Complete the proof that the alternate interior angles of transversals of
parallel lines are congruent.
Note: this proof is for the case where m/1 is less than 90°.
This proof uses the following theorem: Any point on one parallel line is the
same distance from the other line on a perpendicular transversal.
Statement or construction
1 ABCĎ
2 Construct BE perpendicular to such
that point E is on CD
3 Construct CF perpendicular to AB such
that point F is on AB
4 m/CFB=m/BEC = 90°
5 CF=
6
BC= BC
7 ABCF ACBE
8 LFBC ZECB
Reason
Given
All perpendicular angles measure 90° (2.
3).
Any point on one parallel line is the same
distance from the other line on a
perpendicular transversal (1, 2, 3).
They are measures of the same segment.
congruence (4, 6,5)
Corresponding parts of congruent figures
are congruent (7)

Answer: 3,64

Step-by-step explanation:

The proportion of individuals to land area is known as the population density. The Population Density is 148.38 people/m².

The proportion of individuals to land area is known as the population density. People per square kilometre is the metric. The term "population density" refers to the number of people in a given area, typically expressed as "per square kilometre" or "per square mile," and may include or exclude features like glaciers or bodies of water.

The number of members of a species in a given geographic area is known as population density. Demographic data can be measured and examined in relation to infrastructure, environments, and human health using population density data.

Population Density = the nation's population ÷ Area of country

Population of country = 23,023

Area of country = (18*8)+(4*5)

               = 164 m²

Population Density= 23023÷164

                 = 148.38 people/m²

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The domain, ranges and piecewise functions are;

10. D = (-∞, 3], R = [-2, ∞)

11. D = (-∞, -1), R = (-∞, 3)

12. [tex]f(x) =\begin{cases}\frac{2}{5}\cdot x + 4 & \text{ if } x\leq 0 \\x-5 & \text{ if } x > 0 \end{cases}[/tex]

13. [tex]f(x) =\begin{cases}-x-2 & \text{ if } x < -1 \\5 & \text{ if } -1 < x < 3\\ -2\cdot x + 5& \text{ if } 3 \leq x < \infty\end{cases}[/tex]

14. [tex]f(x) = \begin{cases}-2 & \text{ if } x < 4 \\\frac{1}{2} \cdot x + 4 & \text{ if } -4 < x \leq 2 \\-x& \text{ if} 2 < x < \infty\end{cases}[/tex]

A piecewise function comprises of two or more functions that set the definition or rule of the function based on the specified interval.

10. The piecewise function indicates;

p(x) = -3·x + 7 if x ≤ 3

When x = 3,  p(x) = -3 × 3 + 7 = -2

When x approaches -∞, p(x) → ∞

The domain and range in the interval x ≤ 3 are;

Domain = (-∞, 3]

Range = [-2, ∞)

P(x)  = x if 3 < x < 5

The domain and range in the interval 3 < x < 5 are;

Domain; (3, 5)

Range; (3, 5)

P(x) = -1 if x ≥ 5

The domain and range in the interval x ≥ 5 are;

Domain; [5, ∞)

Range; -1

Therefore;

D = (-∞, ∞)

R = [-2, ∞)

11. k(x) = x + 4 if x < -1

k(x) = 5 if -1 < x < 2

k(x) = -(1/2)·x + 1 if x ≥ 2

The domain and range in the interval x < -1 are;

Domain; (-∞, -1)

Range; (-∞, 3)

The domain and range in the interval -1 < x < 2 are;

Domain; (-1, 2)

Range; 5

The domain and range in the interval x ≥ 2 are;

Domain; [2, ∞)

Range; (-∞, 0]

The function is undefined for x = -1

The domain and range of the piecewise function is therefore;

D; (-∞, -1) ∪ (-1, ∞)

R; (-∞, 3) ∪ [5, 5]

12. The points on the graph where x ≤ 0 are; (0, 2), and (-5, 0)

The slope is; 2/5

The y-intercept is; (0, 4)

