Find, correct to six decimal places, the root of the equation cos(x) = x. SOLUTION We first rewrite the equation in standard form: cos(x) − x = 0. Therefore, we let f(x) = cos(x) − x. Then, f '(x) = , so Newton's method becomes xn+1 = xn − cos(xn) − xn −sin(xn) − 1 = xn + cos(xn) − xn sin(xn) + 1 . In order to guess a suitable value for x1, we sketch the graphs of y = cos(x) and y = x in the figure. It appears that they intersect at a point whose x-coordinate somewhat less than 1, so let's take x1 = 1 as a convenient first approximation. Then remembering to put our calculator in radian mode, we get the following. x2 ≈ 0.75036387 x3 ≈ 0.73911289 x4 ≈ 0.73908513 x5 ≈ 0.73908513 Since x4 and x5 agree to six decimal places (eight, in fact), we conclude that the root of the equation, correct to six decimal places, is .

The given circle has a radius of 11.2,

Given radius of the circle (r) = 11.2

The diameter of the given circle = 2*r = 2*11.2 = 22.4

The circumference of the given circle = 2πr = 2*3.14*11.2 = 70.336

The area of the given circle =  πr² = 3.14*(11.2)² = 393.88

From the above analysis, we can conclude that the diameter of the circle is 22.4 and the circumference of the circle is 70.336 and the area of the circle is 383.88.

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Therefore, the solutions of the equation in the interval [0, 2) are: x = π/6, -π/6 (in radians) or x = 30°, -30° (in degrees). Note that the solutions are listed in ascending order.

The equation is:

sec(x) = 2√3/3

First, we can find the values of x for which sec(x) = 2√3/3. Recall that sec(x) = 1/cos(x), so we have:

1/cos(x) = 2√3/3

Multiplying both sides by cos(x), we get:

1 = 2√3/3 cos(x)

cos(x) = 3/(2√3) = √3/2

Now, we can use the unit circle to find the solutions of the equation in the interval [0, 2).

cos(x) = √3/2 when x is π/6 or 11π/6 (in radians), or 30° or 330° (in degrees), since these are the angles in the unit circle where the x-coordinate is √3/2.

However, we need to make sure that these solutions are in the interval [0, 2). Since the period of sec(x) is 2π, we can add or subtract 2π to any solution to get another solution. Therefore, we need to find the solutions in the interval [0, 2π) that correspond to the solutions we found above.

π/6 is already in the interval [0, 2π), so it is a solution in the interval [0, 2). To find the other solution in the interval [0, 2), we can add 2π to 11π/6:

11π/6 + 2π = 23π/6

23π/6 is not in the interval [0, 2), so we need to subtract 2π instead:

11π/6 - 2π = -π/6

-π/6 is in the interval [0, 2), so it is also a solution in the interval [0, 2).

Therefore, the solutions of the equation in the interval [0, 2) are:

x = π/6, -π/6 (in radians) or x = 30°, -30° (in degrees)

Note that the solutions are listed in ascending order.

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The value of b = 4

The function becomes undefined when the denominator goes to 0

3x² - 48 = 0

3x² = 48

x² = 16

x = +/ 4

From the given choices, it's x = 4

Rational expressions that utilize ratios of polynomial expressions are referred to as rational expressions. These can be written in the format p(x)/q(x), where both p(x) and q(x) are polynomials with the constraint that q(x) cannot equal zero.

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The length of the hose to reach any part of the pool surface from there will be 22.36 feet.

Length of hose = √(L² + W²)

The pool is 20 feet long and 10 feet wide, the length of the hose needed would be approximately:

Length of hose = √(20² + 10²) = √500 = 22.36 feet

Therefore, Peter would need a vacuum hose that is approximately 22.36 feet long to reach any part of the pool surface from the outlet.

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The teacher originally had [tex]m*c + 13[/tex] notebooks.

If the teacher handed out m packs of notebooks with c notebooks in each pack, then the total number of notebooks that he gave out would be [tex]m*c[/tex].

