43. (a) Suppose you are given the following (x, y) data pairs.
x 2 3 5
y 4 3 6
Find the least-squares equation for these data (rounded to three digits after the decimal).
ŷ = + x
(b) Now suppose you are given these (x, y) data pairs.
x 4 3 6
y 2 3 5
Find the least-squares equation for these data (rounded to three digits after the decimal).
ŷ = + x
(d) Solve your answer from part (a) for x (rounded to three digits after the decimal).
x = + y

The number of different packages of 4 different sweatshirts would be 70 different packages.

The number of displays possible are 336 different displays.

Utilizing the combination formula allows us to determine the quantity of distinct packages containing four sweatshirts, derived from eight high school football teams. A combination factors in items without regard to their sequence.

C ( n, k) = n ! / ( k ! ( n - k ) ! )

C ( 8 , 4 ) = 8 ! / ( 4 ! ( 8 - 4 ) ! )

C ( 8 ,  4) = 40320 / 576

C ( 8, 4 ) = 70 different packages

The permutation formula enables us to determine the possible number of displays for a sequence of three unique sweatshirts representing different high schools, with great regard to their order. Applying this is essential when the order these items are presented holds significance.

P ( n , k) = n ! / ( n - k )!

P ( 8, 3 ) = 8 ! / ( 8 - 3 ) !

P ( 8, 3 ) = 40320 / 120

P ( 8 , 3 ) = 336

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Answer:

180

Step-by-step explanation:

They need to add to one 180

There will be approximately 2.098 inches of space on either side of the sphere if it is placed on the center of a pedestal 9 inches across.

What is volume of a sphere ?

V = 4/3 π r³,where V is the volume and r is the radius, is the formula for a sphere's volume. A sphere's radius is equal to half of its diameter.

To solve this problem, we first need to find the radius of the sphere:

The volume of a sphere is given by the formula V = (4/3)πr³, where r is the radius of the sphere.

So we have:

28.73 = (4/3)πr³

Multiplying both sides by 3/4π, we get:

r³ = (28.73 × 3/4π)

r³ ≈ 7.177

Taking the cube root of both sides, we get:

r ≈ 1.952 inches

Now we can find the amount of space on either side of the sphere by subtracting the diameter of the sphere (which is twice the radius) from the width of the pedestal, and then dividing by two:

Space on either side = (9 - 2 × 1.952) / 2

Space on either side ≈ 2.098 inches

Therefore, there will be approximately 2.098 inches of space on either side of the sphere if it is placed on the center of a pedestal 9 inches across.

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Answer:

So $700 was invested in the mutual fund that earned a 2% profit, and $500 was invested in the mutual fund that earned a 5% profit.

Step-by-step explanation:

Let x be the amount invested in the mutual fund that earned a 5% profit, and let y be the amount invested in the mutual fund that earned a 2% profit. We know that the total investment was $1,200, so:

x + y = 1200

We also know that the total profit was $39, which can be expressed as a decimal as 0.39 (since profit is calculated as a percentage of the initial investment). The amount of profit earned on the first fund is 5% of x, or 0.05x, and the amount of profit earned on the second fund is 2% of y, or 0.02y. So:

0.05x + 0.02y = 0.39

We now have two equations with two variables:

x + y = 1200

0.05x + 0.02y = 0.39

We can solve for one variable in terms of the other in the first equation, and substitute into the second equation:

x = 1200 - y

0.05(1200 - y) + 0.02y = 0.39

Simplifying and solving for y:

60 - 0.05y + 0.02y = 0.39

0.03y = 0.39 - 60

0.03y = -59.61

y = -59.61 / 0.03

y = 1987

This tells us that $1,987 was invested in the mutual fund that earned a 2% profit. To find the amount invested in the mutual fund that earned a 5% profit, we can substitute into the first equation:

x + y = 1200

x + 1987 = 1200

x = 1200 - 1987

x = -787

This doesn't make sense, since we can't have a negative investment amount. It means that we made a mistake somewhere. Checking our work, we can see that the equation 0.05x + 0.02y = 0.39 should actually be:

0.05x + 0.02y = 39

(without the decimal point). With this correction, we can solve as before:

x + y = 1200

0.05x + 0.02y = 39

x = 1200 - y

0.05(1200 - y) + 0.02y = 39

60 - 0.05y + 0.02y = 39

0.03y = 21

y = 700

So $700 was invested in the mutual fund that earned a 2% profit, and $500 was invested in the mutual fund that earned a 5% profit.

