prove that x = 11.
A linear pair consists of two angles that are adjacent and supplementary, meaning they add up to 180 degrees.
So we can say that m∠1 + m∠2 = 180°.
Substituting the given expression for m∠1 and m∠2, we have 11x - 6 + 4x + 21 = 180.
Simplifying this equation, we get 15x + 15 = 180.
Solving for x, we find that 15x = 165, so x = 11.
Therefore, we have proved that x is indeed equal to 11.
This makes sense because if angles 1 and 2 are a linear pair and add up to 180 degrees, then the value of x should be such that the equation holds true, and that is what we have shown.