Ken exercise at a park first he run 3 miles in 30 minutes.Then he rests for 15 minutes.Then he runs 2 miles in 15 minutes.Finally he walks 1 mile in 15 minutes.Assume each running and walking segment was done at a constant rate.skecth a graph to represent ken's exercing at the park.the fisrt segment should begin at (0,0).
Answers
The horizontal axis represents the time in hours, and the vertical axis represents the number of miles ran.
So for the first part we have 3 miles in 30 minutes (0.5 hour), so we can draw a line from point (0, 0) to point (0.5, 3)
For the second part, he rests for 15 minutes (0.25 hour), so the y-value doesn't change, therefore we have a horizontal line from (0.5, 3) to (0.75, 3).
Then, we have 2 miles in 15 minutes, so we have a line from (0.75, 3) to (1, 5).
And finally, 1 mile in 15 minutes, so a line from (1, 5) to (1.25, 6).
Sketching the graph, we have:
Related Questions
Factories 6ax2-3ay-8bx2+4by
Answers
=[2[tex]x^{2}[/tex]-y][3a-4b] This is the factorized form of 6a[tex]x^{2}[/tex]-3ay - 8 b[tex]x^{2}[/tex] + 4by
What is factorization ?To factorize a number, use the factorization formula. Factorization is the process of dividing a whole into components that, when multiplied together, equal the original number. The factorization approach simplifies any algebraic or quadratic equation by using the basic factorization formula, which represents the equations as the product of factors rather than expanding the brackets. Any equation may have an integer, a variable, or the algebraic expression itself as a factor.
6a[tex]x^{2}[/tex]-3ay - 8 b[tex]x^{2}[/tex] + 4by
Rearrange the expression
6a[tex]x^{2}[/tex] -8bx² -3ay +4by
2[tex]x^{2}[/tex][3a-4b] - y[3a-4b]
=[2[tex]x^{2}[/tex]-y][3a-4b]
This is the factorized form of 6a[tex]x^{2}[/tex]-3ay - 8 b[tex]x^{2}[/tex] + 4by
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Question: how do I figure out if a number is rational or irrational
Answers
A number is said to be rational if it can be written or transformed into the p/q form, where p and q are integers and q is a non-zero number; if it cannot be written in this form, it is said to be irrational.
Any number that can be written or stated in the p/q form, where p and q are integers and q is a non-zero number, is said to be rational.
A number that is irrational, on the other hand, cannot be stated in p/q form, and its decimal expansion is non-repeating and non-terminating.
Example: √2, √7, √11
Hence we get the required answer.
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You are about to enter an elevator with a weight capacity of 500 pounds. There are three people already in the elevator. The weights of the passengers are 107 pounds, 172 pounds and 129 pounds. Assuming that you weigh 136 pounds, is it safe for you to enter the elevator?
Answers
Given
capacity of 500 pounds.
weights of the passengers are 107 pounds, 172 pounds and 129 pounds.
you weigh 136 pounds
Find
is it safe for you to enter the elevator?
Explanation
as total pounds in the elevator = 107 + 172 + 129 = 408 pounds
total pounds included my weight = 408 + 136 = 544 pounds
capacity = 500 pounds
so , extra pounds = 544 - 500 = 44 pounds
so , no it is not safe for you to enter the elevator.
Final Answer
Therefore , No the weigh capacity would be exceeded by 44 pounds
find the lateral area and surface area the lateral area of the prism is __in squared. the surface area of the prism is __ in squared
Answers
Answer
The lateral area of the prism is 900 in squared
The surface area of the prism is 960 in squared
Explanation
Given:
The first side of the triangular base, a = 12 in
The second side of the triangular base, b = 13 in
The height of the prism, h = 30 in
What to find:
The lateral area and surface area of the prism.
Step-by-step solution:
The first step is to find the third side, c of the triangular base using Pythagoras rule.
[tex]\begin{gathered} b^2=c^2+a^2 \\ \\ 13^2=c^2+12^2 \\ \\ c^2=13^2-12^2 \\ \\ c^2=169-144 \\ \\ c^2=25 \\ \\ c=\sqrt{25} \\ \\ c=5\text{ }in \end{gathered}[/tex]Now, the next step is to calculate the lateral area of the prism using the formula below.