The equation is therefore; y = (2/5)·x + 4

Therefore; f(x) = (2/5)·x + 4 if x ≤ 0

The points on the graph when x > 0 are; (0, -5), and (5, 0)

The slope is; 5/5 = 1, the y-intercept is; (0, -5)

The equation is therefore; y = x - 5

Therefore; f(x) = x - 5 if x > 0

The piecewise function is therefore;

[tex]f(x) =\begin{cases} \frac{2}{5}\cdot x +4 & \text{ if } x\leq 0 \\x -5 & \text{ if } x > 0 \end{cases}[/tex]

13. The points on the graph when x < -1 are; (-1, -1), and (-2, 0)

The slope is; 1/-1 = -1

The equation is; y - 0 = -1×(x - (-2)) = -x - 2

y = -x - 2

Therefore, f(x) = -x - 2 if x < -1

The points on the graph when -1 < x < 3 are; (-1, 5), and (3, 5)

The slope is; 0, the equation is; y = 5

Therefore; f(x) = 5 if -1 < x < 3

The points on the graph when 3 ≤ x < ∞ are; (3, -1), and (5, -5)

The slope is; -4/2 = -2

The equation is; y - (-1) = -2×(x - 3) = -2·x + 6

y = -2·x + 6 - 1 = -2·x + 5

y = -2·x + 5

Therefore; f(x) = -2·x + 5 if 3 ≤ x < ∞

The piecewise function is therefore;

[tex]f(x) =\begin{cases} -x -2 & \text{ if } x < -1 \\5 & \text{ if } -1 < x < 3\\-2\cdot x + 5 & \text{ if } 3 \leq x < \infty \end{cases}[/tex]

14. The functions in the piecewise function graph are;

f(x) = -2 if x < -4

The points in the interval -4 < x ≤ 2 are; (-4, 2), (0, 4)and (2, 5)

The slope is; 3/6 = 1/2

The y-intercept is; (0, 4)

The function is therefore; f(x) = (1/2)·x + 4

The points in the interval 2 < x < ∞ are; (2, -2), and (4, -5)

The slope is; -3/2 = -1.5

The equation is; y - (-2) = (-3/2) × (x - 2) = -x + 2

y = -x + 2 - 2 = -x

y = -x

The function is therefore; f(x) = -x

The piecewise function is therefore;

[tex]f(x) =\begin{cases} -2 & \text{ if } x < -4 \\\frac{1}{2}\cdot x + 4 & \text{ if } -4 < x \leq 2\\-x & \text{ if } 2 < x < \infty \end{cases}[/tex]

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That g(x) is a vertical stretch of f(x) by a factor of 3, followed by a vertical shift of 3 units upward.

The function g(x) = 3[tex]2^{x}[/tex] is a transformation of the parent function f(x) = [tex]2^{x}[/tex]. Specifically, g(x) is obtained by first stretching f(x) vertically by a factor of 3, and then shifting it upward by some amount.    

To understand this transformation more clearly, consider the effect of changing the value of x on both functions. For the parent function f(x) = [tex]2^{x}[/tex], increasing x by 1 corresponds to multiplying the output (y-value) by 2. For example, if we evaluate f(x) at x=0, we get f(0) = [tex]2^{0}[/tex] = 1, and if we evaluate it at x=1, we get f(1) =[tex]2^{1}[/tex]  = 2, which is double the value of f(0).

Now, let's consider the function g(x) =3[tex]2^{x}[/tex] . When we evaluate g(x) at x=0, we get g(0) = 3[tex](2)^{0}[/tex] = 3, which is triple the value of f(0). Similarly, when we evaluate g(x) at x=1, we get g(1) = 3[tex](2)^{1}[/tex] = 6, which is triple the value of f(1). This shows that g(x) is a vertical stretch of f(x) by a factor of 3.

