The 10th term of the geometric sequence would be 3 × (-3)¹⁰.
What is a geometric sequence?A geometric sequence is a special type of sequence where the ratio of every two successive terms is a constant. This ratio is known as a common ratio.Given is the geometric sequence -
- 9, 27, - 81 ...
We can write the common ratio as -
r = 27/-9 = -3
{r} = - 3 ... Eq { 1 }
a{10} = (-9)(-3)⁹
a{10} = 3(-3)¹⁰
Therefore, the 10th term of the geometric sequence would be 3 × (-3)¹⁰.
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solve the following initial value problem using the method of undetermined coefficients: (2 points)
The solution to the initial value problem y'' - 2y' + 2y = e^(-x) with y(0) = 0 and y'(0) = 0 is given by y(x) = x - e^(-x) + x^2.
y'' - 2y' + 2y = e^(-x)
y(0) = 0, y'(0) = 0
Let Yp = A + Bx + Cx^2
Yp' = B + 2Cx
Yp'' = 2C
Substituting into the original equation yields:
2C - 2(B + 2Cx) + 2(A + Bx + Cx^2) = e^(-x)
Collecting terms gives:
2C - 2B = 0, 2C - 2A = e^(-x), and 2C = 2Bx
Solving for C yields C = Bx, for B yields B = 2C, and for A yields A = 2C - e^(-x).
Therefore, the general solution is
y = (Bx - e^(-x)) + Bx^2
Using the initial conditions, we get
y(0) = 0 = (B(0) - e^(0)) + B(0)^2
0 = -1 + B(0)^2
B(0) = 1
The solution to the initial value problem is
y(x) = x - e^(-x) + x^2
The solution to the initial value problem y'' - 2y' + 2y = e^(-x) with y(0) = 0 and y'(0) = 0 is given by y(x) = x - e^(-x) + x^2.
The complete question is :
Use the method of undetermined coefficients to solve the following initial value problem:
y'' - 3y' + 2y = cos(2t)
y(0) = 0
y'(0) = 0
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Use the definition of Ax to write the matrix equation as a vector equation. [ 2 7 -5 -1 2 -6 4 8 ] [ 2 -3 ] = [ 17 7 -22 16]
The matrix equation written as a vector equation is .
This is the vector equation equivalent to A = [2 7 -5 -1 2 -6 4 8], x = [2 -3], and b = [17 7 -22 16].
What is matrix equation ?Simultaneous equations can be presented and solved more simply using matrix algebra, a type of mathematical notation. It can be used to generate a mathematical model of the structure and to provide a clear statement of a structural issue.
The vector equation is as follows
r = a(l – t)+ b t
where,
t is some parameter and
for points between A and B then 0≤t ≤ 1.
According to question:
The matrix equation [2 7 -5 -1 2 -6 4 8][2 -3] = [17 7 -22 16] can be written as a vector equation using the definition of the product of a matrix and a vector.
Let's call the matrix [2 7 -5 -1 2 -6 4 8] as matrix A and the vector [2 -3] as vector x. The product of matrix A and vector x is a new vector, which we'll call vector b.
The equation [2 7 -5 -1 2 -6 4 8][2 -3] = [17 7 -22 16] can be written as:
A * x = b
where,
A = [2 7 -5 -1 2 -6 4 8], x = [2 -3], and b = [17 7 -22 16].
This is the vector equation equivalent of the matrix equation, which represents the same linear relationship between the matrix, the vector, and the result vector.
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Which of the following gives the correct range for the graph?
−4 ≤ x ≤ 2
−3 ≤ x ≤ 5
−4 ≤ y ≤ 2
−3 ≤ y ≤ 5
The correct option regarding the range of the function graphed in this problem is given as follows:
−4 ≤ y ≤ 2.
How to obtain the range of a function?The range of a function is the set that contains all the output values that can be assumed by the function.
Hence, on the graph, the range of the function is given by the values of y assumed by the graph of the function.
The minimum and maximum values of y are given as follows:
Minimum of y = -4.Maximum of y = 2.As the function is continuous, the range is given as follows:
−4 ≤ y ≤ 2.
