Use the given function to find the following.
x + 10
5
f (x) =
=
-1
f-¹ (x).
Find
-1
Find f (ƒ-¹ (5)).
(no spaces)

By applying the algebra concept, it can be concluded that Leona ordered 25 boxes of pens.

Algebra is a branch of mathematics that uses symbols and mathematical operations, such as addition, subtraction, multiplication, and division to solve problems.

We have this information:

The price of a box of pens = $8

The price of a box of rubber bands = $4

Total order = 30 boxes

Total payment = $220

Let p symbolize the number of pens and r symbolize the number of rubber bands. Now we have the following equations:

p + r = 30 ....................... (1)

8p + 4r = 220 ............... (2)

To find the value of p and r, we can do it by simplifying the first equation as follows:

p + r = 30

r = 30 - p

Then we can substitute this value into the second equation:

         8p + 4r = 220

8p + 4(30 - p) = 220

8p + 120 - 4p = 220

                 4p = 220 - 120

                      = 100

                   p = 100/4

                      = 25 boxes of pens

So we can calculate the value of r as well:

r = 30 - p

 = 30 - 25

 = 5 boxes of rubber bands

Thus, it can be concluded that Leona ordered 25 boxes of pens.

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You can determine if the inverse of a polynomial function is a function by using the Horizontal Line Test on the inverse.

The Horizontal Line Test is a graphical method that can be used to determine whether a given function is one-to-one, meaning that for each output value, there is at most one input value that maps to it.

If the inverse of a polynomial function is a function, it must pass the Horizontal Line Test, meaning that no horizontal line intersects the graph of the inverse more than once.

In other words, if the inverse of a polynomial function passes the Horizontal Line Test, it is guaranteed to have an inverse, which will be a function itself. If the Horizontal Line Test fails, then the inverse is not a function and does not have an inverse.

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Answer:

So first solve the equation given

X+2y=-14

-x        -x

-----------------

2y=-x-14

----  ---------

2        2

y=- 1/2x-7

this would be your parallel line equation because y=-1/2x-7

Parallel lines have the same slope

Perpendicular lines have a negative reciprocal slope (so -2 would be positive 1/2)

Hope this helps!

Please leave a comment if I made a mistake, I will correct it!

Answer:

Step-by-step explanation:

Take any two pairs of coordinates to find the slope.

The forumla for slope is (y2-y1)/(x2-x1)

Wil will choose the points (2,10) and (4,20)

Therefore:

m = (20-10)/(4-2)

m = 5

The given function is [tex]f(x,y) = x^2 + y^2[/tex].

To find the gradient of f at the point (x0, y0), we take the partial derivatives of f with respect to x and y:

[tex]$$\nabla f(x0,y0) = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right) = \left(2x0, 2y0\right)$$[/tex]

The rate of increase of $f$ at the point (x0, y0) in the direction (u,v) is given by the dot product of the gradient and the direction vector:

[tex]$$\nabla f(x0,y0) \cdot (u,v) = 2x0u + 2y0v$$[/tex]

The direction of maximum rate of increase at (x0, y0) is the direction of the gradient, which is (2x0,2y0). The rate of increase in this direction is the length of the gradient vector, which is given by:

[tex]$$||\nabla f(x0,y0)|| = \sqrt{(2x0)^2 + (2y0)^2} = 2\sqrt{x0^2 + y0^2}$$[/tex]

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The area under the curve y = 3x³ + 4x from x = 1 to x = 2 is 4 square units.

A curve is a continuous, smooth, and often repeating shape in the plane or in space. It is a geometric object that can be described mathematically by an equation or a set of points.

In mathematics, curves are used to model real-world phenomena and to study geometric and mathematical properties. For example, curves can be used to describe the shape of objects, the path of moving objects, or the graph of a mathematical function.

There are many types of curves, including lines, circles, ellipses, parabolas, hyperbolas, and spirals, among others. Each type of curve has unique properties and characteristics, and they are studied and used in different ways in different fields of mathematics and science.