The independent variable grade level with a scale range of 7 to 12 should be placed on the horizontal axis. While the dependent variable math score with a scale of 0% to 100% should be placed on the vertical axis.

On a line graph, the independent variable is typically represented on the x-axis and the dependent variable on the y-axis. The independent variable is the variable that is controlled or manipulated, while the dependent variable is the variable that is being measured or observed and is affected by changes in the independent variable.

From the question, the independent variables are: 7, 8, 9, 10, 11, and 12.

While the dependent variables are: 72, 75, 81, 80, 83, and 91

Therefore, the independent variable grade level with a scale range of 7 to 12 should be placed on the horizontal axis. While the dependent variable math score with a scale of 0% to 100% should be placed on the vertical axis.

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(0,-1), (-4,1), (0,3), (4,5)

Discounted price of the television will be $228.00.

We need to figure out what 5% of $240.00 is in order to determine the dollar amount of the discount. Using the formula, we can accomplish this:

Discount amount = Original price x Discount rate

In the event that a $240.00 TV is discounted by 5%, this implies that the cost of the TV will be scaled down by 5% of its unique cost.

where the discount rate is presented in decimal form rather than as a percentage. Thus, at a discount rate of 5%, we have:

Discount amount = $240.00 x 0.05

Discount amount = $12.00

Therefore, the discount amount in dollars is $12.00. This means that the discounted price of the television will be $240.00 - $12.00 = $228.00.

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The coefficient of variation for the given data is 111.8%, so the answer is Option C.

The coefficient of variation (CV) is a measure of relative variability, which is calculated as the standard deviation divided by the mean, expressed as a percentage. In this case, the mean of the data is (1+1+1+1+6)/5 = 2, and the standard deviation is 2.28. Therefore, the CV = (2.28/2) x 100% = 111.8%.

The CV is useful for assessing the variability of distinct datasets, particularly when their means differ. In this situation, the CV shows that the data has a significant degree of relative variability, which suggests that the mean may not be a suitable representation of the data. It also implies that the outlier value of 6 has a major influence on the data's variability.

Therefore, Option C is the correct answer to the above question.

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This solution is not valid because x must be greater than or equal to 0 in this case. Thus, the only value of a that gives f(x) = 43 is x = -7.

To find the values of a that give f(x) = 43, solve the equations x² - 6 = 43 and 10 - x = 43 separately for x. The correct equation to use is x² - 6 = 43.

There is a typo in the question, as both cases are given for x < 0. Assuming the second case should be for x ≥ 0, we have two equations to solve:

1) x² - 6 = 43 for x < 0
2) 10 - x = 43 for x ≥ 0

For the first equation:
x² - 6 = 43
x² = 49
x = ±√49
x = ±7

Since x must be less than 0, the value of x that gives f(x) = 43 is x = -7.

For the second equation:
10 - x = 43
-x = 33
x = -33

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The probability distribution of Y is geometric with parameter p = 1 - [tex]e^-^\lambda[/tex] is (1 -  [tex]e^-^\lambda[/tex] )^(k-1) *  [tex]e^-^\lambda[/tex]

If X is exponential with rate λ, then Y=[X]+1 is geometric with parameter p=1− [tex]e^-^\lambda[/tex] .