[tex]\begin{gathered} L.A=ha+hb+hc \\ \\ L.A=30\times12+30\times13+30\times5 \\ \\ L.A=360+390+150 \\ \\ L.A=900\text{ }in^2 \end{gathered}[/tex]The lateral area of the prism is 900 in squared
The final step is to calculate the surface area of the prism using the formula below.
[tex]S.A=Lateral\text{ }Area+Base\text{ }Area[/tex]The base area is
[tex]=2(\frac{1}{2}cb)=2(\frac{1}{2}\times5\times12)=2(\frac{60}{2})=60\text{ }in^2[/tex]Therefore, the Surface Area = (900 + 60) = 960 in squared
As part of a charity drive, 50 fourth grade students each donated an equal number of dimes and 30 third grade students each donated the same number of nickels. Let d represent the number of dimes and n represent the number of nickels. Determine which of the expressions shows the total amount (in cents) donated by the 3rd and 4th grade students. A.) 10(50d)+5(30n)B.) 80cC.) 10d+5nD.) 50d+30n
Answers
find the value of x and y that make the quadrilateral
Answers
The quadrilateral have equal opposite sides, for example:
Then, (4x + 6) must be equal to (7x - 3), and (4y - 3) is equal to (3y + 1), therefore
[tex]4x+6=7x-3[/tex]We can solve that equation for x
[tex]\begin{gathered} 4x+6=7x-3 \\ \\ 3x=9 \\ \\ x=\frac{9}{3}=3 \end{gathered}[/tex]Hence, x = 3. Now let's solve the other equation
[tex]\begin{gathered} 4y-3=3y+1 \\ \\ y=4 \end{gathered}[/tex]Then the value of y is 4.
Final answers:
x = 3
y = 4
Thursday
Each 4th grade class at Garza Elementary collected plastic water bottles to recycle.
Each number on the stem and leaf plot represents how many bottles each class
brought in. Record each answer in the blank box next to each question.
Help fill the table please.
1. how many fourth grade classes brought in between 90 and 100 bottles
2. How many classes brought in more than 100 water bottles
3. how many classes brought in less than 99 water bottles
4. How many fourth grade glasses are represented on the stem and leaf plot
Answers
The number of students that collected between 90 and 100 bottles is 3.
The number of students that collected more than 100 bottles are 7.
The number of students that collected less than 90 students are 2
The total number of classes represented is 10.
Here is the completed frequency table
Number of bottles collected Classes
90 - 100 3
101 - 110 6
111 - 120 1
What is a stem and leaf plot?A stem-and-leaf plot is a table that splits a number into a stem and leaf. The stem is the first number in the digit while the leaf is the last number. For example the stem in 45 is 4 and the leaf is 5.
An advantage of the stem-and-leaf plot is that it provides an easy way to present and interpret a dataset. A disadvantage of the stem-and-leaf plot is that it cannot be used for a large dataset.
Bottles collected that are between 90 and 100 are 94, 98, 99
Bottles that are greater than 100 are 105, 107, 107, 113, 114, 119, 120
Bottles that are less than 99 are 94, 98
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Given that f(x) = x sqaured - 10x + 16 and g(x) = x - 8, find f (x) · g(x) andexpress the result in standard form.
Answers
Given the functions
[tex]f(x)=x^2-10x+16[/tex][tex]g(x)=x-8[/tex]You have to calculate
[tex]f(x)\cdot g(x)[/tex][tex](x^2-10x+16)\cdot(x-8)[/tex]To solve this you have to apply the distributive property of multiplications, that is, you have to multiply each term of the first parenthesis with each term of the second parenthesis.
As follows:
[tex]\begin{gathered} (x^2-10x+16)\cdot(x-8) \\ (x^2\cdot x)+(x^2)\cdot(-8)+(-10x\cdot x)+(-10x)\cdot(-8)+(16\cdot x)+(16\cdot(-8)) \\ x^3-8x^2-10x^2+80x+16x-128 \end{gathered}[/tex]Now simplify the like terms
[tex]\begin{gathered} x^3+(-8x^2-10x^2)+(80x+16x)-128 \\ x^3-18x^2+96x-128 \end{gathered}[/tex]The result in standard form is
[tex]f(x)\cdot g(x)=x^3-18x^2+96x-128[/tex]
(-8.8)(-9) =
(0.5)(-0.5)=
(-0.2)(-5)
Answers
Answer:
(-8.8)(-9)= 79.2
(0.5)(-0.5)= -0.25
(-0.2)(-5)= 1
Find a_10 5, 12, 19, 26, 33...
