To find the formula for the nth term of a geometric sequence, we can use the formula:
a_n = a_1 * r^(n-1)
where a_1 is the first term of the sequence, r is the common ratio, and n is the position of the term.
In this case, the first term of the sequence is a_1 = 7 and the common ratio is r = (-21)/7 = -3. Plugging these values into the formula, we get:
a_n = 7 * (-3)^(n-1)
Therefore, the formula for the nth term of the geometric sequence is:
a_n = 7 * (-3)^(n-1)
Answer:
[tex]a_n=7\left(-3\right)^{n-1}[/tex]
Step-by-step explanation:
[tex]\boxed{\begin{minipage}{5.5 cm}\underline{Geometric sequence}\\\\$a_n=ar^{n-1}$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\\phantom{ww}$\bullet$ $r$ is the common ratio.\\\phantom{ww}$\bullet$ $a_n$ is the $n$th term.\\\phantom{ww}$\bullet$ $n$ is the position of the term.\\\end{minipage}}[/tex]
Given geometric sequence:
7, -21, 63, ...To find the common ratio, divide a term by the previous term:
[tex]\implies r=\dfrac{a_3}{a_2}=\dfrac{63}{-21}=-3[/tex]
Substitute the found common ratio and given first term into the formula to create an equation for the nth term:
[tex]a_n=7\left(-3\right)^{n-1}[/tex]
(Equivalent Algebraic Expressions MC)
25x¹⁰/y⁶ is equivalent to expression to (5x⁻⁵ y³)⁻².
Define expression.Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation. An expression, often known as a mathematical expression, is a finite collection of symbols that are well-formed in accordance with context-dependent principles. Any mathematical statement that includes numbers, variables, and an arithmetic operation between them is known as an expression or algebraic expression. In the phrase 4m + 5, for instance, the terms 4m and 5 are separated from the variable m by the arithmetic sign +.
Given
Expression
(5x⁻⁵ y³)⁻²
25x¹⁰ y⁻⁶
25x¹⁰/y⁶
25x¹⁰/y⁶ is equivalent to expression to (5x⁻⁵ y³)⁻².
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7=y/3
Solve and create new equation
Write the equation of the line that is perpendicular to the x-axis and passes through the point (6,-2).
The slope of a line that is perpendicular to the x-axis is undefined, so we cannot use the standard form of the equation of a line, y = mx + b, where m is the slope of the line. Instead, we can use the point-slope form of the equation of a line, which is given by y - y1 = m(x - x1). In this case, m is undefined, so we can simplify the equation to y = x1 - y1, where x1 and y1 are the coordinates of the point that the line passes through. In our case, this gives us the following equation:
y = 6 - (-2)
= 8
Therefore, the equation of the line that is perpendicular to the x-axis and passes through the point (6,-2) is y = 8.
Jacob is traveling down a level roadway with a speed of 9.5 m/s. He slams on the breaks and encounters a coefficient of friction of 0.847. What distance does his 985 kg truck skid before stopping?
To find the distance that Jacob's truck skids before coming to a stop, we can use the formula for the distance traveled while braking, which is:
distance = initial velocity^2 / (2 * coefficient of friction * acceleration due to gravity)
In this case, the initial velocity is 9.5 m/s, the coefficient of friction is 0.847, and the acceleration due to gravity is 9.8 m/s^2. Plugging these values into the formula, we get:
distance = (9.5 m/s)^2 / (2 * 0.847 * 9.8 m/s^2)
Solving for distance, we get:
distance = 8.76 meters
Therefore, Jacob's truck will skid a distance of 8.76 meters before coming to a stop.
Determine the inverse of the function f(x) = log2(3x + 4) − 5.
f inverse of x is equal to 2 to the power of x plus 3 all over 2
f inverse of x is equal to the quantity x plus 5 end quantity squared minus 4 all over 3
f inverse of x is equal to 2 to the power of the quantity x plus 5 end quantity minus 4 all over 3
f inverse of x is equal to 2 to the power of the quantity x minus 5 end quantity plus 4 all over 3
The inverse function of the function f(x) = log₂(3x + 4) − 5 is y = [2^(x + 5) - 4]/3
How to determine the inverse of the function?From the question, we have the following equation that can be used in our computation:
f(x) = log2(3x + 4) − 5.
Rewrite as
f(x) = log₂(3x + 4) − 5.
Express f(x) as y
This gives
y = log₂(3x + 4) − 5.
Swap the positions of x and y
So, we have the following representation
x = log₂(3y + 4) − 5.