Let X be an exponential random variable with rate λ, then the probability density function of X is fX(x) = λe^(-λx) for x ≥ 0.
Now, let Y = [X] + 1, where [X] is the largest integer less than or equal to X.
The probability that Y = k, where k is a positive integer, is given by:

P(Y = k) = P([X] + 1 = k)

= P(k-1 ≤ X < k)

= ∫_(k-1)^k λ [tex]e^-^\lambda^x[/tex] dx

= e^(-λ(k-1)) - e^(-λk)

= (1 -  [tex]e^-^\lambda[/tex] )^(k-1) *  [tex]e^-^\lambda[/tex]

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The given sequence is geometric because the common ratio between is -3

A sequence is an ordered list of numbers or other mathematical objects that follow a specific pattern or rule. Each term in a sequence is determined by the previous terms, and the order of the terms is usually significant. Sequences can be finite or infinite, and they are often represented using either a formula or a recursive definition.

This sequence is geometric because each term is obtained by multiplying the previous term by -3.

To see this, notice that:

-2 * (-3) = 6

6 * (-3) = -18

-18 * (-3) = 54

Therefore, the common ratio between consecutive terms is -3, which is a constant. Thus, this sequence is geometric with a first term of -2 and a common ratio of -3.

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the probability that a newborn elephant weighs at least 384 pounds is approximately 0.0002, or 0.02%.

To answer your question, we will use z-value calculations. First, we need to calculate the z-score for a newborn elephant weighing 384 pounds.

The z-score formula is:

z = (X - μ) / σ

where X is the observed value (384 pounds), μ is the mean (232 pounds), and σ is the standard deviation (43 pounds).

Plugging in the values, we get:

z = (384 - 232) / 43
z = 152 / 43
z ≈ 3.53 (rounded to 2 decimal places)

Now, we need to find the probability of a newborn elephant weighing at least 384 pounds, which means finding the area under the normal distribution curve to the right of z = 3.53. This can be done using a z-table or a calculator with a normal distribution function.

Looking up z = 3.53 in the z-table or using a calculator, we find that the area to the left of z = 3.53 is approximately 0.9998. Since we are interested in the area to the right, we subtract this value from 1:

P(X ≥ 384) = 1 - 0.9998
P(X ≥ 384) ≈ 0.0002 (rounded to 4 decimal places)

So, the probability that a newborn elephant weighs at least 384 pounds is approximately 0.0002, or 0.02%.

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D. All of the above.binary integer programming problems can answer a variety of questions, such as whether a project should be undertaken, or an investment should be made, or a plant should be located at a particular location. By setting up the BIP problem and solving it, the best solution to the problem can be determined.

Binary integer programming (BIP) is a type of optimization problem that seeks to find an optimal solution to a decision-making problem, where the decision variables must be restricted to discrete values (i.e. binary values such as 0 or 1).

BIP problems can answer a variety of questions, such as whether a project should be undertaken, or an investment should be made, or a plant should be located at a particular location. By working out the various parameters associated with the problem, and then solving the BIP problem, the best solution to the problem can be determined.

For example, a company may be faced with deciding which of two potential projects to undertake. To solve this problem, the company could define the decision variables (which project to choose) as binary integers, and then use the BIP problem formulation to determine which project would be the most profitable. This would involve considering all the relevant parameters such as expected revenue, cost, and time frame, and then solving the BIP problem to determine which project would yield the highest overall return.

In conclusion, binary integer programming problems can answer a variety of questions, such as whether a project should be undertaken, or an investment should be made, or a plant should be located at a particular location. By setting up the BIP problem and solving it, the best solution to the problem can be determined.

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The volume of a cylinder is given by the formula V = πr²h, where r is the radius and h is the height. Since the diameter of the rod is 8 cm, the radius is 4 cm.

The given rod has a circular cross-section with a radius of 4 centimeters and a length of 30 centimeters. The volume of this rod can be calculated using the formula for the volume of a cylinder, which is V = πr²h, where r is the radius of the circular base and h is the height or length of the cylinder.

Substituting the given values into the formula, we get:

V = π(4²)(30)

Simplifying the expression, we get:

V = 480π cubic centimeters

This is the volume of one rod. To find the total amount of steel needed to make 113 rods, we simply multiply the volume of one rod by 113, since all rods are of the same size and shape.

total steel needed = 113 × V

total steel needed = 113 × 480π cubic centimeters

total steel needed = 54,240π cubic centimeters

Therefore, the total amount of steel needed to make 113 rods is 54,240π cubic centimeters.