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What percentage commission does a salesperson make if they earn $151.34 from selling $658 worth of goods?
Answer:
23%
Step-by-step explanation:
Brody calculated the area of a square to be 16 36 square foot. Which shows the side length of the square? A. 2 9 ft B. 1 3 ft C. 4 9 ft D. 2 3 ft
The side length of the square be 4/6 foot.
What is meant by square?A square is a regular quadrilateral with four equal-length sides and four equal-length angles. The square's angles are 90 degrees or at right angles. In addition, the square's diagonals are equally spaced and split at a 90-degree angle.
Having four sides, a square is a quadrilateral. Each side measures the same length. A square has 360 degrees of angles total. In a square, every angle is a right angle.
Let the equation be
Area of a square = length²
If the area of a square is known, find the square root of the area to get the length.
√16 / 36 = 4/6 foot
Therefore, the side length of the square be 4/6 foot.
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Here we work in the system of integer polynomials. Those are polynomials of the Form f(x)=rnxn+···+r1x+r0 where every coefficient is an integer. General question: When does some combination of the polynomials ax + b and cx + d equal 1 ? That is, when do there exist integer polynomials P(x) and Q(x) with P(x)·(ax + b) + Q(x)·(cx + d) = 1 ? We concentrate here on cases when c = 0. (a) Prove: No combination of 2x + 5 and 3 can equal 1. That is, no integer polynomials P (x), Q(x) can satisfy: P (x)·2x + 5 + Q(x)·3 = 1. (b) Find a combination of 2x + 5 and 4 that equals 1. (c) Does some combination of 15x+9 and 25 equal 1? How about 15x+9 and 20? Explain your reasoning. (d) Investigate further examples of ax + b and d, deciding in each case whether 1 is a combination. What patterns do you detect? Can you prove that some of your observed patterns always hold true?
It appears that for any ax + b, a combination of ax + b and 2a+d equals 1. This can be proven by setting P(x) = -ax - b and Q(x) = 2a+d.
(a) No combination of 2x + 5 and 3 can equal 1.
Let P(x) and Q(x) be two integer polynomials such that P(x)·(2x + 5) + Q(x)·3 = 1.
This can be written as P(x)·2x + P(x)·5 + Q(x)·3 = 1.
Rearranging, we get P(x)·2x + (P(x)·5 + Q(x)·3) = 0
This implies that P(x)·2x = -(P(x)·5 + Q(x)·3).
Since both P(x) and Q(x) are integer polynomials, it follows that P(x)·2x and (P(x)·5 + Q(x)·3) are both integers.
However, this is impossible since the left hand side is an even integer, while the right hand side is an odd integer.
Therefore, no combination of 2x + 5 and 3 can equal 1.
(b) A combination of 2x + 5 and 4 that equals 1 is P(x) = -2x - 5 and Q(x) = 4.
(c) No combination of 15x+9 and 25 equals 1. However, a combination of 15x+9 and 20 equals 1. This can be seen by setting P(x) = -15x - 9 and Q(x) = 20.
(d) Some examples of ax + b and d combinations that equal 1 are:
2x + 5 and 4
3x + 4 and 7
4x + 3 and 9
5x + 2 and 11
It appears that for any ax + b, a combination of ax + b and 2a+d equals 1. This can be proven by setting P(x) = -ax - b and Q(x) = 2a+d.
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Penny's parents have agreed to loan her $4500 to pay her tuition. They are charging her an interest rate of
3% per annum, compounded monthly. Penny has arranged to pay them $160 per month to pay off the loan.
a) how long it takes Penny to pay off the loan b) the amount of her final payment
Answer:
$157.92.
Step-by-step explanation:
a) To calculate how long it takes Penny to pay off the loan, we can use the formula:
n = log(P/A) / log(1 + r/12),
where n is the number of months, P is the initial loan amount ($4500), A is the monthly payment ($160), and r is the annual interest rate (3%).
n = log(4500/160) / log(1 + 0.03/12)
n = log(28.125) / log(1.0025)
n = 3.7062
Rounding up, it takes Penny 4 months to pay off the loan.
b) To calculate the amount of Penny's final payment, we can use the formula:
P = A * (1 - (1 + r/12)^-n),
where P is the remaining balance, A is the monthly payment ($160), n is the number of months (4), and r is the annual interest rate (3%).