Finally, notice that the function g(x) has the same shape as f(x), but is shifted upward by an amount of 3 units. We can see this by comparing the graphs of the two functions. The graph of f(x) starts at the point (0,1) and increases rapidly as x gets larger. The graph of g(x) starts at the point (0,3) and increases at the same rate as f(x). This shows that g(x) is a vertical stretch of f(x) by a factor of 3, followed by a vertical shift of 3 units upward.

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The 99% confidence interval for the mean GPA of all accounting students at this university is approximately (3.12272, 3.27728). This means we are 99% confident that the true mean GPA of all accounting students at this university falls between 3.12272 and 3.27728.

To construct the 99% confidence interval for the mean GPA of all accounting students at this university, follow these steps:
Identify the sample size, mean, and standard deviation: In this case, the sample size (n) is 25, the mean (x) is 3.20, and the standard deviation (s) is 0.15.
Determine the confidence level: The problem states we need a 99% confidence interval, so the confidence level is 99%.
Find the critical value (z-score) for the confidence level: For a 99% confidence interval, the critical value (z) is 2.576 (you can find this value in a standard z-score table).
Calculate the standard error (SE) of the sample mean: SE = s / √n = 0.15 / √25 = 0.15 / 5 = 0.03.
Calculate the margin of error (ME): ME = z * SE = 2.576 * 0.03 = 0.07728.
Find the lower and upper limits of the confidence interval:
- Lower limit = x - ME = 3.20 - 0.07728 = 3.12272.
- Upper limit = x + ME = 3.20 + 0.07728 = 3.27728.

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To check if Line L1 and Line L2 are parallel, we need to check if their slopes are equal.

Given L1 is y=2x+1

The slope of L1 is the coefficient of x, which is 2.

since L1 is in the form of y=mx+c where m is the slope

Given  L2 is 4y-8x+1=0

To find the slope of L2, we need to rearrange it to slope-intercept form

y = mx + b, where m is the slope and b is the y-intercept.

4y - 8x + 1 = 0

4y = 8x - 1

y = 2x - 1/4

Now the above equation is in the form of y=mx+c

The slope of L2 is also 2.

Since both lines have the same slope, we can conclude that they are parallel. Therefore, we can say that Line L1 and Line L2 are parallel.

The value of y is 4.

What is an inverse function?

The inverse function of a function f in mathematics is a function that reverses the operation of f. If f is bijective, then and only if it is, the inverse of f exists.

Here, we have

Given: If f varies inversely as x, and y= 2 when x= 2, find y when x= 1.

We have to find the value of f.

f ∝ 1/g

f₁ ×g₁ = f₂ ×g₂

Let the required value of f = x and g = y

Inserting in equation (1) and we get

2 ×2 = 1 ×y

4 = y

Hence, the value of y is 4.

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the discontinuities of a rational function involves finding the values of x that make the denominator equal to zero, while finding the asymptotes of a rational function involves examining the behavior of the function as x approaches certain values.

Describe the function.

There will be questions on every subject, including created and real places as well as algebraic variable design, on the midterm test. a schematic illustrating the connections between various components that work together to produce the same outcome. A service is made up of many unique parts that work together to produce unique outcomes for each input.

To find the discontinuities of a rational function, we need to determine where the function is undefined. In general, a rational function is a function of the form:

f(x) = p(x) / q(x)

where p(x) and q(x) are polynomials in x, and q(x) is not the zero polynomial. The rational function f(x) is undefined at any value of x that makes the denominator q(x) equal to zero, since division by zero is undefined.

Therefore, to find the discontinuities of a rational function, we need to solve the equation q(x) = 0. The values of x that make q(x) equal to zero are called the "zeros" or "roots" of the denominator q(x). These values of x are the discontinuity points of the function, since the function is undefined at those points.

On the other hand, to find the asymptotes of a rational function, we need to examine the behavior of the function as x approaches certain values. In general, a rational function may have three types of asymptotes: horizontal, vertical, and oblique (also called slant).

Vertical asymptotes occur when the function approaches positive or negative infinity as x approaches a certain value, typically where the denominator q(x) equals zero.