In physics, curves are used to describe the motion of objects and the behavior of physical systems. For example, the trajectory of a projectile can be modeled by a parabolic curve, while the path of a planet around the sun can be modeled by an elliptical curve.

To find the area under the curve y = 3x^3 + 4x from x = 1 to x = 2, we can use the definite integral:

∫[1,2] (3x^3 + 4x) dx

To evaluate this integral, we can use the power rule for integration: ∫x^n dx = (x^(n+1))/(n+1) + C, where C is the constant of integration.

Using this rule, we get:

∫ (3x³ + 4x) dx = ∫ 3x³ dx + ∫ 4x dx = x^4/4 + 2x² + C

Evaluating this definite integral from x = 1 to x = 2, we get:

[tex][x^{\frac{4}{4}+2x^{2} } ]2-[x^{\frac{4}{4}+2x^{2} } ]1=(2^{\frac{4}{4} }+22^{2} ) -(1^{\frac{4}{4} } +21^{2} )=\frac{21}{4}-\frac{16}{4}=4[/tex]

So the area under the curve y = 3x^3 + 4x from x = 1 to x = 2 is 4 square units.

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The value of the double integral is [tex]$-\frac{\pi}{12}$[/tex].

We are given that;

∫∫D^y^dA where d is the set of points (x, y)

0 ≤ 2x/ π ≤ y, y ≤ sin x

Now,

We can see that x varies from 0 to [tex]\pi[/tex], and for each fixed x, y varies from [tex]\frac{2x}{\pi} to \sin x[/tex]. Therefore, we can write the double integral as:

[tex]$$\iint_D y \, dA = \int_0^\pi \int_{\frac{2x}{\pi}}^{\sin x} y \, dy \, dx$$[/tex]

Now we can evaluate the inner integral with respect to y:

[tex]$$\int_{\frac{2x}{\pi}}^{\sin x} y \, dy = \frac{y^2}{2} \bigg|_{\frac{2x}{\pi}}^{\sin x} = \frac{\sin^2 x}{2} - \frac{2x^2}{\pi^2}$$[/tex]

Plugging this result into the outer integral, we get:

[tex]$$\iint_D y \, dA = \int_0^\pi \left( \frac{\sin^2 x}{2} - \frac{2x^2}{\pi^2} \right) dx = \frac{1}{4} \int_0^\pi (1 - \cos 2x) dx - \frac{2}{\pi^2} \int_0^\pi x^2 dx$$[/tex]

Using the antiderivatives of [tex]\cos 2x[/tex] and x^2, we get:

[tex]$$\iint_D y \, dA = \frac{1}{4} (x - \frac{1}{2} \sin 2x) \bigg|_0^\pi - \frac{2}{\pi^2} (\frac{x^3}{3}) \bigg|_0^\pi =\frac{\pi}{4} - 0 - 0 + 0 - \frac{2}{\pi^2} (\frac{\pi^3}{3}) + 0 =\frac{\pi}{4} - \frac{2\pi}{3} =-\frac{\pi}{12}$$[/tex]

Therefore, by integral the answer will be [tex]$-\frac{\pi}{12}$[/tex].

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There are 55 senators and 235 representatives are in this state's legislature.

An equation is a mathematical statement that shows the equality of two expressions. Equations can be used to represent relationships between variables and to solve problems. They are written using an equal sign (=) and can include numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division.

For example, the equation 2x + 3 = 7 is an equation that states that the expression on the left side (2x + 3) is equal to the expression on the right side (7). The goal is to find the value of the variable x that satisfies the equation. In this case, x = 2 is a solution to the equation.