P = 160 * (1 - (1 + 0.03/12)^-4)
P = 160 * (1 - (1.0025)^-4)
P = $157.92
Therefore, it takes Penny 4 months to pay off the loan and her final payment is $157.92.
When u and v are nonzero vectors, Spanlu,v) contains only the line through u and the line through v and the origin. OA. False. Span(u,v) includes linear combinations of both u and v. 0 B. False. Span(u,v) will not contain the origin. ° C. True. Span(u,v) is the set of all scalar multiples of u and all scalar multiples of v
False. Span(u,v) includes linear combinations of both u and v.
The correct option is A
Now, According to the question:
When u and v are nonzero vectors, Span {u, v} contains only the line through u and the origin, and the line through v and the origin. b. Any list of five real numbers is a vector in ℝ5 .
Span(u,v) is the set of all linear combinations of u and v, meaning any combination of u and v multiplied by scalars. This includes scalar multiples of both u and v, but does not include the origin. To calculate Span(u,v), first write u and v as vectors u = (u1,u2,...,un) and v = (v1,v2,...,vn). Then, any linear combination of u and v can be written as c1u + c2v, where c1 and c2 are scalars, and the set of all such linear combinations is Span(u,v). For example, if u = (1,2) and v = (3,4), then Span(u,v) = {(1,2), (3,4), (4,6), (5,8), (7,10), ...}. In other words, Span(u,v) is the set of all scalar multiples of u and all scalar multiples of v.
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How many solutions to the equation 5x=x+3
The equation 5x = x + 3 has 1 solution.
The solution to the equation 5x = x + 3 is x = 3.
7. use induction to prove the following equation: 1 4 9 ... n2 = n(n 1)(2n 1)/6 where n ≥ 1 (8 points).
By using the induction, the equation 1+ 4 + 9 + ...........+n² = n(n-1)(2n-1)/6 where n ≥ 1 is proved.
The term induction in math is defined as a technique of proving a statement, theorem or formula which is thought to be true, for each and every natural number n.
In this case, we want to use induction to prove the equation:
=> 1 + 4 + 9 + ... + n² = n(n + 1)(2n + 1)/6 for all integers n ≥ 1.
To start the induction proof, we first prove that the statement is true for the base case n = 1.
The left side of the equation is 1, and the right side is
=> 1 * (1 + 1) * (2 * 1 + 1) / 6 = 1 * 2 * 3 / 6 = 1,
so the equation is true for n = 1.
Next, we assume that the equation is true for some integer k, where k ≥ 1. This means that:
=> 1 + 4 + 9 + ... + k² = k(k + 1)(2k + 1)/6.
Now, we use this information to prove that the statement is true for k + 1. The left side of the equation for k + 1 is:
=> 1 + 4 + 9 + ... + k² + (k + 1)².
This is equal to k(k + 1)(2k + 1)/6 + (k + 1)² by the induction hypothesis.
The right side of the equation for k + 1 is:
=> (k + 1)(k + 2)(2k + 3) / 6.
We can simplify this to:
=> (k² + 3k + 3) * (k + 2) / 6 = (k² + 3k + 3)(k + 2) / 6.
Finally, we need to show that these two expressions are equal:
=>k(k + 1)(2k + 1)/6 + (k + 1)² = (k² + 3k + 3)(k + 2) / 6.
Expanding both sides and cancelling common terms, we find that:
=> k³ + 3k² + 2k + k² + 3k + 3 = 2k³ + 9k² + 12k + 6.
This shows that the equation is true for k + 1, and by induction, it is true for all integers n ≥ 1.
