Proofs

demha · 2013-09-26 05:13:50

Yet again, I have another lesson I need help with. As usual there are 20 problems to be solved but I will be posting 5 at a time. This is what my lesson says based on this topic,
"One of the more traditional tools of Geometry is called a proof. Proofs are usually set up so that there is a problem on the left half of a page, and on the right half is space for you to justify each step. In essence, you need to defend, using theorums, postulates, and definitions,why it is you're able to do what you just did."

PROOFS
1. If I have two coplanar lines, I must have a plane. - My answers is A. (there is not enough information to prove so.)
A unfounded
B Definition of a point
C Definition of a plane
D Given
E Definition of a line
F Definition of radius

2. There are two adjacent angles whose outside edges form a straight line. The measure of the first angle is 100 degrees, so the measure of the other must be 80 degrees. - I choose B (we all know that supplementary angles must sum up to 180 degrees)
A Given
B Definition of supplementary angles
C Definition of an octagon
D unfounded
E Definition of radius
F 1267200 inches

3. I have drawn a polygon with eight sides, so it must be an octagon. - Choose F. (the only polygon with 8 sides is an octagon)
A Definition of supplementary angles
B unfounded
C Definition of radius
D 1267200 inches
E Definition of an octagon
F Given

4. A square has two diagonals. - I choose D (proof is given a square can only have two diagonals)
A Given
B unfounded
C Definition of an octagon
D The number of diagonals is (n-3)n/2, where n is the number of sides, which is 4 for a square.
E 1267200 inches
F Definition of supplementary angles

5. If the diameter of the circle is 12, the radius must be 6. - I choose F (the diameter is always twice the radius)
A Definition of radius
B 1267200 inches
C Definition of an octagon
D Definition of supplementary angles
E unfounded
F Given

Bob · 2013-09-26 07:56:04

hi demha

PROOFS
1. If I have two coplanar lines, I must have a plane. - My answers is A. (there is not enough information to prove so.)
A unfounded
B Definition of a point
C Definition of a plane
D Given
E Definition of a line
F Definition of radius

Correct.

2. There are two adjacent angles whose outside edges form a straight line. The measure of the first angle is 100 degrees, so the measure of the other must be 80 degrees. - I choose B (we all know that supplementary angles must sum up to 180 degrees)
A Given
B Definition of supplementary angles
C Definition of an octagon
D unfounded
E Definition of radius
F 1267200 inches

Correct.

3. I have drawn a polygon with eight sides, so it must be an octagon. - Choose F. (the only polygon with 8 sides is an octagon)
A Definition of supplementary angles
B unfounded
C Definition of radius
D 1267200 inches
E Definition of an octagon
F Given

What you have in the bracket is correct but you have chosen the wrong answer.

4. A square has two diagonals. - I choose D (proof is given a square can only have two diagonals)
A Given
B unfounded
C Definition of an octagon
D The number of diagonals is (n-3)n/2, where n is the number of sides, which is 4 for a square.
E 1267200 inches
F Definition of supplementary angles

Correct.

5. If the diameter of the circle is 12, the radius must be 6. - I choose F (the diameter is always twice the radius)
A Definition of radius
B 1267200 inches
C Definition of an octagon
D Definition of supplementary angles
E unfounded
F Given

What you have in the bracket is correct but you have chosen the wrong answer.

Bob

demha · 2013-09-26 08:12:43

3. I have drawn a polygon with eight sides, so it must be an octagon. - My New Answer is E. (the only polygon with 8 sides is an octagon)
A Definition of supplementary angles
B unfounded
C Definition of radius
D 1267200 inches
E Definition of an octagon
F Given

So it would be the definition but to prove it further information is required, yes?

---

5. If the diameter of the circle is 12, the radius must be 6. - My New Answer is A (the diameter is always twice the radius)
A Definition of radius
B 1267200 inches
C Definition of an octagon
D Definition of supplementary angles
E unfounded
F Given

Similar to #3, correct?

Bob · 2013-09-26 08:57:17

Both correct now.