Let's call the number of senators in the state legislature "s" and the number of representatives "r". We know the following:

s + r = 290 (the combined number of senators and representatives is 290)

r - s = 180 (the difference of the numbers is 180)

We can use the first equation to solve for one of the variables in terms of the other:

r = 290 - s

Next, we can substitute this expression for r into the second equation:

290 - s - s = 180

Simplifying the left-hand side:

290 - 2s = 180

Adding 2s to both sides:

290 = 180 + 2s

Subtracting 180 from both sides:

110 = 2s

Dividing both sides by 2:

s = 55

So there are 55 senators in the state legislature. To find the number of representatives, we can use the first equation:

r = 290 - s = 290 - 55 = 235

So there are 235 representatives in the state legislature.

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The area of the rectangle is 72 cm^2.

Let's call the side length of each square s.

Since the perimeter of each square is 24 cm, then each side measures 24 cm / 4 = 6 cm.

So the length of the rectangle is 2s = 2 * 6 cm = 12 cm.

And the width of the rectangle is s = 6 cm.

Therefore, the area of the rectangle is A = l * w = 12 cm * 6 cm = 72 cm^2.

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The area of the rectangle is 72 square centimeters.

The rectangle is formed by placing two identical squares side by side the perimeter of each square is 24 cm.

Remember that for a square of sidelength S, the perimeter is P = 4*S

Then the sidelength of the squares is

24cm = 4*S

24cm/4 = 6cm = S

When we place these two squares together, we will have a rectangle of 6cm by 12cm, and the area is the product of the dimensions, then the area is:

A = 6cm*12cm = 72cm²

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The angle relations in a cyclic quadrilateral are discussed above.

Given is that ABCD is a cyclic quadrilateral.

A cyclic quadrilateral is a four sided shape that can be inscribed into a circle.

Each vertex of the quadrilateral lies on the circumference of the circle and is connected by four chords.

The opposite angles of a cyclic quadrilateral have a total of 180°.

For a cyclic quadrilateral with successive sides a, b, c, d, semi - perimeter s, and angle A between sides a and d, the angle relations for a cyclic quadrilateral as -

        [tex]$\tan {\frac {\theta }{2}}={\sqrt {\frac {(s-b)(s-d)}{(s-a)(s-c)}}}.[/tex]

       [tex]${\displaystyle \cos {\frac {\varphi }{2}}={\sqrt {\frac {(s-b)(s-d)(b+d)^{2}}{(ab+cd)(ad+bc)}}}}[/tex]

Therefore, the angle relations in a cyclic quadrilateral are discussed above.

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The equation to see how far Zach travels is y = 21x and Zach traveled 336 miles using 16 gallons of gas.

Number of miles traveled Zach's car traveled= 21 miles

Quantity of gallons used = 1 gallon

y = kx

Where,

x = total gallons used

k = unit rate of miles per gallon

y = Number of miles traveled

So,

21 = k × 1

21 = k

y = 21x

Hence,

if he uses 16 gallons of gas

y = kx

y = 21 × 16

y = 336 miles

In conclusion, Zach's car traveled 336 miles.

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The probability that a randomly selected female high school basketball player will be drafted by a WNBA team is 0.000351 or approximately 0.04%.

The probability of a female high school basketball player being drafted into the WNBA can be taken as a combination of two probabilities.

The probability that she will play basketball in the NCAA, which is equal to 3.9%, and the probability that she will be drafted into the WNBA given that she plays in the NCAA, which is equal to 0.9%, are both equal.

The probability can be calculated by using the formula:

P(WNBA draft) = P(NCAA) × P(WNBA draft | NCAA)

Where, P(NCAA) = 3.9 ÷ 100 which is equal to 0.039
and, P(WNBA draft | NCAA) = 0.9 ÷ 100 which is equal to 0.009

Hence, by substituting the values in the equation we get

P(WNBA draft) = 0.039 × 0.009

Which is equal to 0.000351 or approximately 0.04%

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Answer:

Below

Step-by-step explanation:

See diagram below   solve the angles in order 1-2-3-4-5-6

as shown in the diagram:

Answer:

85 degrees

Step-by-step explanation:

Okay, let's have a look at the triangle on the left.
We can see an exterior angle of 115 degrees, adjacent to the interior angle.
Thus, as they both form a line (180 degrees), Interior angle = 180 - 115
= 65 degrees
We know that all the angles of the triangle equal 180.
Now to find the remaining angle of the triangle on the left,
180 - (85 + 65) = 30 degrees

Moving towards the center, we have our interior triangle angle (30 degree), a 90 degree angle and an interior angle of the triangle on the right. So, as they all form a line, the interior angle of the triangle on the right will be
180 - (90 + 30) = 60 degrees

Also, on top of the triangle on the right, we have a 115 degree exterior angle again.
Repeating our first step, the interior angle would be 65 degree

Now to find the remaining angle of the triangle on the right,
180 - (30 + 65) = 95 degrees

This angle forms a line with another angle exterior to this one, so again,
180 - 95 = 85 degrees

Hope this helps

An equation that represents the Kim's and her sister's age combined is,

x + k = n.

Two algebraic expressions having same value and symbol '=' in between are called as an equation.

Let k represents Kim's age, and x represents her sister's age.

Let the combined age is n.

Then,

an equation that represents the Kim's and her sister's age combined is,

x + k = n.

Therefore, the equation is x + k = n.

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To convert cups to pounds, you can use the following conversion factor   1 cup = 0.5 pounds

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The Fahrenheit value is 97.52

How To Convert 36.4 °C to °F?

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The number of students that answered Malachi's question is 20 students.

A histogram is a graphical representation of data that shows the frequency distribution of a set of continuous or discrete data. It is a bar graph that displays the number of data points within specified ranges or bins, allowing for the observation of patterns, trends, and anomalies in the data.

The height of each bar in the histogram represents the frequency of data within the corresponding bin or range, providing a visual representation of the distribution of the data.

In this case, we have that the number of students that answered the question are;

1 + 4 + 10 + 4 + 1 = 20 students

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Answer:

20

Step-by-step explanation:

i got it correcttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttt

Answer:

Step-by-step explanation: just / the $7.04 by 8 since there is 8 bars then boom

Answer:

B. y = -3x -1

Step-by-step explanation:

Is this one because of the part of maths that studies the way in which words and the groups they form are combined to express meanings, as well as the relationships that are established between all these units. A set of rules defining the correct sequences of the elements of a programming science and maths formulas.

There are 640 students are expected to choose chocolate as their favorite ice cream.

How many students are expected to choose chocolate as their favorite ice cream flavor if there are 700 students in the school?

Total number of student =700

no of students are expected to choose chocolate as their favorite ice cream be x

school are randomly selected and asked about their favorite ice cream flavor =60

60+x=700

x=700-60=640

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The total number of arrangements possible if A and B must not sit together:

=  3600

Now, According to the question:

The no. of circular arrangements of n distinct items = (n-1)!

If A,B,C,D,E,F,G and H are the 8 persons to be seated around the table and A,B the 2

particular persons who must not sit together.

The total no.of circular arrangements for 8 persons = (8-1)! = 7! = 5040 --(1)

The no. of circular arrangements possible if A and B were to sit together = 6! (considering A and B as a single pair).

But, A and B can interchange their positions.

Hence, the no. of arrangements =6!×2=720×2=1440−−−−>(2)

So, the total number of arrangements possible if A and B must not sit together, = (1) - (2)

= 5040-1440

= 3600

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The weight of a hamburger can be considered a continuous random variable.

A random variable is a mathematical representation of a random phenomenon.

In probability and statistics, the weight of a hamburger can be used to illustrate the concepts of discrete and continuous random variables.

A discrete random variable is one that can only take on specific, distinct values.

For example, the number of hamburgers sold at a fast food restaurant in one day could be considered a discrete random variable, as the number of hamburgers sold can only be a whole number.

On the other hand, a continuous random variable is one that can take on an infinite number of values within a given range.

The weight of a hamburger could be considered a continuous random variable, as the weight of a hamburger can take on any value within a certain range, such as between 0.1 and 0.5 kilograms.