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Zach’s car travels 21 miles on 1 gallon of gas. Write an equation to represent the relationship between the gas Zach’s car uses and the distance he travels. Then solve the equation to see how far Zach travels on a trip if he uses 16 gallons of gas
The function that represents the context is y = 21x and Zach's car can travel 336 miles using 16 gallons of gas.
What is a function?A function can be defined as the outputs for a given set of inputs.
The inputs of a function are known as the independent variable and the outputs of a function are known as the dependent variable.
Given, Zach’s car travels 21 miles on 1 gallon of gas.
Let, x represents the number of gallons of gas and y represents the distance traveled.
Therefore, The function can be formed as,
y = 21x or f(x) = 21x.
The distance covered by Zach's car using 16 gallons of gas would be,
y = 21×16.
y = 336 miles.
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PLEASE HELP ILL GIVE BRAINLYIST
What is the volume of the composite solid? Use 3.14 for \pi and round your answer to the nearest cm^(3)
Answer: i believe it is B
Step-by-step explanation:
if we do 3.14 for pi and round our answer to the nearest cm then with how i did it you get the answer of B
if the system has a radius of 40 cm, what is the tangential acceleration (in m/s2) of a point on its outermost edge at t = 0.5 s?
The angular acceleration of the system times the system's radius, which is 40 cm, or 0.4 m, gives rise to the tangential acceleration at the system's farthest edge at time t = 0.5 s. Thus, 0.4 m/s2 is the tangential acceleration.
The radius of the system multiplied by the system's angular acceleration gives the tangential acceleration of a point at its furthest edge. The system's radius in this instance is 40 cm, or 0.4 metres, and the time is 0.5 seconds. As a result, the tangential acceleration at time t = 0.5 s is equal to the system's angular acceleration times its radius, which is 0.4 m. We multiply the angular acceleration by the system's radius, which is 0.4 m, to determine the tangential acceleration. Thus, 0.4 m/s2 is the tangential acceleration at time t = 0.5 s. This indicates that a point on the system's outermost edge accelerates by 0.4.
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Find the ordered pair that is a member of both y = 18 + x and y = -6x 52 or indicate if it does not exist or there are infinite possibilities.
Please help me with math problem!! Will give brainliest!! :) It's due tonight!! 20 POINTS!!!
The required they could have saved 21.97 feet of cable, as per the given conditions.
What are Pythagorean triplets?In a right-angled triangle, its sides, such as hypotenuse, and perpendicular, and the base is Pythagorean triplets.
Here,
Now, perpendicular p = 30 ft base b = 50 ft hypotenuse h = x ft
According to Pythagoras' theorem h² = p² + b²
x² = 50² + 30²
x = √2500 + 900
x = √3400
x = 58.3 feets
Wire saved = [30 + 50] - 58.3
Wire saved = 21.97
Thus, the required they could have saved 21.97 feet of cable, as per the given conditions.
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Which integral would give the total volume of this solid?
The integral to calculate the total volume of the solid would be a triple integral over all the three dimensions of the solid. This integral would be in the form of an iterated integral of the form:
To calculate the total volume of a solid, we need to use a triple integral over all three dimensions of the solid. This triple integral is an iterated integral, meaning that it is composed of three integrals nested within each other. This integral has the form [tex]$\int \int \int f(x,y,z) dx dy dz$[/tex]. In this equation, the function [tex]$f(x,y,z)$[/tex] is a function that describes the shape of the solid. In other words, this function describes the properties of the solid as we move through each of the three dimensions. To calculate the total volume, we then need to calculate the integral of this function over all three dimensions. This gives us the total volume of the solid.
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Please help I can’t find Jonathan’s mistake.
The shortest distance between Tim and Jan is given as follows:
19.2 miles.
(they are 19.2 miles apart).
Jonathan's mistake is that the shortest distance is not obtained passing through the home, but walking diagonally from Tim's position to Jonathan's position or vice versa.
What is the Pythagorean Theorem?The Pythagorean Theorem states that for a right triangle, the length of the hypotenuse squared is equals to the sum of the squared lengths of the sides of the triangle.