Horizontal asymptotes occur when the function approaches a constant value as x approaches positive or negative infinity.

Oblique asymptotes occur when the degree of the numerator is exactly one greater than the degree of the denominator. In this case, the function approaches a straight line (i.e., a slant asymptote) as x approaches positive or negative infinity.

To summarize, finding the discontinuities of a rational function involves finding the values of x that make the denominator equal to zero, while finding the asymptotes of a rational function involves examining the behavior of the function as x approaches certain values.

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Answer:

-2i² - 4i + 4

Step-by-step explanation:

(1−2i) ⋅ (4+i)

= 4 + i - 8i - 2i²

= -2i² - 7i + 4

So, the answer is -2i² - 7i + 4

To find the circle and radius of equation x2 + y2 +12x +6y +20=0, we need to complete the square for both x and y terms.

First, we can simplify the equation by rearranging the constant term to the right-hand side:

x^2 + y^2 + 12x + 6y = -20

Next, we can complete the square for the x-terms by adding (12/2)^2 = 36 to both sides of the equation:

x^2 + 12x + 36 + y^2 + 6y = -20 + 36

Simplifying further:

(x + 6)^2 + (y + 3)^2 = 13

This is now in the standard form of a circle: (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.

So the center of the circle is (-6, -3) and the radius is the square root of 13. Therefore, the equation x^2 + y^2 + 12x + 6y + 20 = 0 represents a circle with center (-6, -3) and radius sqrt(13).

Step-by-step explanation:

For the number to be even, it needs to end in an even number ,,,,there are no even numbers that are primes ( because the number would be divisible by 2)

[tex]y\sqrt[3]{6y}-14\sqrt[3]{48y}~~ = ~~-11y\sqrt[3]{6y} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \square y\sqrt[3]{6y}-14\sqrt[3]{48y^{\square }}\implies \square y\sqrt[3]{6y}-14\sqrt[3]{2^3\cdot 6y^{\square }}\implies \square y\sqrt[3]{6y}-28\sqrt[3]{6y^{\square }} \\\\\\ \underline{17} y\sqrt[3]{6y}-28\sqrt[3]{6y^{\underline{4}}}\implies 17y\sqrt[3]{6y}-28\sqrt[3]{6y^3\cdot y} \\\\\\ \stackrel{ \textit{like-terms} }{17y\sqrt[3]{6y}-28y\sqrt[3]{6y}}\implies \boxed{-11y\sqrt[3]{6y}}[/tex]

If the representative fraction scale is 1:48000, then the verbal scale from inches to feet is  1 inch represents 0.00025 feet.

We have to find the verbal (written) scale from inches to feet that represents a map with a representative-fraction (RF) scale of 1:48,000,

We use the formula:

⇒ Verbal scale = RF × Inches per foot,

We know that the RF scale is 1:48,000, which means that one unit on the map represents 48,000 units in the real world.

There are 12 inches in a foot,

So, we can convert the verbal scale to inches-per-foot by dividing by 12:

⇒ Inches per foot = (Verbal scale)/(12),

⇒ Inches per foot = (1/48000) × (12/1) = 0.00025

Therefore, the verbal scale from inches to feet is : 1 inch represents 0.00025 feet.

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Answer:

the answer is 3) 6r-6+5r+20

The length of arc TU is approximately 0.3935 cm.

Calculation of circle ?

To find the length of arc TU, we first need to determine the measure of angle TAU. Since TV is a diameter, we know that angle TSV is a right angle (90 degrees). Since S is the midpoint of TV, angle TSU is half of angle TSV, which means angle TSU is 45 degrees (90/2 = 45). Therefore, angle TAU is the complement of angle TSU, which is 90 - 45 = 45 degrees.

Next, we need to use the formula for the length of an arc of a circle, which is L = rθ, where L is the length of the arc, r is the radius of the circle, and θ is the measure of the central angle of the arc in radians. Since we have the measure of the central angle in degrees, we need to convert it to radians by multiplying by π/180.