Bob

demha · 2013-09-26 10:00:34

Cool! On to the next 5:

The follow questions #6 - #10 will be using this picture:

6. In the figure above, line segment MC is equal to imaginary line segment MI. - My Answer is F (does not exactly explain, so I choose 'F' instead of 'A')
A Given
B unfounded
C Definition of supplementary angles
D 1267200 inches
E Definition of an octagon
F Definition of a circle: all points are equidistant from the center

7. In the figure above, line segment EJ is equal to line segment JM - My Answer is B (not enough information or proof is given)
A Definition of radius
B unfounded
C Definition of an octagon
D 1267200 inches
E Given
F Definition of supplementary angles

8. In the figure above, the measure of angle AMC is 90 degrees. - My Answer is C (not enough information or proof is given)
A Given
B Definition of radius
C unfounded
D Definition of an octagon
E 1267200 inches
F Definition of supplementary angles

9. In the figure above, the measure of arc AC is 90 degrees. - My Answer is C (i thought about 'A' but then realized that this is not always true, the arc may be shorter or longer than the central angle that subtends it [for example the 'EA' arc])
A The angle measure of an arc is equal to the measure of the central angle that subtends it.
B 126498 inches
C unfounded
D Definition of an octagon
E Definition of supplementary angles
F 1267200 inches

10. In the figure above the measure of angle AME is x degrees, then the measure of angle EMB is 180-x degrees. - My Answer is A (proof is given for this one which led me to choose 'A - Given')
A Given
B unfounded
C Definition of an octagon
D Definition of supplementary angles
E 1267200 inches
F Definition of radius

Last edited by demha (2013-09-26 21:54:40)

Bob · 2013-09-26 19:13:56

hi demha

That image isn't accessible without a password. Would you be able to produce an image by another method?

Bob

demha · 2013-09-26 21:54:56

I have edited my post. It should be appearing now.

anonimnystefy · 2013-09-26 22:17:59

Hi demha

6 and 7 are correct. In Q8, you put that it is unfounded. I'd say that the information is given.

For Q9: What do you mean, it can be shorter or longer?

Q10: Shouldn't it be D?

The answers above are what I am seeing, but you should definitely wait for confirmation from someone else.

demha · 2013-09-26 23:09:40

8. In the figure above, the measure of angle AMC is 90 degrees. - So new answer would be 'A'? I said 'C' because not enough proof/information was given to support the answer.
A Given
B Definition of radius
C unfounded
D Definition of an octagon
E 1267200 inches
F Definition of supplementary angles

9. In the figure above, the measure of arc AC is 90 degrees. - My Answer is C - answer 'A' says that the measure of an arc is equal to the measure of the angle that subtends it, which is not always true. The arc can be of different lengths sometimes. But in THIS case, would 'A' be the correct answer?
A The angle measure of an arc is equal to the measure of the central angle that subtends it.
B 126498 inches
C unfounded
D Definition of an octagon
E Definition of supplementary angles
F 1267200 inches

10. In the figure above the measure of angle AME is x degrees, then the measure of angle EMB is 180-x degrees. - So my new answer would be 'D'? Now that I think of it, it does seem like two supplementary angles. Since the image is a circle I guess I wasn't expecting them to implement supplementary angles into it.
A Given
B unfounded
C Definition of an octagon
D Definition of supplementary angles
E 1267200 inches
F Definition of radius

Last edited by demha (2013-09-26 23:11:01)

Bob · 2013-09-27 00:20:42

Revised answers look OK now.

the measure of an arc is equal to the measure of the angle that subtends it, which is not always true.

The way this is worded is a little strange to me but I cannot think of a time when this is not true. Can you give an example?

EDIT: I have checked the definition of 'subtends'. A LINE (not necessarily straight) subtends an ANGLE, not he other way round.

Bob

Last edited by Bob (2013-09-27 00:24:11)

demha · 2013-09-27 00:59:26

Well with this picture above, look at the arcs EMA and EMB create. They are not the same length as the angle that holds them. But I might be wrong which makes me reconsider the answer for #9. 'A' is correct, yes?