Answers
Given,
The progression is 5, 12, 19, 26, 33...
The first term of the series is, a = 5.
The common difference of the series is,
d = 12 - 5 = 7
The 10th term of the series is,
[tex]\begin{gathered} a_{10}=a+(10-1)\times d \\ =5+9\times7 \\ =5+63 \\ =68 \end{gathered}[/tex]Hence, the 10th term of the series is 68.
i just want the answers, i dont want an explanation.
Answers
Given:
Required: Evaluation
Explanation:
Use BODMAS rule to evaluate all
71.
[tex]\begin{gathered} 16+4(-3)-7=16-12-7 \\ =-3 \end{gathered}[/tex]72.
[tex]\begin{gathered} (-24)\div(4)+16(2)=-6+32 \\ =26 \end{gathered}[/tex]73.
[tex]\begin{gathered} -5-8+6(-3)=-13-18 \\ =-31 \end{gathered}[/tex]74.
[tex]\begin{gathered} 30-6\div3+4=30-2+4 \\ =32 \end{gathered}[/tex]75.
[tex]\begin{gathered} 21-45+8\div2=21-45+4 \\ =25-45=-20 \end{gathered}[/tex]Final Answer:
71 - E
72 - AE
73 - A
74 - BE
75 - B
Select the correct answer from each drop-down menu.
A triangle FGH, right angle at H is shown. Base GH has length labeled 3 units. Height FH has length labeled 4 units, and hypotenuse FG has length 5 units.
Use the figure to complete the following statements.
sin(F) =
cos(F) =
sin(G) =
cos(G) =
Answers
The value of given expressions will be sin(F) = 3/5, cos(F) = 4/5, sin(G) = 4/5 and cos(G) = 3/5.
A triangle may be defined as a closed figure having three sides and three angle, the sum of which is equal to 180°. A right-angled triangle is the one which has one angle as 90°. We consider a right-angled triangle FGH which is right-angled at H. The sine of any angle is given by the division of Perpendicular to its Hypotenuse. The cosine of any angle is given by the division of Base to its Hypotenuse. Now, we need to find
sin(F) = Perpendicular/Hypotenuse, for angle at F perpendicular will be GH and Hypotenuse will be FG. So, sin(F) = 3/5
cos(F) = Base/Hypotenuse, for angle at F Base will be FH and Hypotenuse will be FG. So, cos(F) = 4/5
sin(G) = Perpendicular/Hypotenuse, for angle at G perpendicular will be FH and Hypotenuse will be FG. So, sin(F) = 4/5
cos(G) = Base/Hypotenuse, for angle at G Base will be GH and Hypotenuse will be FG. So, cos(F) = 3/5
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Which of the following statements best describe the branches of the hyperbola?Select one:a.It is not possible to determine how the branches of the hyperbola open.b.The branches of the hyperbola are positioned at a 30⁰ angle.c.The branches of the hyperbola open up and down.d.The branches of the hyperbola open side to side.
Answers
Answer:
Explanation:
The first step is to plot the graph of the parabola. It is shown below
Looking at the graph, the branches open up and down. Thus, the correct option is
c. The branches of the hyperbola open up and down.
Does the graph represent a function? 4 5 6 7 -3+ no O yes
Answers
To prove:
The given graph is a function.
The given graph is a function if and only if no x value has more than one value of y, or we can say that a graph is a function iff no vertical line intersects the graph in more than one point.
Thus, by the given graph it is clear that for x = 7 their are two values for y that is y = -6 , -7
So, the given graph is not a function
The average amount of time spent grocery shopping is 45 minutes with a standard deviation of 12 minutes. Using the Empirical Rule, what percent should spend less than 33 minutes?
Answers
You have:
mean = 45
standard deviation = 12
Use the following formula:
If P is the centroid of the triangle JKL, JK = 22, KN = 13, and OL = 18, find each measure.