To obtain the shortest distance between Tim and Jan, we consider a right triangle in which:
The height is of 16 - 4 = 12.The base is of 12 + 3 = 15.Hence the distance is obtained as follows:
d² = 12² + 15²
d = square root(12² + 15²)
d = 19.2 miles.
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What is 200 meters in feet?
To convert 200 meters to feet, you can use the following conversion factor 1 meter = 3.2808 feet
What is 200 meters in feet?Meters and feet are units of length used to measure distances and lengths.
The conversion factor of 1 meter to 3.2808 feet is based on the definition of the foot as being equal to approximately 0.3048 meters.
To convert from meters to feet, simply multiply the number of meters by the conversion factor of 3.2808.
For example, 200 meters is equal to 200 x 3.2808 = 656.1664 feet.
It's important to note that this conversion factor is an approximation and may not be entirely accurate in all situations.
For more precise conversions, it's best to use a conversion calculator or a reference guide that provides exact conversion factors.
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Bob bought a computer in California the computer cost $600 California has an 8% sale tax how much did bob pay for the computer including sale tax
Answer:
Below
Step-by-step explanation:
600 = cost
.08 * 600 = tax
Add the two together 600 + .08 (600) = 1.08 * 600 = $ 648
Use the image below to determine what relation ship exists between AGB and BGD
A: adjacent angles that form AGC
B: vertical angles
C: complementary angles
D: supplementary angles
find the lateral surface area of the cylinder used to play for for pie and round to the nearest whole number
Answer:
lateral Surface Area(LSA)=2×22/7×4.9(4.9+11.2)
LSA=2×22/7×4.9×16.1
LSA=495.88
The nearest whole number is 496
What is the volume of the solid formed by revolving the region bounded by the graphs y = √x and y = x2 about the x-axis? Use the Shell method.
The volume of the solid formed by revolving the region bounded by the graphs y = √x and y = x^2 about the x-axis is approximately 2.667 cubic units.
What is the shell method for finding the volume of the solid?
The shell method for finding the volume of a solid formed by revolving a region about an axis is a method used to find the volume of the resulting solid by adding up the volumes of an infinite number of thin cylindrical shells.
In this case, the region bounded by the graphs y = √x and y = x^2 is revolved about the x-axis. To use the shell method, we first need to determine the height and radius of each cylindrical shell. The height of each shell is equal to the difference between the upper and lower bounds of the region, which is x^2 - √x. The radius of each shell is equal to the value of y at that point, which is either √x or x^2.
The volume of each shell can then be calculated as V = 2πr * h, where r is the radius and h is the height. To find the total volume, we need to integrate the volume of each shell over the interval of x.
The definite integral for the volume is given by:
∫_{0}^{1} 2π x * (x^2 - √x) dx
Evaluating this definite integral gives a volume of approximately 2.667.
Hence, the volume of the solid formed by revolving the region bounded by the graphs y = √x and y = x^2 about the x-axis is approximately 2.667 cubic units.
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you read 7 pages in 12 minutes how many pages can you read in 3 hours?
Answer:
105 pages
Step-by-step explanation:
We can solve this using ratios
Changing 3 hours to minutes
3 * 60 = 180 minutes
7 pages x pages
--------------- = ----------------
12 minutes 180 minutes
Using cross products
180 * 7 = 12 x
Divide each side by 12
180*7/12 = x
105 =x
105 pages
Answer:If you can read 7 pages in 12 minutes then you will be able to read 105 pages in 3 hours.
Step-by-step explanation: you multiply 12x15 to get 180 since 180 is 3 hours in minutes then you multiply 7x15= 105. So you will read 105 pages every 3 hours.
100th Answer!!!!
Picaro’s Packaging Plant wishes to design boxes with a volume of not more than 100 . Squares are to be cut from the corners of a 12-in. by 15-in. piece of cardboard (see figure), with the flaps folded up to make an open box. What size squares should be cut from the cardboard?