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∫(0 to π/2) ∫(0 to sqrt(4-z)) ∫(0 to 4-r²) 9(r³*cos(θ)³ + r³*cos(θ)*sin(θ)²)r dr dθ dz. By solving this triple integral, we'll find the value of the given expression in cylindrical coordinates.

To evaluate the given integral in cylindrical coordinates, we first need to convert the given expression and region of integration from Cartesian coordinates to cylindrical coordinates. In cylindrical coordinates, we have x = r*cos(θ), y = r*sin(θ), and z = z. The given expression becomes:

9(x³ + xy²)dV = 9(r³*cos(θ)³ + r³*cos(θ)*sin(θ)²)r dr dθ dz

Now, let's find the bounds of integration for the solid E. Since it lies in the first octant and beneath the paraboloid z = 4 - x² - y², we have:

0 ≤ z ≤ 4 - r²*cos(θ)² - r²*sin(θ)²
0 ≤ r ≤ sqrt(4 - z)
0 ≤ θ ≤ π/2

Now we can set up and evaluate the integral:

∫(0 to π/2) ∫(0 to sqrt(4-z)) ∫(0 to 4-r²) 9(r³*cos(θ)³ + r³*cos(θ)*sin(θ)²)r dr dθ dz

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Since this PDF is only defined for y > 0, we have:

We can use the transformation method to find the PDF of Y.

Let g(x) = [tex]e^{x}[/tex] be the transformation function. Then, we have:

Y = g(X) = [tex]e^{x}[/tex]

To find the PDF of Y, we need to find the cumulative distribution function (CDF) of Y and then take its derivative.

F_Y(y) = P(Y <= y) = P(e^X <= y) = P(X <= ln(y))

Using the standard normal distribution, we can calculate this probability as:

P(X <= ln(y)) = Φ((ln(y) - μ) / σ)

where Φ is the cumulative distribution function of the standard normal distribution, μ = E[X] = -5, and σ = [tex]sqrt(Var[X]) = 3[/tex].

Therefore, we have:

F_Y(y) = Φ((ln(y) + 5) / 3)

To find the PDF of Y, we take the derivative of F_Y(y) with respect to y:

f_Y(y) = d/dy (F_Y(y)) = d/dy (Φ((ln(y) + 5) / 3))

Using the chain rule, we have:

f_Y(y) = Φ'((ln(y) + 5) / 3) / y

where Φ' is the probability density function of the standard normal distribution, which is given by:

Φ'(x) = [tex](1/sqrt(2π)) * e^(^-x^2/2^)[/tex]

Substituting this expression into our equation for f_Y(y), we get:

f_Y(y) = [tex](1/sqrt(2π)) * e^(-(ln(y) + 5)^2 / (2*3^2)) / y[/tex]

Simplifying the exponent, we get:

f_Y(y) = [tex](1/(y3sqrt(2π))) * e^(-(ln(y) + 5)^2 / 18^)[/tex]

Finally, since this PDF is only defined for y > 0, we have:

f_Y(y) = [tex]{ y > 0, (1/(y3sqrt(2π))) * e^(-(ln(y) + 5)^2 / 18^)[/tex] otherwise }

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Answer:

y(x)=4*x^2+8*x+-3

y(x)=4*(x^2+2*x+-3/4) ( Factor out )

y(x)=4*(x^2+2*x+(1)^2+-1*(1)^2+-3/4) ( Complete the square )

y(x)=4*((x+1)^2+-1*(1)^2+-3/4) ( Use the binomial formula )

y(x)=4*((x+1)^2+1*-7/4) ( simplify )

y(x)=4*(x+1)^2+-7 ( expand )

Step-by-step explanation:

hope this helps:)

Answer:

y=4(x+1)^2 -7

Step-by-step explanation:

If you’re having trouble converting these equations into vertex form I suggest using math-way. com. It is extremely helpful for me when I’m in math class

Answer:

D

Step-by-step explanation:

0.034x1000=34

4.8x10=48

Answer:

D.