Answers
Given
JK = 22
KN = 13
OL = 18
Procedure
Centroid formulation
KM = JK/2 = 22/2 = 11
KM = 11
NL = KN
NL = 13
KL = 2KN = 2(13) = 26
KL = 26
JO = OL = 18
JO = 18
JL = 2OL = 2(18)
JL = 36
For j(x) = 3x − 1, find j of the quantity x plus h end quantity minus j of x all over h period a 3 to the power of the quantity x minus 1 end quantity times the quantity 3 to the power of h end quantity all over h b 3 to the power of the quantity x minus 1 end quantity times the quantity 3 to the power of h minus 1 end quantity all over h c 3 to the power of the quantity x minus 1 end quantity times the quantity 3 to the power of h plus 1 end quantity all over h d the quantity x minus 1 end quantity times the quantity 3 to the power of h plus 1 end quantity all over h
Answers
The resulting value of the function [j(x+h)-j(x)/h] is 3⁽ˣ⁻¹⁾ ([tex]3^{h}[/tex]-1)/h
What is an exponential function?An exponential function is defined as a function whose value is a constant raised to the power of an argument is called an exponential function.
It is a relation of the form y = aˣ in mathematics, where x is the independent variable
Given the function expressed as;
j(x) = 3⁽ˣ⁻¹⁾
Required value of the function [j(x+h)-j(x)]/h
We have to determine the function j(x+h) and j(x)
j(x+h) = [tex]3^{(x+h) -1}[/tex]
Substitute the values,
[j(x+h)-j(x)]/h = [ [tex]3^{(x+h) -1}[/tex] - 3⁽ˣ⁻¹⁾ ]/h
⇒ [ [tex]3^{(x-1) +h}[/tex] - 3⁽ˣ⁻¹⁾ ]/h
⇒ [ [tex]3^{(x-1)} \times3^{h}[/tex] - 3⁽ˣ⁻¹⁾ ]/h
⇒ 3⁽ˣ⁻¹⁾ ([tex]3^{h}[/tex]-1)/h
Hence, the equivalent value of the function is 3⁽ˣ⁻¹⁾ ([tex]3^{h}[/tex]-1)/h
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Melissa brought 6 apples for $1.20 .if each apple cost the same amount, how much would 20 apples cost
Answers
The cost of 20 apples is such that the ratio of apples to cost maintained or each apple costs the same amount is $4.
What are the ratio and proportion?The ratio is the division of the two numbers.
For example, a/b, where a will be the numerator and b will be the denominator.
As per the given,
Melissa brought 6 apples for $1.20.
The ratio of cost to apple ⇒
1.20/6
Now let's suppose the cost of 20 apples is x dollars.
The ratio of cost to apple ⇒
x/20
Since the cost of each apple is the same, therefore, both ratios must be the same.
x/20 = 1.20/6
x = 20(1.20/6)
x = 4
Hence "The cost of 20 apples is such that the ratio of apples to cost maintained or each apple costs the same amount is $4".
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2. A new car dealership invites 500 people to a promotional event. Only 330 people attend the event. What percentage of the people attend the event?
Answers
since
[tex]550X=330[/tex]then
[tex]X=\frac{330}{500}=0.66[/tex]then it is the 100x0.66= 66%
answers pleaseeee
im struggling
Answers
Answer:
y=3/4x-2
Step-by-step explanation:
slope=3/4x
y-intercept=-2
Find the standard form for the equation of a circle ( x − h )^2 + ( y − k )^2 = r^2 with a diameter that has endpoints ( − 4 , − 8 ) and ( 6 , − 4 ) .h=k=r=
Answers
Answer:
[tex](x-1)^2+(y+6)^2=29[/tex][tex]\begin{gathered} h=1 \\ k=-6 \\ r=\sqrt{29} \end{gathered}[/tex]Explanation:
Given:
Endpoints of the diameter of a circle as (-4, -8) and (6, -4)
To find:
Equation of a circle in standard form
The equation of a circle in standard form is generally given as;
[tex](x-h)^2+(y-k)^2=r^2[/tex]where (h, k) is the coordinate of the center of the circle and r is the radius of the circle.
We'll go ahead and determine the coordinates of the midpoint of the endpoints of the diameter which will be the coordinates of the center of the circle as seen below wi;
[tex]h=\frac{x_1+x_2}{2}=\frac{-4+6}{2}=\frac{2}{2}=1[/tex][tex]k=\frac{-8+(-4)}{2}=\frac{-12}{2}=-6[/tex]So the center of the circle has coordinates (1, -6)
Since h = 1, and k = -6 and we have that x1 = -4 and y1 = -8, we can go ahead and solve for r as seen below;
[tex]\begin{gathered} (-4-1)^2+(-8-(-6))^2=r^2 \\ (-5)^2+(-2)^2=r^2 \\ 25+4=r^2 \\ 29=r^2 \\ \therefore r=\sqrt{29} \end{gathered}[/tex]We can now write the equation of the circle in standard form as;
[tex]\begin{gathered} (x-1)^2+(y-(-6))^2=29 \\ (x-1)^2+(y+6))^2=29 \end{gathered}[/tex]Gravel is being dumped from a conveyor belt at a rate of 30 ft^3/min, and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 ft high?