As we take care of that I would like to post the next 5:
11. In a right triangle where one side is 3, and the hypotenuse is 5, the remaining side must be 4. - My answer is E
A Given
B unfounded
C 45-45-90 Theorem
D Definition of supplementary angles
E Pythagorean Theorem
F Definition of a triangle

12. In a triangle, if I have two angles that add up to 50 degrees, the remaining angle must be 130 degrees. - I would say D for this one
A Given
B unfounded
C Definition of supplementary angles
D Sum of angles in a triangle
E Definition of triangle inequality
F Definition of radius

13. The diameter of a circle always passes through the midpoint of the circle. - A little confused which would be correct, I go with C on this one
A Given
B unfounded
C Definition of diameter
D Definition of supplementary angles
E Definition of a circle
F Definition of radius

14. If a central angle is 30 degrees, then the arc it defines is also 30 degrees. - I would go with A
A Given
B Definition of an inscribed angle
C unfounded
D Properties of a central angle
E Properties of an arc
F Definition of radius

15. The area of a sphere is 4 times the area of a circle with the same radius. - My answer is A
A Given
B Definition of a radius
C Definition of a circle
D Definition of a sphere
E Formula for area of a sphere
F unfounded

Agnishom · 2013-09-27 01:23:51

15 is E; 14 is what I'm confused about.
Rest are all okay

demha · 2013-09-27 03:09:37

Alright so #15 my new answer will be 'E'. As for #14, I am starting to reconsider my answer, I was thinking of 'D'.

Bob · 2013-09-27 05:30:44

hi demha,

This business of arcs and angles and things subtending other things is leaving me mystified. I don't really understand what your teacher is trying to say. I have, as usual, made a diagram (see below).

Here we have a circle, centre B and a chord AC. The angle subtended by the chord is ABC.

I have marked the arc AC in red. It also subtends an angle of the same size. I think there may be a theorem somewhere in Euclid's Elements that prove that the angle subtended by the chord, and the angle subtended by the arc, are the same. Most people would probably look at the diagram and say it's obvious that they are the same and what is all the fuss about? I wouldn't waste any more time on it; you have the right answer already.

Now for the other questions:

Q11. E is correct.

Q12. D is correct.

Q13. To decide between C, E and F I would have to know how a circle is being defined, how a 'midpoint of a circle' is being defined and how radius and diameter are being defined. If the question gives no more clues then I'd have to guess between these three possibilities, Sorry, I think it's a poor question.

EDIT: I looked it up

http://www.mathsisfun.com/definitions/diameter.html

Q14. As I said at the beginning these are the same as a result of a theorem ( => a property) rather than a given.

Q15. Revised answer E is good.

Bob

Last edited by Bob (2013-09-27 05:36:09)

anonimnystefy · 2013-09-27 06:07:37

Hi

I'd go for C with Q13.

demha · 2013-09-27 06:37:26

So for #13 I am sticking with 'C'

#9 i just want to make sure, 'A' is correct? Here is the question again:
9. In the figure above, the measure of arc AC is 90 degrees.
A The angle measure of an arc is equal to the measure of the central angle that subtends it.
B 126498 inches
C unfounded
D Definition of an octagon
E Definition of supplementary angles
F 1267200 inches

#14 so the true answer would be 'D', correct?

Just want to make sure with all of these before moving on to the final 5.

anonimnystefy · 2013-09-27 07:18:40

Those are all correct.

demha · 2013-09-27 09:05:55

Alright that's good to hear! Last 5:

16. If a radius bisects a chord, then the lengths of the parts of the radius on either side of the chord are equal. - I would go with 'C'
A Given
B Definition of a chord
C unfounded
D Definition of supplementary angles
E Definition of a bisector
F Definition of radius

17. If a circle has a central point M, and both point A and point D are on the circle, then ls_MA and ls_MD will be equal. - I'm thinking of 'E' on this
A Given
B unfounded
C Definition of a line segment
D Definition of supplementary angles
E Definition of a bisector
F Definition of radius

18. If I have two points, (-2, -3) and (-4, 4) then the distance between them is sqrt(53). - I'll go with 'C'
A Distance Formula
B Definition of a point
C Definition of a distance
D Definition of a line
E Definition of coordinates
F Definition of radius