The radius of the circle is half the diameter, which is 0.5 cm. Therefore, we have:

L = rθ

L = 0.5 × (45 × π/180)

L = 0.5 × (0.25π)

L = 0.125π

To get a numerical value, we can use an approximation of π, such as 3.14. Therefore, the length of arc TU is approximately:

L ≈ 0.125π ≈ 0.125 × 3.14 ≈ 0.3935 cm

So the length of arc TU is approximately 0.3935 cm.

The formula of a circle relates its radius, diameter, and circumference.

The diameter of a circle is the distance across the circle passing through its center, and is given by:

D = 2r

where r is the radius of the circle.

The circumference of a circle is the distance around its outer edge, and is given by:

C = 2πr

where r is the radius of the circle, and π (pi) is a mathematical constant approximately equal to 3.14159.

The area of a circle is the region enclosed by the circle, and is given by:

A = πr²

where r is the radius of the circle, and π (pi) is a mathematical constant approximately equal to 3.14159.

These formulas can be used to solve various problems related to circles, such as finding the area, circumference, radius, or diameter of a circle, given one or more of these values.

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Answer:

7

Step-by-step explanation:

Since Nancy has 4 pencils, we know Byron has 3 times that amount

4 groups of 3 is equivalent to 12.

We know that is how many pencils Byron has. Mac has 5 less.

So, 12-5 = 7.

Answer:

49 it all good if was 29 or 39 thank me axter

27/2 or 13.5 pieces.

To find the quotient, we need to divide the length of the sub sandwich by the length of each piece she wants to cut it into:

9 ÷ (2/3)

We can simplify this by multiplying the numerator by the reciprocal of the denominator:

9 ÷ (2/3) = 9 × (3/2)

Multiplying straight across:

9 × (3/2) = 27/2

So Jenny will be able to cut the sub sandwich into 27/2 or 13.5 pieces.

Step-by-step explanation:

Cutting a 9 inch sub into 2/3 inch pieces ?

9 inch / 2/3 inch / piece = 9 * 3/2= 27/2 = 13.5 pieces ~ 13 with a bit left over

Answer:

A, D, and E

Hope this helps<3

The correct solution to a system of linear equations is:

Find where the two lines intersect.

To find the solution to a system of linear equations on a graph, you can follow these steps:

1. Graph the two equations on the same coordinate system. This will give you two lines that represent the equations.

2. Look for the point of intersection between the two lines. This point represents the solution to the system of equations.

3. If the lines do not intersect, then the system of equations has no solution. If the lines overlap or coincide with each other, then the system of equations has infinitely many solutions.

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The solution of a system of equations can be obtained from their graph by the points where the graph intersects.

The correct option is therefore;

A system of equations are a set of two or more equations that share common variables.

The solution of a system of equations is a set of values of the input variables that satisfies the equations in the system of equations

The graph of an equation is the set of points that satisfies the relationship between the variables in the equation.

Therefore, the solution of a system of equations can be obtained from a graph by finding the point of intersection of the graphs of the equations in the system, which is the point that agrees with all the equations in the equation system.
The correct option is therefore; Find where the two lines intersect

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Answer:

c. [tex]p^2+2p-24[/tex]

Step-by-step explanation:

[tex](p + 6)(p - 4)\\\\= (p)(p) + (6)(p) + (-4)(p) + (6)(-4)\\\\= p^2+6p-4p-24\\\\=p^2+2p-24[/tex]

The smallest positive integer of the set of the positive integers divisible by 15 and 35 is 105.

The set of all those positive integers that are divisible by both 15 and 35 is infinite because there is no limit to the numbers which are divisible by 15 as well as 35.

We have to find the least positive integer of this set.

In order to do so we will find the least common multiple of 15 and 35.

The LCM of 15 and 35 is 105 so this LCM will be the smallest positive integer that is divisible by 15 and 35.

The reason why the LCM is the smallest positive integer is because the LCM is the first value that is common in the tables of 15 and 35.