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The Sun's diameter is 864,000 miles. 864,000 x 108 = 93,000,000 kilometers separate the earth from the sun. This is equivalent to 109 times the size of Earth.

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Option is not mentioned, The ratio of the diameter of the earth to the sun is di = 1.8 × 10-3m.

The Sun is 1,391,000 kilometres (864,400 miles) across. It is the same as Earth being 109 times its size. The Sun's mass is roughly 333,000 times that of the Earth.

The Sun's diameter is equal to the diameter of 108 Earths. The Sun is 864,000 kilometres across. The earth is 93,000,000 kilometres from the sun, or 864,000 times 108 kilometres. It is the same as Earth being 109 times its size.

The Sun is 864,000 kilometres across. The earth is 93,000,000 kilometres from the sun, or 864,000 times 108 kilometres. It is the same as Earth being 109 times its size.

The Sun is 1,391,000 kilometres (864,400 miles) across. This is the same as 109 times the size of Earth. .

The angle of reflection is θ.

D is the distance between the mirror's centre and the sun's centre, where d is the sun's diameter. d/D the equation is equal to 0.009

Angle of incidence = angle of reflection according to the rules of reflection. Consequently, θ2 = θ, = θ, d D The ratio tan θ;=0.009 tan θ =0.009 is another option.

We will assume that because the sun is far away from the mirror, the image will form at the mirror's focus. Therefore, distance OB = f, where f is the concave mirror's focal length

The focal length of a spherical concave mirror is equal to half of its radius of curvature. Given that the radius of the curveturer is 0.4 m, f= 42 = 0.4 m 2 = 0.2 m.

Taking into account the image's diameter as d2, tan 0, di f tan 0 di 0.2 = Taking the value of tan from above, 0.009 di 0.2 = di 0.009 x 0.2 = As a result, the image's diameter is d; = 0.0018m di = 1.8 × 10-3m.

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The coach made a mistake in converting the average rate from miles per hour to feet per second. To convert miles per hour to feet per second, the formula is (miles per hour) x (5280 feet/mile) / (60 minutes/hour) = (feet per second).

However, the coach used a different conversion factor, which resulted in an incorrect value. This incorrect value led the coach to conclude that Aliza was not running fast enough, when in reality, the actual rate could be higher than 8.2 feet per second.

Therefore, the coach made an error in the conversion factor used and in the conclusion based on the incorrect converted value.

Rounding off to the nearest whole number, the value of the investment at maturity would be $15,108.

What is the whole number?

A whole number is a positive integer that has no fractional or decimal component. Whole numbers include 0, 1, 2, 3, 4, and so on.

The formula to calculate the future value of an investment is given by: y = a (1 + r)t, where y is the future value, a is the initial investment, r is the rate of return as a decimal, and t is the time in years.

In this case, the initial investment is $7,000, the rate of return is 10% (or 0.10 as a decimal), and the time in years is 14. So, the calculation would be:

y = $7,000 * (1 + 0.10)^14

y = $7,000 * 1.10^14

y = $7,000 * 2.151373

y = $15,108.16

Rounding off to the nearest whole number, the value of the investment at maturity would be $15,108.

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The two equations are y = - 1 and y = (1/5)x - 12/5 and the solution to the system is (7, - 1).

We know lines have various types of equations, the general type is

Ax + By + c = 0, and the equation of a line in slope-intercept form is

y = mx + b.

Where slope = m and b = y-intercept.

the slope is the rate of change of the y-axis with respect to the x-axis and the y-intercept is the (0,b) where the line intersects the y-axis at x = 0.

The first line passes through (- 1, 0) and (2, - 1).

Slope(m) = (- 1 + 1)/(2 - 0).

Slope(m) = 0.

- 1 = 0(2) + b.

b = - 1.

y = - 1.

A horizontal line at y = - 1.

The second line passes through (4, - 2) and (- 3, - 3).

Slope(m) = (- 3 + 2)/(- 3 - 4).

Slope(m) = - 1/- 5.

Slope(m) = 1/5.