If he gave out m packs of notebooks with c notebooks in each pack and has [tex]13[/tex] notebooks left, then the total number of notebooks he originally had would be:

[tex]m*c + 13[/tex]

Therefore, the expression for the total number of notebooks originally had by the teacher is [tex]m*c + 13[/tex].

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The initial value of the account is: 6320

Compound interest is defined as the interest you earn on interest. This can be illustrated by using basic math: if you have $100 and it earns 10% interest each year, you'll have $110 at the end of the first year.

The general formula to find compound interest is:

A = P(1 + r/n)^t

where:

A is final amount

P is initial principal balance

r is interest rate

n is number of times interest applied per time period

t is number of time periods elapsed

We are given the equation as:

f(t) = 6320(1.054)^(t)

Thus, the initial value is 6320

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There are 400,000 grains of sugar in a small bag of sugar that weighs 0.24 kilograms.

To find out how many grains of sugar are there in a small bag of sugar that weighs 0.24 kilograms, based on Marcus' estimate, follow these steps:

1. Convert the mass of the bag of sugar from kilograms to grams: 0.24 kg * 1000 g/kg = 240 g.
2. Use Marcus' estimate of the mass of a grain of sugar: 6 x 10^-4 g.
3. Divide the total mass of the bag of sugar by the mass of a single grain of sugar: 240 g / (6 x 10^-4 g/grain).

Now, let's perform the calculation:
240 g / (6 x 10^-4 g/grain) = 240 g / 0.0006 g/grain = 400,000 grains.

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The mean of the thickness of paper is 39 inches, and the standard deviation is 0.1 inches.

To find the mean and standard deviation of the thickness of paper after dividing by 0.0001, you should follow these steps:

1. Divide the mean by 0.0001:
Mean after dividing = 0.0039 inches / 0.0001
Mean after dividing = 39 inches

2. Divide the standard deviation by 0.0001:
Standard deviation after dividing = 0.00001 inches / 0.0001
Standard deviation after dividing = 0.1 inches

The mean of the thickness of paper after dividing by 0.0001 is 39 inches, and the standard deviation is 0.1 inches.

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The sector angle formed by the cone when it is opened is 54°.

V = 95.4 cm³

r = 2.4 cm

Calculating the height of the cone using the volume formula,

V= 1/3 x π x r² x h

Substituting the values -

95.4 = 1/3 x 3.14 x 2.4² x h

95.4 = 6.03 x h

h = 15.8 cm

Calculating the slant height using  the Pythagoras theorem -

l = √(h² + r²)

Substituting the values -

l = √(15.8² + 2.4²)

l = 16

Calculating the curved surface area of the cone -

= πrl

= π(2.4)(16)

= 120.6 cm².

Calculating the sector angle of the sector formed -

The curved surface area of the cone = area of the sector formed by the cone

= 120.6 cm².

Area of a sector in a circle = ∅/360 × πr²,

120.6 = ∅/360 × (3.14)(16²)

120.6 = ∅/360 × 803.84

(120.6)(360) = (∅)(803.84)

43,416/803.84 = ∅

∅ = 54°

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The degree of the monomial expression -3xy²z(-2x²yz) is derived to be equal to 8.

A monomial is a type of algebraic expression that consists of only one term. It is an expression in which the variables and their exponents are multiplied together, with no addition or subtraction involved. The degree of a monomial is the sum of the exponents of its variables.

Given the monomial:

-3xy²z(-2x²yz)

We can simplify it by multiplying the coefficients and adding the exponents of the variables:

-3(-2)x^(2+1) y^(2+1) z^(1+1)

= 6x³y³ z²

Therefore, the degree of the given monomial expression is: 3 + 3 + 2 = 8.

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The probability that the mean cost of the 9 weddings is more than the mean cost of the 6 showers is approximately 0.0207 or 2.07%.

The probability that the mean cost of the 9 weddings is more than the mean cost of the 6 showers can be found using the Z-score and the difference between the means of two normally distributed variables.