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Answer:

A) The slope is negative.  It is -30

B)The slope is positive.  It is 2.5

Step-by-step explanation:

Graph A:

I selected two points on the graph.

(1, 220) and (7, 40)

The slope is the change in y over the change in x.  The y values are 40 and 220.  The x values are 7 and 1.  You find the change by subtracting.

[tex]\frac{40 - 220}{7-1}[/tex] = [tex]\frac{-180}{6}[/tex] = -30

Graph B:

This graph is proportional because it is a straight line and it goes through the original.  This means that the slope can be found with any ordered pair in the form [tex]\frac{y}{x}[/tex].  I am going to use the point (2, 5).  [tex]\frac{5}{2}[/tex] = 2.5

Helping in the name of Jesus.

The answers are:

The directrix, which is made up of all locations (x, y) in a plane that are evenly spaced from it, and the focus, which is a fixed point but not on the directrix, are collectively referred to as a parabola. The graph of the parabola has the standard shape of a parabola with vertex (0,0) as well as the x-axis as its axis of symmetry.

Apollonius, who found many properties of conic sections, is responsible for the term "parabola." The word has the meaning "application," and as Apollonius had shown, this idea of "application of areas" is related to this curve. Pappus is responsible for the focus-directrix characteristic of the parabola and other conic sections.

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If you repeat this experiment 600 times, you predict that 300 repetitions will result in picking the same color marble twice.

To predict the number of repetitions that will result in picking the same color marble twice in 600 experiments, we need to find the probability of this event occurring and then multiply it by the total number of experiments.

Calculate the probability of picking the same color marble twice:

Assuming there are two colors (A and B) and an equal number of each

color, the probability of picking the same color twice is:

P(AA) = P(A) × P(A|A) = (1/2) × (1/2) = 1/4

P(BB) = P(B) × P(B|B) = (1/2) × (1/2) = 1/4

Add the probabilities of picking two marbles of the same color:

P(same color twice) = P(AA) + P(BB) = 1/4 + 1/4 = 1/2

Multiply the probability by the total number of experiments:

Predicted repetitions = Probability × Total experiments

                                   = (1/2) × 600

                                   = 300

So, if you repeat this experiment 600 times, you predict that 300

repetitions will result in picking the same color marble twice.

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Answer:

We know that when 5(2x-3) = 25 x = 4 because when you plug 4 back into the equation you get 25=25. To solve this equation first you will distribute 5 into 2x -3 you will then get 10x-15 = 25. Then you will add 15 to both sides and get 10x = 40, next divide 10 from both sides and you will get x=4. To check this you can plug 4 back into the equation so it will look like this. 5(2(4)-3) = 25, first you will solve what is in the parenthesis, you will set 5(5)=25. Then multiply 5 by 5 and you will get 25=25. This proves that when 5(2x-3) = 25 x = 4

Step-by-step explanation:

Answer:

256ft²

Step-by-step explanation:

White squares=

2 (8x8) =

2 (64) = 256

Whole Rectangle=

8 + 8 = 16

A = bh = 16 x 32 = 512

512 - 256 = 256

It will take approximately 3.33 hours for both persons working together to complete the job.

What is distance?

Distance is the measure of how far apart two objects or locations are from each other. It is usually measured in units such as meters, kilometers, miles, or feet.

To solve this problem, we can use the following formula:

1 / T = 1 / t1 + 1 / t2

Where T is the time it takes for both persons working together to complete the job, t1 is the time it takes for the first person to complete the job alone, and t2 is the time it takes for the second person to complete the job alone.

Plugging in the given values, we get:

1 / T = 1 / 5 + 1 / 10

Simplifying the right side, we get:

1 / T = 3 / 10

Multiplying both sides by 10T, we get:

10 = 3T

Dividing both sides by 3, we get:

T = 10 / 3 = 3.33 hours

Therefore, it will take approximately 3.33 hours for both persons working together to complete the job.

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Answer:

61

Step-by-step explanation:

11^2 + 60^2 = 3721

square root of that is 61

put it in a calculator