Step-by-step explanation:

To find the equivalent expression of 0.034 x 4.8, we can use a calculator or perform the multiplication by hand. Multiplying 0.034 and 4.8 gives:

0.034 x 4.8 = 0.1632

So, we're looking for the expression that equals 0.1632. Let's simplify each of the expressions given in the problem to see which one is equal to 0.1632:

A. 34 x 48 x 1/10 = 163.2

B. 34 x 48 x 1/100 = 16.32

C. 34 x 48 x 1/1000 = 1.632

D. 34 x 48 x 1/10000 = 0.1632

From the simplification, we can see that option D, 34x48x1/10000, is equal to 0.1632, which is what we calculated earlier.

Therefore, the correct answer is D: 34x48x1/10000.

The value of the integral is 32.

The integral you want to evaluate is ∫∫∫E(xz−y³)dV, where E = (x, y, z) with -1≤x≤3, 0≤y≤4, and 0≤z≤1.

To evaluate the integral using three different orders of integration, we will proceed with the following steps:

1. Order: dxdydz
  ∫(from -1 to 3) ∫(from 0 to 4) ∫(from 0 to 1) (xz - y³) dz dy dx

2. Order: dxdzdy
  ∫(from -1 to 3) ∫(from 0 to 1) ∫(from 0 to 4) (xz - y³) dy dz dx

3. Order: dydxdz
  ∫(from 0 to 4) ∫(from -1 to 3) ∫(from 0 to 1) (xz - y³) dz dx dy

After solving these integrals, you will find that all three orders of integration yield the same result: the value of the integral is 32.

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The third equation and substituting a1 in terms of [tex]a0[/tex], we can solve for [tex]a3[/tex]in terms

To find the power series solution of the given differential equation (x² + 1)y" + xy' - y = 0 about the ordinary point x = 0, we assume that the solution can be written as a power series:

y(x) = [tex]a0 + a1x + a2x² + a3x³ + ...[/tex]

We can then differentiate this power series twice to find expressions for y' and y'':

[tex]y'(x) = a1 + 2a2x + 3a3x² + ...[/tex]

[tex]y''(x) = 2a2 + 6a3x + ...[/tex]

We can then substitute these expressions into the differential equation and equate the coefficients of like powers of x to obtain a set of recursive equations for the coefficients. Specifically, we have:

(x^2 + 1)(2a2 + 6a3x + ...) + x(a1 + 2a2x + 3a3x² + ...) - (a0 + a1x + a2x² + a3x³ + ...) = 0

Expanding the terms and equating coefficients, we get:

[tex]a0 + 2a2[/tex] = [tex]0[/tex]

[tex]a1 - a0[/tex] = [tex]0[/tex]

[tex]2a2 + a1[/tex] = [tex]0[/tex]

[tex]6a3 + a2[/tex] = [tex]0[/tex]

Using the first equation, we can solve for [tex]a2[/tex] in terms of [tex]a0[/tex]:

[tex]a2[/tex] = -[tex]a0/2[/tex]

Using the second equation, we can solve for [tex]a1[/tex] in terms of [tex]a0[/tex]:

[tex]a1[/tex] = [tex]a0[/tex]

Using the third equation and substituting [tex]a2[/tex] in terms of [tex]a0[/tex], we can solve for [tex]a1[/tex] in terms of [tex]a0[/tex]:

[tex]a1 = -a0/2[/tex]

Using the fourth equation and substituting [tex]a2[/tex] in terms of [tex]a0[/tex], we can solve for [tex]a3[/tex] in terms of [tex]a0[/tex]:

[tex]a3 = a0/24[/tex]

Thus, the power series solution of the differential equation about x = 0 is:

y(x) = a0(1 -[tex]x^2/2[/tex] + [tex]x^4/24[/tex] - [tex]x^6/720[/tex] + ...)

where a0 is an arbitrary constant.