- 3 = (1/5)(- 3) + b.

b = 3/5 - 3.

b = (3 - 15)/5.

b = - 12/5.

y = (1/5)x - 12/5.

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the integral value of = ∫ √3 9 arctan( 1 x) dx is: 9 {√3x/6 - π/4 + 1/2 ln 2}

Integration in mathematics is a technique for integrating or adding up the parts to get the total. By breaking down the functions into their component pieces, differentiation is carried out in reverse. The summation under a broad scale is found using this approach. The process of combining smaller parts or information from various subsystems into a single operational unit is known as integration. By breaking down the functions into their component pieces, differentiation is carried out in reverse. The summation under a broad scale is found using this approach.

Given that,

= [tex]\int\limits^{\sqrt{3}}_1{9arc\tan(1/x)} \, dx[/tex]

= [tex]9\int\limits^{\sqrt{3}}_1{tan^{-1}(1/x)} \, dx[/tex]

= [tex]9\int\limits^{\sqrt{3}}_1{cot^{-1}(x)} \, dx[/tex]

[As here we use tan⁻¹(1/x) = cot⁻¹x]

We integrate cot⁻¹x by parts by taking cot⁻¹x as I function.

= [tex]9{[cot^{-1}(x)\int\limits\, dx]{^\sqrt{3}} _1 - \int\limits^\sqrt{3}} _1 {d/dx (cot^{-1}x \int\limits\, dx }[/tex]

= 9 {√3x/6 - π/4 + 1/2 ln 2}

Thus, the integral value of = ∫ √3 9 arctan( 1 x) dx is: 9 {√3x/6 - π/4 + 1/2 ln 2}.

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We can see that p-q = 1, which implies that p and q are directly related.

(a) If x and y are inversely prοpοrtiοnal, then we knοw that xy = k fοr sοme cοnstant k.

If [tex]x^p[/tex] and [tex]y^q[/tex] are alsο inversely prοpοrtiοnal, then we can write [tex](x^p)(y^q) = k'[/tex] fοr sοme cοnstant k'. Using the inverse prοpοrtiοnality relatiοn, we get:

[tex](x^p)(y^q) = k'[/tex]

[tex](xy)^pq = k'[/tex]

[tex](k)^pq = k'[/tex]

[tex]k^(pq-1) = k'[/tex]

Frοm this equatiοn, we can see that pq-1 = 0, which implies that pq = 1. Therefοre, p and q are inversely related.

(b) If x and y are inversely prοpοrtiοnal, then we knοw that xy = k fοr sοme cοnstant k.

If [tex]x^p[/tex] and [tex]y^q[/tex] are directly prοpοrtiοnal, then we can write [tex](x^p) = k'(y^q)[/tex] fοr sοme cοnstant k'. Using the direct prοpοrtiοnality relatiοn, we get:

[tex](x^p) = k'(y^q)[/tex]

[tex](xy)^p = k'(y^q)(k)[/tex]

[tex]k^p = k'(y^{(q-1)})[/tex]

[tex]k^{(p-q)} = k'[/tex]

Frοm this equatiοn, we can see that p-q = 1, which implies that p and q are directly related.

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The two column proof showing that m∠AKG = m∠HKB is as below and explained.

A two-column proof is defined as a geometric proof consists of a list of statements, and the reasons that we know those statements are true.

The two column proof showing the vertical angles are explained below.

Statement 1: Segment GH Intersects Segment AB at K

Reason 1: Given

Statement 2: m∠AKG + m∠GKB = 180°

m∠GKB + m∠HKB = 180°

Reason 2:  Definition of supplementary angles

Statement 3: m∠AKG + m∠GKB = m∠GKB + m∠HKB

Reason 3: Substitution property

Statement 4: m∠AKG = m∠HKB

Reason 4: Subtraction property

Now, supplementary angles are defined as angles that sum up to 180 degrees and that is why m∠AKG + m∠GKB = 180° and m∠GKB + m∠HKB = 180°

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