Add 5 to both sides
log₂(3y + 4) = x + 5
So, we have
3y + 4 = 2^(x + 5)
Subtract 4 from both sides
3y = 2^(x + 5) - 4
Divide b y3
y = [2^(x + 5) - 4]/3
Hence, the inverse function is y = [2^(x + 5) - 4]/3
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part a: find the right reimann sum estimate of the integral from 1 to 5 of f of x dx, based on the subintervals given in the table. (4 points)
A right Riemann sum has the formula A=∑ni=1Δxf(xi) A = ∑ i = 1 n Δ x f ( x i ) where x is the width of each of the n rectangles and f(xi) is the height.
what is is Riemann sum?A specific method of approximating an integral with a finite sum in mathematics is known as a Riemann sum. Riemann is the name of a German mathematician who lived in the 19th century. Approximating the area of functions or lines in graphs, as well as the length of curves and other approximations, is a highly typical application. The picked points in each subinterval of the partition are all in the upper Riemann sum, given an interval partition. The interval is often separated into subintervals of equal size.
Riemann sum formula is A=∑f(xi)Δx A = ∑ f ( x i ) Δ x ,
where A is the curve's area under the evaluable interval,
f(xi) f ( x i ) is each rectangle's height (or the average of the two heights in the case of a trapezoid)
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f(x) = 3x^2 + 5
g(x) = 4x - 2
h(x) = x^2 - 3x + 1
Find f(x) + g(x) - h(x).
The value of the expression f(x) + g(x) - h(x) will be 2x² + 7x + 2. Then the correct option is B.
What is a function?A function is an assertion, concept, or principle that establishes an association between two variables. Functions may be found throughout mathematics and are essential for the development of significant links.
The functions are given below.
f(x) = 3x² + 5
g(x) = 4x - 2
h(x) = x² - 3x + 1
The expression is given as,
⇒ f(x) + g(x) - h(x)
⇒ (3x² + 5) + (4x - 2) - (x² - 3x + 1)
⇒ 3x² + 5 + 4x - 2 - x² + 3x - 1
⇒ 2x² + 7x + 2
The value of the expression f(x) + g(x) - h(x) will be 2x² + 7x + 2. Then the correct option is B.
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Assume that the distribution
A statistics instructor randomly selected four bags of oranges, each bag labeled 10 pounds, and
weighed the bags. They weighed 10.8, 10.7, 10.2, and 10.8 pounds.
of weights is Normal. Find a 95% confidence interval for the mean weight of all bags of oranges. Use
technology for your calculations. Answer parts a and b below.
a. Choose the correct interpretation of the confidence interval below and, if necessary, fill in the
answer boxes to complete your choice.
and. A.There is a 95% chance that all intervals will be between
B. We are 95% confident the population mean is between
C. We are 95% confident that the sample mean is between
D. The requirements for constructing a confidence interval are not satisfied.
(Type integers or decimals rounded to the nearest thousandth as needed. Use ascending order.)
95% confident the population mean is between (9.1,9.9).
We are given the following data set:
9.3, 9.7, 9.2, 9.7
What is Standard Deviation?
Standard Deviation [tex]$=\sqrt{\frac{\sum\left(x_i-\bar{x}\right)^2}{n-1}}$[/tex]
where[tex]$x_i$[/tex] are data points, [tex]$\bar{x}$[/tex]is the mean and [tex]$\mathrm{n}$[/tex] is the number of observations.
Mean [tex]$=\frac{\text { Sum of all observations }}{\text { Total number of observation }}$[/tex]
Mean [tex]$=\frac{37.9}{4}=9.475$[/tex]
Sum of squares of differences [tex]$=0.2075$[/tex]
S.D[tex]$=\sqrt{\frac{0.2075}{3}}=0.262$[/tex]
[tex]$95 \%$[/tex] Confidence interval:
[tex]$\bar{x} \pm t_{\text {critical }} \frac{s}{\sqrt{n}}$[/tex]
Putting the values, we get,
[tex]$t_{\text {critical }}$[/tex] at degree of freedom 3 and[tex]$\alpha_{0.05}=\pm 3.182$[/tex]
[tex]$9.475 \pm 3.182\left(\frac{0.262}{\sqrt{4}}\right)=9.475 \pm 0.416842=(9.058158,9.891842) \approx(9.1,9.9)$[/tex]
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The ratio of the number of model cars that Jim owns to the number of model cars Terrence owns is 8 : 6. Terrence owns 36 models cars. How many model cars does Jim own?