To use the power series method to solve the initial-value problem y" – xy' - y = 0, y(0) = 2, y'(0) = -1, we assume that the solution can be written as a power series:

[tex]y(x) = a0 + a1x + a2x² + a3x³ + ...[/tex]

We can then differentiate this power series twice to find expressions for y' and y'':

[tex]y'(x) = a1 + 2a2x + 3a3x² + ...[/tex]

[tex]y''(x) = 2a2 + 6a3x + ...[/tex]

We can then substitute these expressions into the differential equation and equate the coefficients of like powers of x to obtain a set of recursive equations for the coefficients. Specifically, we have:

[tex]2a2 + a0 = 0[/tex]

[tex]a1 - a0 = -1[/tex]

[tex]6a3 - a1 = 0[/tex]

[tex]2a4 - 6a3 - a2 = 0[/tex]

Using the first equation, we can solve for a2 in terms of a0:

[tex]a2 = -a0/2[/tex]

Using the second equation, we can solve for a1 in terms of a0:

[tex]a1 = a0 - 1[/tex]

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a) The given series is[tex]\sum (-3n/n+1)^{2n }[/tex]where n starts from 1.

b) the limit of n√|an| as n approaches infinity is 9/7.

c) the series is divergent.

a) The given series is[tex]\sum (-3n/n+1)^{2n }[/tex]where n starts from 1.
b) We can use the Root Test to determine the convergence or divergence of the given series. Let's find the limit of the nth root of the absolute value of the nth term as n approaches infinity.

[tex]lim_{ n = oo}\sqrt{(-3n/n+1)^{2n| }}=\\ lim _n[(3n^2)/(n+1)^2] \\\\= lim (3n^(3/2))/n^(3/2+2)/(n+1)^(3/2)\\= lim 3(n+1)^(3/2)/n^(7/2+1)[/tex]
We can now use L'Hopital's Rule to evaluate the above limit. Taking the derivative of the numerator and denominator with respect to n, we get:

[tex]lim 3(3/2)(n+1)^(1/2)/[(7/2)n^(5/2+1)]\\= lim (9/7) (n+1)^(1/2)/n^(7/2)\\= 9/7[/tex]

Therefore, the limit of n√|an| as n approaches infinity is 9/7.
c) Since the limit of n√|an| is greater than 1, by the Root Test, the series is divergent.

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Answer:

978 in

Step-by-step explanation:

(15 x 8) ÷ 2 = 60

60 x 2 = 180

15 x 21 = 315

315 x 2 = 630

8 x 21 = 168

180 + 630 + 168 = 978

No, Σ(an bn) is not necessarily divergent.

The product of two divergent series can converge, as long as their terms cancel each other out to some degree. For example, if an = 1/n and bn = n, then Σan and Σbn are both divergent, but Σ(an bn) = Σ1 is a convergent series.

A series is a convergent (or converges) if the sequence

[tex]{\displaystyle (S_{1},S_{2},S_{3},\dots )}[/tex]of its partial sums tends to a limit; that means that, when adding one

a{k} after the other in the order given by the indices, one gets partial sums that become closer and closer to a given number. More precisely, a series converges, if there exists a number

 such that for every arbitrarily small positive number

there is a (sufficiently large) integer

N such that for all

n>= N,

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This situation illustrates neither correlation nor causation.

Correlation refers to a relationship between two variables that are associated with each other. Causation, on the other hand, refers to a situation where one variable directly affects another variable and causes a change in it.

In the given situation, there is no direct relationship between the outcome of a particular football game and the outcome of the presidential election. It is highly unlikely that the outcome of a football game would have any causal effect on the outcome of a presidential election. Therefore, there is no correlation or causation between the two variables.

It is possible that this is simply a coincidence or that the two variables are indirectly related through some other factor that is not mentioned in the statement. However, without further evidence, we cannot make any conclusions about correlation or causation in this situation.

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Answer:

x = 40 because parallel lines cut by a transversal form congruent alternate interior angles.