Answers
The height of the pile increasing at the rate of 4.56 inches per minute when the pile is 10 ft high.
In given situation, the rate of change of height is equal to the rate of change of volume, divided by the base area.
First we find the base area.
base area = (π/4)d²
base area = (π/4)10²
base area = 25π ft²
base area ≈ 78.5 ft²
Then the rate of change of height would be,
(30 ft³/min)/(78.5 ft²)
≈ 0.38 ft/min
= 4.56 inches / minute
Therefore, the height of the pile increasing at the rate of 4.56 inches per minute when the pile is 10 ft high.
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it takes 126 cubes that have edge lengths of 1/3feet to completely fill this plastic bin what is the area of the base of bin
Answers
Answer: 1) Each of the 126 cubes has a volume = (1/3)3 = 1/3 × 1/3 × 1/3 = 1/27 cubic feet.
2) It takes 126 cubes to fill the bin, so the bin has a volume of 126 × 1/27 = 126/27 cubic feet. (Don't do this division yet - leave it as an improper fraction)
3) The volume of the bin is also given by the formula Volume = Area of Base × Height. Plus we know the volume is 126/27 cubic feet and the height is 2 1/3 = 7/3 of a foot, so:
Step-by-step explanation: Volume = Area of Base × Height
126/27 = Area of Base × 7/3
126/27 × 3/7 = Area of Base
The area of the base of bin is 2 feet².
What is Volume of Cube?The formula of volume of the cube is given by: Volume = a³, where a is the length of its sides or edges.
Given:
Edge = 1/3 feet
Volume of 1 cube= l³= (1/3)³ = 1/27
Totals cubes= 126
So, Volume of 126 cube= 126 x 1/27 = 126/ 27
Then, Volume of Plastic Bin= Area of base x height
126 / 27 = Area of base x 7/3
Area of base = 126 /27 x 3 /7
Area of base = 2 ft²
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22. A 250W carrier is to be modulated at an 85% modulation level. What is the total transmitted power? a. 340.3 W b. 25.32 dB c. 0.340 kW d. 55.32 dBm e. All of the above.
Answers
The total transmitted power is a. 340.3 W.
The transmitted and carrier power is related with modulation through the formula -
[tex] P_{t} = P_{c}(1 + \frac{ {m}^{2} }{2} )[/tex]
where [tex]P_{t}[/tex] is transmitted power, [tex]P_{t}[/tex] is carrier power and m is modulation.
Keep the values in formula to find the value of total transmitted power.
[tex]P_{t}[/tex] = 250 (1 + 85%²/2)
Taking square of percentage
[tex]P_{t}[/tex] = 250 (1 + 0.7225/2)
Performing division in the bracket
[tex]P_{t}[/tex] = 250 (1 + 0.36125)
Performing addition in the bracket
[tex]P_{t}[/tex] = 250×1.36125
Performing multiplication
[tex]P_{t}[/tex] = 340.3125
Thus, the total transmitted power when carrier power is 250 W and modulation is 85% is a. 340.3 W.
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how many cubic blocks with a side length of 1/6 cm are needed to fill the volume of this prism?The answer choices are 4,8,9,32.
Answers
Given the figure, we can deduce the following information:
Height=1/2 cm
Length=1/2 cm
Width= 1/6 cm
To determine the number of cubic blocks with a side length of 1/6 cm needed to fill the volume of the given prism, we first note that a cube has equal sides. So, the dimensions of the cube must be:
Height= 1/6 cm
Length=1/6 cm
Width=1/6 cm
Next, we get the volume of the cube by using the formula:
[tex]\begin{gathered} Volume\text{ of the cube}=(Height)(Length)(Width) \\ =(\frac{1}{6})(\frac{1}{6})(\frac{1}{6}) \\ Simplify \\ =\frac{1}{216}\text{ }cm^3\text{ } \end{gathered}[/tex]Then, we get the volume of the given prism using the same formula:
[tex]\begin{gathered} Volume\text{ of the given prism}=(He\imaginaryI ght)(Length)(W\imaginaryI dth) \\ =(\frac{1}{2})(\frac{1}{2})(\frac{1}{6}) \\ =\frac{1}{24}\text{ }cm^3\text{ } \end{gathered}[/tex]Now, we find the number cubic blocks by using the formula:
[tex]\begin{gathered} Number\text{ }of\text{ cubic blocks}=\frac{Volume\text{ of the given prism}}{Volume\text{ of the cube}} \\ =\frac{\frac{1}{24}}{\frac{1}{216}} \\ Simplify \\ =9 \end{gathered}[/tex]Therefore, the answer is 9.