19. The given points (4, -8), (4, -5), and (-2, -6) make a right triangle. - I choose 'D'
A Distance Formula
B Definition of a right triangle
C Definition of a triangle
D unfounded
E Pythagorean Theorem
F Definition of radius

20. The given points (2, -3), (-7, -7), (2, -7), and (-7, -2) make a square. - I think 'F' would be the answer
A Definition of coordinate
B Pythagorean Theorem
C Definition of a square
D Definition of supplementary angles
E Distance Formula
F unfounded

anonimnystefy · 2013-09-27 09:15:29

demha wrote:

Alright that's good to hear! Last 5:
16. If a radius bisects a chord, then the lengths of the parts of the radius on either side of the chord are equal. - I would go with 'C'
A Given
B Definition of a chord
C unfounded
D Definition of supplementary angles
E Definition of a bisector
F Definition of radius
Correct.
17. If a circle has a central point M, and both point A and point D are on the circle, then ls_MA and ls_MD will be equal. - I'm thinking of 'E' on this
A Given
B unfounded
C Definition of a line segment
D Definition of supplementary angles
E Definition of a bisector
F Definition of radius
I don't see what the question has to do with bisectors. I'd say this one would be F, because MA and MD are radii.
18. If I have two points, (-2, -3) and (-4, 4) then the distance between them is sqrt(53). - I'll go with 'C'
A Distance Formula
B Definition of a point
C Definition of a distance
D Definition of a line
E Definition of coordinates
F Definition of radius
Here, you are using the Distance formula to calculate distance, not the definition.
19. The given points (4, -8), (4, -5), and (-2, -6) make a right triangle. - I choose 'D'
A Distance Formula
B Definition of a right triangle
C Definition of a triangle
D unfounded
E Pythagorean Theorem
F Definition of radius
Correct.
20. The given points (2, -3), (-7, -7), (2, -7), and (-7, -2) make a square. - I think 'F' would be the answer
A Definition of coordinate
B Pythagorean Theorem
C Definition of a square
D Definition of supplementary angles
E Distance Formula
F unfounded
Correct.

demha · 2013-09-27 12:26:29

Ok, so #17 is F.

As for #18, it would be A.

Now I submitted my lesson and got 19 out of 20. #1 was wrong:
1. If I have two coplanar lines, I must have a plane. - My original answer was 'A'. If that is so, I was thinking it couldn't be 'D' because not enough proof/information is given. So would this be 'C', definition of a plane?
A unfounded
B Definition of a point
C Definition of a plane
D Given
E Definition of a line
F Definition of radius

anonimnystefy · 2013-09-27 13:40:02

Yes, I think Q1 is C.

Good job!

Math Is Fun Forum

#1 2013-09-26 05:13:50

Proofs

#2 2013-09-26 07:56:04

Re: Proofs

#3 2013-09-26 08:12:43

Re: Proofs

#4 2013-09-26 08:57:17

Re: Proofs

#5 2013-09-26 10:00:34

Re: Proofs

#6 2013-09-26 19:13:56

Re: Proofs

#7 2013-09-26 21:54:56

Re: Proofs

#8 2013-09-26 22:17:59

Re: Proofs

#9 2013-09-26 23:09:40

Re: Proofs

#10 2013-09-27 00:20:42

Re: Proofs

#11 2013-09-27 00:59:26

Re: Proofs

#12 2013-09-27 01:23:51

Re: Proofs

#13 2013-09-27 03:09:37

Re: Proofs

#14 2013-09-27 05:30:44

Re: Proofs

#15 2013-09-27 06:07:37

Re: Proofs

#16 2013-09-27 06:37:26

Re: Proofs

#17 2013-09-27 07:18:40

Re: Proofs

#18 2013-09-27 09:05:55

Re: Proofs

#19 2013-09-27 09:15:29

Re: Proofs

#20 2013-09-27 12:26:29

Re: Proofs

#21 2013-09-27 13:40:02

Re: Proofs

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