All categories

3 years ago

How do I understand two column proofs?

Mathematics

1 answer:

Anna007 [38]3 years ago

7 0

Explanation:

When you solve an equation, you use the rules of algebra. Before you begin solving equations, you learn the rules of algebra. Specifically, you learn about rules relating to addition, multiplication, equality, identity elements, order of operations, and inverse operations.

Whenever you "show work" solving an equation, you are demonstrating at each step that you know how to apply these rules to get one step closer to a solution.

A 2-column proof is a list of "statements" in one column, and associated "reasons" in the other column.

The first statement is generally a list of all of the things that are "given" in the particular problem. The first reason is generally, "Given".

The last statement is generally a statement of what you are trying to prove. The last reason is a description of the postulate or theorem you used to conclude the last statement is true, based on the previous statement.

Here's a brief example:

Suppose we have line segment RT with point S on the segment. Suppose the lengths are given: RS = 3, ST = 4. We are asked to prove that RT = 7. The proof might look like this:

Statement . . . . Reason

Point S lies on RT; RS = 3; ST = 4 . . . . Given

RT = RS +ST . . . . segment addition postulate

RT = 3 + 4 . . . . substitution property of equality

RT = 7 . . . . properties of integers

So, creating or filling in 2-column proofs requires you have a good understanding of the theorems and postulates you are allowed (or expected) to choose from, and an understanding of logical deduction. Essentially, you cannot make a statement, even if you "know it is true", unless you can cite the reason why you know it is true. Your proof needs to proceed step-by-step from what you are given to what you want to prove.

It might be useful to keep a notebook or "cheat sheet" of the names and meanings of the various properties and theorems and postulates you run across. Some that seem "obvious" still need to be justified. X = X, for example, is true because of the reflexive property of equality.

It can be helpful to read and understand proofs that you see in your curriculum materials, or that you find online--not just skim over them. This can help you see what detailed logical steps are needed, and the sorts of theorems and postulates that are cited as reasons. It is definitely helpful to pay attention when new relationships among geometrical objects are being introduced. You may have to use those later in a proof.

_____

Additional comment

As in the above proof, you may occasionally run across a situation where you're asked to "justify" some arithmetic fact: 3+4=7 or 2×3=6, for example. I have never been quite clear on the justification that is appropriate in such cases. In the above, I have used "properties of integers", but there may be some better, more formal reason I'm not currently aware of. This is another example of the "obvious" needing to be justified.

You might be interested in

Suppose that the sample standard deviation was s = 5.1. Compute a 98% confidence interval for μ, the mean time spent volunteerin

NISA [10]

Answer:

The 95% confidence interval would be given by (5.139;5.861)

Step-by-step explanation:

1) Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

$\bar X$ represent the sample mean for the sample

$\mu$ population mean (variable of interest)

s represent the sample standard deviation

n represent the sample size

2) Confidence interval

Assuming that $\bar X =5.5$ and the ranfom sample n=1086.

The confidence interval for the mean is given by the following formula:

$\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}$ (1)

In order to calculate the critical value $t_{\alpha/2}$ we need to find first the degrees of freedom, given by:

$df=n-1=1086-1=1085$

Since the Confidence is 0.98 or 98%, the value of $\alpha=0.02$ and $\alpha/2 =0.01$ , and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.01,1085)".And we see that $t_{\alpha/2}=2.33$

Now we have everything in order to replace into formula (1):

$5.5-2.33\frac{5.1}{\sqrt{1086}}=5.139$

$5.5+2.333\frac{5.1}{\sqrt{1086}}=5.861$

So on this case the 95% confidence interval would be given by (5.139;5.861) We are 98% confident that the mean time spent volunteering for the population of parents of school-aged children is between these two values.

5 0

3 years ago

Which statements are true about David's work? Check all that apply. The GCF of the coefficients is correct. The GCF of the varia

Anon25 [30]

Answer:

The GCF of the coefficients is correct.

The variable c is not common to all terms, so a power of c should not have been factored out.