The number of Jim's model car is 48
How to determine the number of JIm's carFrom the question, we have the following parameters that can be used in our computation:
Ratio of the number of model cars that Jim owns to the number of model cars Terrence owns is 8 : 6
This means that
Jim : Terrence = 8 : 6
Also from the question, we have
Terrence owns 36 models cars
This means that
Terrence = 36
Substitute the known values in the above equation, so, we have the following representation
Jim : 36 = 8 : 6
Multiply by 6
Jim : 36 = 48 : 36
So, we have
Jim = 48
Hence, the number of car is 48
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a map has a scale of 1cm to 12km
on a map ,the distance between cardiff and london is 20cm
what is the real distance,in km, between cardiff and london
By using the given scale, we will see that the real distance is 240km
What is the real distance between Cardiff and London?We know that the scale of the map is 1 cm to 12km, so we just need to take the distance in centimeters on the map, and multiply that number by 12 km.
We know that on the map, the distance between Cardiff and London is 20cm, then the real distance is 12km times that, so we will get:
real distance = 20*12km = 240km
The real distance between Cardiff and London is 240 kilometers.
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Given m is parallel to n, find the value of x
Answer:
x = 44
Step-by-step explanation:
Given that line m is parallel to line n, the angles that we see are same-side exterior angles, meaning that these two angles add up to 180°.
Set up the equation: [tex]112 + (2x - 20) = 180[/tex] and solve for x.
. . . . . . . . . . . . 112 - 112 + 2x - 20 = 180 - 112
. . . . . . . . . . . . 2x - 20 + 20 = 68 +20
. . . . . . . . . . . . 2x /2 = 88 /2
. . . . . . . . . . . . ∴ x = 44
Which linear function represents the table?
X
4
-2
1
8
-3
y
26
-10
8
50
-16
y = 6x + 2
Oy=6x-2
y=-6x + 2
y=-6x-2
Answer:
A.) y=6x+2
Step-by-step explanation:
I got this answer by first finding the slope. To find the slope you use the slope formula which is y2-y1 over x2-x1. When I set up the equation I got -10-26 over -2-4 and I got -36/-6 which is 6. So your slope is 6. So then you would mark out answer c and d because their slope is negative and the slope I got is positive. To find the y-int I put in two points online and I put in the slope. I did this twice with four different points just to make sure that it was right and then it gave me that the y-int is 2. So then you would mark out b because the y-int is not negative but positive. (Hope this helps in some way...if not then I'm sorry..)
A spinner is divided into five equal sections labeled 1, 2, 3, 4, and 5. Another spinner is divided into two equal sections labeled A and B. Nigel will spin each spinner one time.
How many of the possible outcomes have a number less than 3 or an A?
Hint: first write out the sample space (you should have 10)
Which samples have a number less than 3 or an A?
9/10 is the possible outcome that has a number less than 3 or an A.
What is probability?
The proportion of possible outcomes to outcomes in an exhaustive set of equally likely alternatives that result in a certain event.
Here, we have
A spinner is divided into five equal sections labeled 1, 2, 3, 4, and 5. Another spinner is divided into two equal sections labeled A and B.
We have to find how many of the possible outcomes have a number less than 3 or an A.
Denote A the first event and B the second event:
P(A∪B) = P(A) + P(B)
A = Get a number less than 3
P(A) = 2/5
B = Get section A
P(B) = 1/2
Then,
P(A∪B) = 2/5 + 1/2
P(A∪B) = 9/10
Hence, 9/10 is the possible outcome that has a number less than 3 or an A.
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Rewrite 9x + 81 using a common factor.
9x(x + 81)
9x(x + 9)
9(x + 81)
9(x + 9)
Answer:
9 ( x + 9 )
Step-by-step explanation:
81 = 9 × 9
9x = 9 × x
9x + 81 = 9 ( x + 9 )
Socrates was alive from 469 to 399 B.C.E , Plato from 427 to 347 B.C.E ,Aristotle from 384 to 322 B.C.E. What years were they all alive simultaneously
(2x4-2x³ +7+ 8x²-x) / (x²+2)
The Alternative form of the ''(2x4-2x³ +7+ 8x²-x) / (x²+2)''
is 2[tex]x^{2}[/tex] - 2x + 4 + [tex]\frac{3x - 1}{x^{2} + 2}[/tex]
How to solve algebra equation ?One of the many branches of mathematics is algebra. Algebra, which is a common thread throughout practically all of mathematics, is broadly defined as the study of mathematical symbols and the rules for manipulating these symbols in formulae.One-step equations can be solved in four different ways: by adding, subtracting, multiplying, and dividing. In an equation, both sides will stay equal if we add the same amount to either side.The addition rule and the multiplication/division rule are the two methods for solving equations that are introduced to us in algebra 1. According to the equation addition rule, an equation's solution set can have the same amount added to both sides without altering.Find the attachment answer
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Divide 8.88 divided by 10 . Explain how you determine the placement of the decimal point in your quotient and how that changed the value of each digit
With the decimal point in the proper place, dividing decimals is equivalent to dividing entire numbers.