A manufacturer of ski clothing makes ski pants and ski jackets. The profit on a pair of ski pants is $3 and on a jacket is $2. Both pants and jackets require the work of sewing operators and cutters. There are 60 minutes of sewing operator time and 48 minutes of cutter time available. It takes 8 minutes to sew one pair of ski pants and 4 minutes to sew one jacket. Cutters take 4 minutes on pants and 8 minutes on a jacket. The manufacturer wants to make a minimum of 4 ski pants and 2 ski jackets, Let x represent the number of ski pants. Let y represent the number of ski jackets.
Answers
For the sewing operator:
[tex]8x\text{ + 4y }\leq\text{ 60}[/tex]For cutter:
[tex]4x\text{ + 8y }\leq48[/tex]The two above equation is solve to obtain the graph
The solution of the graph is where the two lines meet
The coordinate is (10, 4), at the this point the maximum profit is made
So 10 ski pants and 4 ski jackets is made to get maximum profit
The profit made is given by
P = 3x + 2y
x =10 , y = 4
P = 3(10) + 2(4)
P = 30 + 8 = $38
A trendline is given to depict the enrollment at a local college where x is the year and y represents enrollment. y=178.09x−353194Use the trendline to estimate the enrollment for 2009. Round to the nearest whole number.
Answers
To determine the enrollment in 2009 we need to plug x=2009 in the equation given to determine the value of y:
[tex]\begin{gathered} y=178.09(2009)-353194 \\ y=357782.81-353194 \\ y=4588.81 \end{gathered}[/tex]Therefore, in the year 2009, there were 4589 students enrolled
6.4 The ratio of length and breadth of a rectagular field is 4:3. If perimeter of the field is 700m, find (i) length and breadth of the field (ii) area of the field.
Answers
Hope it helped !! :)
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Which of the following equations represents a line that is perpendicular toy=-3x+6 and passes through the point, (3, 2)?A. y=-3x+1B. y=(1/3x)+3C. y=(-1/3x)+1OD. y=(1/3x)+1
Answers
Step 1
For perpendicular lines
[tex]m_1=-\frac{1}{m_2}[/tex][tex]m_1=-3[/tex][tex]\begin{gathered} -3=-\frac{1}{m_2} \\ m_2=-\frac{1}{-3} \\ m_2=\frac{1}{3} \end{gathered}[/tex]Step 2
[tex]\begin{gathered} Given\text{ points \lparen3,2\rparen} \\ y=mx+b---\text{ Standard equation of a line} \\ y=\frac{1}{3}x+b \end{gathered}[/tex][tex]2=\frac{1}{3}(3)+b[/tex]Find b, the y-intercept
[tex]\begin{gathered} 2=1+b \\ b=2-1=1 \end{gathered}[/tex]Answer; Option D
[tex]y=\frac{1}{3}x+1[/tex]IXL Transversals of parallel lines: prove angle relationships 6QF for geometry, please help
Answers
4) [tex]m\angle XWY=m\angle HGY[/tex] (definition of congruent angles)
5) [tex]\angle HGY \cong \angle GTU[/tex] (corresponding angles theorem)
6) [tex]m\angle HGY=m\angle GTU[/tex] (definition of congruent angles)
7) [tex]\angle GTU[/tex] and [tex]\angle UTR[/tex] are supplementary (linear pair)
8) [tex]m\angle GTU+m\angle UTR=180^{\circ}[/tex] (definition of supplementary angles)
9) [tex]m\angle HGY+m\angle UTR=180^{\circ}[/tex] (substitution)
10) [tex]m\angle RTU+m\angle XWY=180^{\circ}[/tex] (substitution)