In step 6, David applied the distributive property.

Step-by-step explanation:

Given the polynomial :

80b⁴ – 32b²c³ + 48b⁴c

The Greatest Common Factor (GCF) of the coefficients:

80, 32, 48

Factors of :

80 : 1, 2, 4, 5, 8, 10, 16, 20, 40, and 80

32 : 1, 2, 4, 8, 16, and 32

48 : 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.

GCF = 16

b⁴, b², b⁴

b⁴ = b * b * b * b

b² = b * b

b⁴ = b * b * b * b

GCF = b*b = b²

GCF of c³ and c

c³ = c * c * c

c = c

GCF = c

We can see that David's GCF of the coefficients are all correct

From the polynomial ; 80b⁴ does not contain c ; so factoring out c is incorrect

In step 6 ; the distributive property was used to obtain ; 16b²c(5b² – 2c² + 3b²)

6 0

3 years ago

Read 2 more answers

Nedra purchased apples and oranges at the grocery store. Apples are $0.55 each and oranges are $0.75 each. If she spent a total

Thepotemich [5.8K]

Answer:

The two linear equation

x + y = 19.... Equation 1

0.55x + 0.75y = 12.65... Equation 2

Nedra purchased

9 Apples and 11 Oranges

Step-by-step explanation:

Nedra purchased apples and oranges at the grocery store. which two linear equations can be used to find the number of apples and oranges?

Let the number of apples be represented by x

The number of oranges by represented by y.

Hence,

Apples are $0.55 each and oranges are $0.75 each. If she spent a total of $12.65 for 19 pieces of fruit,

Hence:

x + y = 19..... Equation 1

x = 19 - y

$0.55 × x + $0.75 × y = $12.65

0.55x + 0.75y = 12.65.....Equation 2

Hence: we substitute 19 - y for x in Equation 2

0.55(19 - y) + 0.75y = 12.65

10.45 - 0.55y + 0.75y = 12.65

-0.55y+ 0.75y = 12.65 - 10.45

0.20y = 2.2

y = 2.2/0.20

y = 11 oranges

x = 19 - y

x = 19 - 11

x = 8 Apples

6 0

3 years ago

Help out?? Mathematics image.

Brilliant_brown [7]

Answer: (12) ∠1 = 20° (13) ∠2 = 50° (14) ∠3 = 15° (15) UV = 80° (16) AB = 40° (17) ABC or 180° - CD (18) BC - 140° (19) ABC = 150°

Step-by-step explanation:

12) $\frac{1}{2}$ (UV - ST) = ∠1

$\frac{1}{2}$ (80 - 40) = ∠1

$\frac{1}{2}$ (40) = ∠1

20 = ∠1

13) $\frac{1}{2}$ (UV + ST) = ∠2

$\frac{1}{2}$ (70 + 30) = ∠2

$\frac{1}{2}$ (100) = ∠2

50 = ∠2

14) $\frac{1}{2}$ (VB - BS) = ∠3

$\frac{1}{2}$ (60 - 30) = ∠3

$\frac{1}{2}$ (30) = ∠3

15 = ∠3

15) $\frac{1}{2}$ (UV - ST) = ∠1

$\frac{1}{2}$ (UV - 20) = 30

UV - 20 = 60

UV = 80

16) ∠1 = arc AB

∠1 = 40

arc AB = 40

17) arc AB + arc BC = arc AC

= 180 = arc CD

18) ∠1 + ∠2 + ∠3 = 180

20 + ∠2 + 20 = 180

∠2 + 40 = 180

∠2 = 140

19) ∠1 + ∠ 2 = arc ABC

∠1 + ∠2 + ∠3 = 180

arc ABC + 30 = 180

arc ABC = 150

4 0

3 years ago

The complement of a right angle is an obtuse angle. true or false.

Genrish500 [490]

False because the obtuse angle is to large to complement the right angle to diffrent angles can not be both of their compliments

6 0

3 years ago