1: Calculate the difference in means and standard deviations.
Δμ = μ_weddings - μ_showers = $82.30 - $65 = $17.30
Δσ = sqrt((σ_weddings²/n_weddings) + (σ_showers²/n_showers)) = sqrt((18.20²/9) + (17.73²/6)) = $8.47

2: Calculate the Z-score.
Z = (Δμ - 0) / Δσ = (17.30 - 0) / 8.47 ≈ 2.04

3: Determine the probability using a Z-table.
P(Z > 2.04) ≈ 0.0207

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The length of one side of the square is x - 6 units

The formula for calculating the area of a square is expressed as;

A = a²

Such that the a is the length of its side

From the information given, we have that;

Area = x^ 2 − 12 x + 36

solve the quadratic expression, we have that;

x² - 6x - 6x + 36

group in pairs

(x²- 6x) - (6x + 36)

factorize the terms

x(x - 6) - 8(x - 6)

Then, we have;

(x - 6) and (x - 6) units

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Answer: 1/5

Step-by-step explanation:

There are 10 slips of paper.

The only numbers less than three are 1 and 2

The probability that he will pick up a slip of paper less than three is 2 since only 1 and 2 are less than three.

Therefore the probability is 2/10, and when simplified, it is 1/5.

Therefore the answer is 1/5.

If you have any more questions feel free to ask in the comments! I'd be happy to help!

the volume of the basketball is approximately 490 cubic inches. we can get this answer by using volume formula of volume

what is approximately ?

"Approximately" means almost, but not exactly. It is used to indicate that a value or quantity is very close to the true or exact value, but there may be a small difference or error. In mathematical terms, an approximate value is an estimate or a rounded value that is used

In the given question,

To find the volume of the basketball, we first need to find its radius.

Circumference of a sphere = 2πr

29.516 = 2 * 3.14 * r

r = 29.516 / (2 * 3.14) ≈ 4.7 inches (rounded to one decimal place)

Now, we can use the formula for the volume of a sphere:

Volume of sphere = (4/3) * π * r^3

Volume of basketball = (4/3) * 3.14 * (4.7)^3

Volume of basketball ≈ 490 cubic inches (rounded to the nearest whole number)

Therefore, the volume of the basketball is approximately 490 cubic inches..

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If you are earning $12.50/hr in a part-time job, then your gross "base-pay" after deductions is $253.48.

Your base pay is $12.50/hr and you worked for 18 hours, so the base pay would be:

⇒ $12.50/hr × 18 hrs = $225;

The tips are $4.60/hr on average, so the "total-tips" earned would be:

⇒ $4.60/hr × 18 hrs = $82.80;

To calculate the "total-deductions", we first find the total percentage deducted, which is the sum of FICA and Federal Tax Withholding:

⇒ 7.65% + 10% = 17.65%,

To find the total amount deducted, we multiply the total percentage deducted by the "base-pay" + "tips";

⇒ 17.65% x ($225 + $82.80) = $54.32,

So the pay, before any deductions, is the sum of "base-pay" and "tips";

⇒ $225 + $82.80 = $307.80;

To find the "net-pay", we subtract "total-deductions" from pay;

We have,

⇒ $307.80 - $54.82 = $253.48,

Therefore, your gross base pay is $253.48.

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Answer:

add the tips and pay together , this equals 17.10

then multiply by the 18 hours and you’ll get 307.80

Step-by-step explanation:

Answer:

To find the area of a triangle given its vertices, we can use the formula:

Area = 1/2 * |(x1*(y2-y3) + x2*(y3-y1) + x3*(y1-y2))|

where (x1,y1), (x2,y2), and (x3,y3) are the coordinates of the three vertices.

Using this formula, we can calculate the area of the triangle with vertices at (5,0), (-3,1), and (2,6) as follows:

Area = 1/2 * |(5*(1-6) + (-3)(6-0) + 2(0-1))|

= 1/2 * |-25 - 18 - 2|

= 1/2 * |-45|

= 22.5

Therefore, the area of the triangle is 22.5 units squared. So the answer is option C: 22.5 units squared.