The size squares that should be cut from the cardboard is 0.692 inch
What size squares should be cut from the cardboard?Represent the cut-out with x
So, we have the following volume equation:
V(x) = (12 - 2x)(15 - 2x)x
The volume is 100
So, we have
(12 - 2x)(15 - 2x)x = 100
Using a graphing calculator, we have
x = 0.692
Hence, the solution is 0.692
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Which proportion satisfies the geometric mean (altitude) theorem for the triangle?.
We are instructed to select the ratio that fulfills the triangle's geometric mean (altitude) theorem by 2/h = h/3.
What is the triangle's geometric theorem?We are instructed to select the ratio that fulfills the triangle's geometric mean (altitude) theorem.The geometric mean of the line segments formed by the altitude on the hypotenuse is what the right triangle altitude theorem states the height on the hypotenuse to be. Two identical right triangles are created for a right triangle when a perpendicular is traced from the vertex to the hypotenuse.According to the geometric mean (altitude) theorem, the hypotenuse is divided into two segments by an altitude drawn at a right angle to it, and the product of these two segments equals the altitude times the altitude.2/h = h/3.
The Complete Question.
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suppose that you pull out a toy at random, and you observe only the color, noting that it is red. conditional on just this information, what is the probability that the toy is not cool?
Without additional information, it is impossible to determine the probability that the toy is not cool.
Without additional information, it is impossible to determine the probability that the toy is not cool. This is because the color of the toy does not provide any information about its coolness. For example, a red toy could be a cool action figure or a boring stuffed animal. Therefore, the color of the toy does not provide any insight into its coolness, and thus any probability assigned to the toy being not cool would be purely speculative and not based on any information. In order to determine the probability that the toy is not cool, we would need additional information, such as what the toy is, what its features are, or who makes it.
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Lisa has $580 budgeted for a trip she wants to spend five eights of it at her budget and travel expenses to the nearest cent what is she willing to spend on travel expenses?
Answer:
since it is 5/8ths of 580, multiply 5/8 by 580: 580(5/8) = 362.5 Then it says round to the nearest cent which is 0.5. So she is willing to spend $362.5 of her budget.
x is taken away by 1/6, then taken away by 1/13 and left with 1,000.
what is x?
So the initial value of x was 11142.86 when divided by 1/6, then divided by 1/13, and finally divided by 1,000.
What is equation?An algebraic equation, also known as a polynomial equation, is a mathematical equation of the form P=0, where P is a polynomial with coefficients in some field, most commonly the field of rational numbers. In its most basic form, an equation is a mathematical statement that indicates that two mathematical expressions are equal. 3x + 5 = 14, for example, is an equation in which 3x + 5 and 14 are two expressions separated by a 'equal' sign.
Here,
Let's call the original value of x as X. Then, we can write the following equation:
X - X/6 - X/13 = 1000
Expanding the first two terms on the left-hand side:
X - X/6 - X/13 = 1000
7X/78 = 1000
X = 78 * 1000 / 7
X = 11142.85714
So, the original value of x was 11142.86 when x is taken away by 1/6, then taken away by 1/13 and left with 1,000.
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tom pays R1250
every month what is his annual rent if his rent increased by 8,5% at the end of the year what will his increased monthly rent be
Tom's increased monthly rent after the 8.5% increase will be R1356.25.
How to determine the increased monthly rentFrom the question, we have the following parameters that can be used in our computation:
Monthly rent = R1250
Yearly rate = 8.5%
If Tom pays R1250 every month, then his annual rent is:
Annual rent = 1250 * 12 = R15000.
If his annual rent increased by 8.5%, his new annual rent will be
New = 15000 * (1 + 0.085) = 16275.
Divide by 12 to get the increased monthly rent
Increased monthly rent = 16275 / 12 = R1356.25.
Hence, the rent is R1356.25.
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Given mn find the value of x.
Because lines m and n are parallel, we conclude that x° = 57°
How to find the value of x?Ok, remember that when we have an intersection of two lines, the vertical angles (the ones connected by the vertex) have the same measure.
Here we know that lines m and n are parallel lines, then the angles in the two intersections are the same angles.
That means that x° and the 57° angle are vertical angles, so these have the same measure, then we can conclude that:
x° = 57°
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