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______________________________

_______________________________

Answer:

it would be at point (6,3)

Step-by-step explanation:

If you were to reflect it over the x-axis you would get (-6,3)

The best representation of the buttons in the box is the third sample.

To determine which sample is the best representation of the button in the box, we need to examine the samples and compare them to the known ratios of the colors of buttons in the box.

First sample: 6 red, 2 green, 1 blue, 1 yellow (out of 10)

Second sample: 3 red, 3 green, 3 blue, 1 yellow (out of 10)

Third sample: 2 red, 2 green, 5 blue, 1 yellow (out of 10)

To compare the samples to the known ratios, we can calculate the percentage of each color in each sample and compare it to the percentage of each color in the box.

The percentages of each color in the box are:

Red: 10/100 = 10%

Green: 30/100 = 30%

Blue: 40/100 = 40%

Yellow: 20/100 = 20%

First sample:

Red: 6/10 = 60%

Green: 2/10 = 20%

Blue: 1/10 = 10%

Yellow: 1/10 = 10%

Second sample:

Red: 3/10 = 30%

Green: 3/10 = 30%

Blue: 3/10 = 30%

Yellow: 1/10 = 10%

Third sample:

Red: 2/10 = 20%

Green: 2/10 = 20%

Blue: 5/10 = 50%

Yellow: 1/10 = 10%

Based on these percentages, we can see that the third sample is the best representation of the buttons in the box, as it is closest to the known ratios of colors in the box.

The first sample is skewed towards red buttons and away from blue and green buttons, while the second sample has equal percentages of red, green, and blue buttons, which is not reflective of the known ratios in the box.

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Answer:

3/y

Step-by-step explanation:

15x÷5xy

15x/5xy....you will cancel x by x and simplify by 5(GCF)

3/y ...... it is the simplify form of the given equetion.

Answer:

12

Step-by-step explanation:

multiply 4x18 then divide by 6

Since sin(a) is negative and a is in the third quadrant, we can use the Pythagorean identity to find cos(a):

[tex]cos^2(a) + sin^2(a) = 1[/tex]

[tex]cos^2(a) + (-4/5)^2 = 1[/tex]

[tex]cos^2(a) = 9/25[/tex]

cos(a) = -3/5 (since a is in the third quadrant)

Similarly, since sec(B) = 5/3, we can use the definition of secant to find cos(B):

sec(B) = 1/cos(B) = 5/3

cos(B) = 3/5

(a) To find sin(a + B), we can use the sum formula for sine:

sin(a + B) = sin(a) cos(B) + cos(a) sin(B)

= (-4/5)(3/5) + (-3/5)(4/5)

= -12/25 - 12/25

= -24/25

(b) To find tan(a + B), we can use the sum formula for tangent:

tan(a + B) = (tan(a) + tan(B)) / (1 - tan(a) tan(B))

To find tan(a), we can use the identity: [tex]tan^2(a) + 1 = sec^2(a)[/tex]

[tex]tan^2(a) = sec^2(a) - 1 = (5/3)^2 - 1 = 16/9[/tex]

tan(a) = -4/3 (since a is in the third quadrant)

To find tan(B), we can use the identity: tan(B) = sin(B) / cos(B) = 4/3

Plugging these values into the formula for tan(a + B), we get:

tan(a + B) = (-4/3 + 4/3) / (1 + (-4/3)(4/3))

= 0 / (1 - 16/9)

= 0

(c) To determine the quadrant containing a + B, we need to consider the signs of sin(a + B) and cos(a + B).

From part (a), we know that sin(a + B) is negative. To determine the sign of cos(a + B), we can use the Pythagorean identity:

[tex]sin^2(a + B) + cos^2(a + B) = 1[/tex]

Substituting sin(a + B) = -24/25, we get:

[tex](-24/25)^2 + cos^2(a + B) = 1[/tex]

[tex]cos^2(a + B) = 1 - (-24/25)^2[/tex]

cos(a + B) = ±7/25

Since cos(a + B) is positive in the first and fourth quadrants, and negative in the second and third quadrants, we can conclude that a + B is in the third quadrant, since cos(a + B) is negative and sin(a + B) is negative.

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The function shows the relationship between y, the total number of reams of printer paper remaining in the closet, and x, the number of boxes in the closet is y = 12x - 4

Let's start by considering the initial amount of printer paper in the closet before any boxes are used. Since each box contains 12 reams of printer paper, if there are x boxes in the closet, then the total number of reams of paper is given by 12x.

Now, if a worker uses 4 reams from one of the boxes, then the total number of reams of paper remaining in the closet is (12x - 4). If we define y as the total number of reams of paper remaining in the closet, then we have:

y = 12x - 4

This function shows the relationship between y, the total number of reams of printer paper remaining in the closet, and x, the number of boxes in the closet.

As x increases, the total number of reams of paper in the closet increases as well. However, each time a worker uses 4 reams of paper from a box, the total number of reams of paper in the closet decreases by 4.

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Answer:

y=4x+4

Step-by-step explanation:

The slope formula is:
[tex]y=mx+b[/tex]

with m being the slope and b being the y-intercept

Given: slope=4, y-intercept=4

We can substitute the slope and the y-intercept into the question:

y=4x+4

Hope this helps! :)

The equation of the line in slope-intercept form of a line with slope 4 and y-intercept 4 is [tex]\text{y} = 4\text{x} + 4[/tex].

The ratio that y increase as x increases is the slope of a line. The slope of a line reflects how steep it is, but how much y increases as x increases. Anywhere on the line, the slope stays unchanged (the same).

[tex]\text{m}=\dfrac{\text{y}_2-\text{y}_1}{\text{x}_2-\text{x}_1}[/tex]

[tex]\text{m}=\dfrac{(\text{y}\bar{\text{a}}-\text{y}\bar{\text{a}})}{(\text{x}\bar{\text{a}}-\text{x}\bar{\text{a}})}[/tex]

It is given that:

A line with slope 4 and y-intercept 4.

The linear equation in one variable can be made:

As we know,

The standard equation of the line is:

[tex]\text{y} = \text{mx} + \text{c}[/tex]

Here m is the slope and c is the y-intercept.

[tex]\text{m} = 4[/tex]

[tex]\text{c} = 4[/tex]

[tex]\boxed{\bold{y = 4x + 4}}[/tex]

Thus, the equation of the line in slope-intercept form of a line with slope 4 and y-intercept 4 is [tex]\text{y} = 4\text{x} + 4[/tex].

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For every 3 squares, there are 5 total shapes. Therefore, the simplified ratio of squares to total shapes is 3 : 5.

If the unsimplified ratio of squares to total shapes is 6 : 16, then we can express this as 6/16, then,we can simplify this ratio by dividing both the numerator and denominator by 2, giving us 3/8.

This means that for every 3 squares, there are 5 total shapes. Therefore, the simplified ratio of squares to total shapes is 3 : 5.

Full question "Unsimplified ratio of squares to total shapes: 6 : 16. For every 3 squares, there are ____ total shapes; Therefore, the simplified ratio of squares to total shapes is __ : __.

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Answer:

Step-by-step explanation:

16 - (y^2/sqrt(49)) +12

at y = -5

16 - ( (-5)^2 / sqrt(49)) +12    

16 - ( 25 / 7) + 12

28 - (25/7)

(28*7 - 5) / 7    (LCM)

(196 - 5) / 7

(191/7)

27.2857 = Ans

Derivative of y w.r.t t is dy/dt = 16 at x = -1.

We need to find dy/dt at x = -1, given y = -2x² + 4 and dx/dt = 4.

Step 1: Differentiate y with respect to x.
Since y = -2x² + 4, we have:

dy/dx = -4x

Step 2: Substitute x = -1 into the dy/dx equation.
When x = -1, we get:

dy/dx = -4(-1) = 4

Step 3: Use the Chain Rule to find dy/dt.
The Chain Rule states that dy/dt = (dy/dx)(dx/dt). We know dy/dx = 4 and dx/dt = 4, so we have:

dy/dt = (4)(4) = 16

Thus, dy/dt = 16 at x = -1.

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For two parametric equations, x = cos(t)− sin(t) ; y = cos(t) + sin(t) ; 0≤t≤2π

a) Cartesian equation for the parametric curve is represented by x² + y² = 2.

b) The sketch for this parametric curve, with arrows in the direction of curve tracing is present above figure.

A parametric curve in the x-t plane has the equations x=x(t), y=y(t). The curve associates a point of the plane (x,y) to a value of the parameter t. The rectangular form of the curve can be determined by eliminating the parameter t, i.e. determine the parameter in one equation and Substituting this value in the other equation. We have the following parametric equations,

x = cos⁡t - sin⁡ty = cos⁡(t)+ sin⁡t, 0 ≤ t ≤ 2π

(a) we have to eliminate parameter t to determine a cartesian equation for the parametric curve, use x²+ y² = (cos⁡(t) − sin⁡(t))²+ (cos⁡(t) + sin⁡(t))²

=> x² +y² = cos²t + sin²t - 2cos⁡t sin⁡t + cos²t + sin²t + 2cos⁡t sin⁡t

=> x² + y² = 2 ( sin²t + cos²t) = 2

which represents a circle curve centered at the origin and having radius √2.

(b) A sketch of this parametric curve is shown above figure and arrows are used to indicate the direction of curve trace.

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Option C is correct. No, the least value is smaller than the critical value.

Null hypothesis (H₀): The average first-year teacher salary is $52,000.

Alternative hypothesis (Ha): The average first-year teacher salary is not $52,000.

The significance level is 5% (or 0.05), which means we will reject the null hypothesis if the probability of obtaining the observed result is less than 5%.

Calculate the standard error of the mean:

Standard Error = Standard Deviation / √n

where n is the number of samples (n = 25 in this case).

Standard Error = $1500 / √25

= $1500 / 5

= $300

Now, perform the hypothesis test using a t-test since the sample size is relatively small (n < 30) and the population standard deviation is unknown.

t-score = (Sample Mean - Population Mean) / Standard Error

t-score = ($52,525 - $52,000) / $300

t-score = $525 / $300

t-score = 1.75

To find the critical value at a 5% significance level with 24 degrees of freedom (n - 1), we can consult a t-table. At a 5% significance level (two-tailed test), the critical t-value is approximately ±2.064.

Since the calculated t-score (1.75) is not greater than the critical t-value (2.064), we fail to reject the null hypothesis.

Therefore, option C is correct. No, the least value is smaller than the critical value.

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Complete question:

The average first year teacher salary in a certain state is known to be $52,000 with a standard deviation of $1500. A researcher tests this claim by averaging the salaries of 25 first year teachers and finding their average salary to be $52,525. Is there significant evidence to suggest that the claim is wrong at the 5% significance level?

A. Yes, the least value is greater than the critical value

B. No, the least value is larger than the critical value

C. No, the least value is smaller than the critical value

D. yes, the least value is smaller than the critical value

To determine a second pair of corresponding angles from KJN and LJM that are congruent, we can start by identifying the first pair of corresponding angles.

angle JKN is  harmonious to angle LJM. thus, we need to find another brace of corresponding angles that involve these same two angles.   One possibility is to look at the  perpendicular angles formed by the  crossroad of KJ and JM. Angle KJM is  perpendicular to angle NJL. therefore, angle KJM in KJN corresponds to angle NJL in LJM. thus, these two angles are  harmonious.  

We can prove that these two angles are  harmonious using the  perpendicular angles theorem, which states that  perpendicular angles are always  harmonious. Since KJ and JM  cross at point J, angles KJM and NJL are  perpendicular angles and must be  harmonious.   thus, we've shown that the alternate brace of corresponding angles from KJN and LJM that are  harmonious are angle KJM and angle NJL.

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