How to Divide Decimals?While dividing decimals, we must remember to place the decimal point correctly in the quotient. This is similar to how we divide whole numbers. The decimal point, which separates a decimal number's whole number and fractional parts, is a dot. A value less than 1 can be found in the digits following the decimal point. 24 is a good example. A decimal number, 15, has a fractional component of 15 and a whole number component of 24. There are two situations in which dividing decimal numbers might occur: the first calls for dividing decimals by whole numbers, and the second requires dividing decimals by decimals.To Learn more About dividing decimals refer To:
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about 4% of the population has a particular genetic mutation. 100 people are randomly selected. round your answer to three decimal places. (a) Find the mean for the number of people with the genetic mutation in such groups of 600. (b) Find the standard deviation for the number of people with the genetic mutation in such groups of 600.
We got the mean value 4.167 and standard deviation 19.56.
What is mean value and standard deviation in statistics?In statistics, a mean value is the average value of all the available data. And it can be determined by adding all the data by dividing the amount of data.
A standard deviation is an estimation that implies how far each available data apart from the mean value.
How do we calculate the mean value and standard deviation?we are given, 4% of the population has a genetic mutation
Total people in the group =600
Now, total number of people who have genetic mutation = (4/100) ×600
= 24
the number of randomly selected people =100= X
Mean = Number of selected people/ people who have mutation
Mean = X/n = 100/24 = 4.167
the standard deviation = √∑(X-mean) ²/n
= √∑(selected people - mean) ²/ mutation number
=√(100-4.167) ²/24
standard deviation = 19.56
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Please help me!!!!!!! You guys are my only hope!!! Correct answer gets brainliest
In Exercises 1-3, decide whether enough information is given to prove that the triangles are congruent using either the SSS Congruence Theorem (Theorem 5.8) or the HL Congruence Theorem (Theorem 5.9). Explain.
The congruency proofs that can be used for each of the triangles are;
1) SSS Congruency
2) None
3) HL Congruence
How to prove the triangle congruency postulate?
1) The SSS congruency postulate states that If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.
Now, we see that we are shown 2 corresponding sides that are congruent. However, the third side is automatically congruent using the reflexive property of congruence.
Thus, we conclude that the congruency can be proved using SSS Congruency postulate.
2) In this one, we are given 2 congruent sides and the included angle and so it is SAS Congruency and neither SSS nor HL congruency can be used.
3) The HL congruence Postulate states that if the hypotenuse and leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, then the two triangles are congruent.
In this case, we see that the hypotenuse is congruent to itself by reflexive property of congruence while the one leg is also congruent and so we can use HL Congruence Theorem
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can someone help? i don't understand cos at all
Cos W as a fraction in the simplest terms is 2/3
Trigonometric ratios in a Right angled triangle:Triangle side length ratios are known as trigonometric ratios. In trigonometry, these ratios show how the ratio of a right triangle's sides to each angle.
From a Right-angled triangle with an angle θ
sin θ = Opposite side/Hypotenuse
cos θ = Adjacent side/Hypotenuse
tan θ = Opposite side /Adjacent side
Here we have
A triangle UVW
where VW = 10 units
And WU = 15 units
As we know
cos θ = Adjacent side / Hypotenuse
cos W = VW/WU = 10/15
cos W = 2/3 [ divided by 5 ]
Therefore,
Cos W as a fraction in the simplest terms is 2/3
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Cai drove 189 miles in 7 hours. If he continued at the same rate, how far would he travel in 12 hours?
324 Miles
If you divide 189 by 7, you get 27. If you then multiply 27(miles per hour) by 12(hours) you get 324.
An elephant weighed 14,000 pounds. The handlers realized the
elephant was overweight and put it on a special meal plan. The
elephant lost 2,100 pounds. What was the percent of decrease in its
body weight?
14000
The value of the decrease percentage of elephant weight is 15%.
What is percentage?A percentage is a value that indicates 100th part of any quantity. A percentage can be converted into a fraction or a decimal by dividing it by 100.
The original weight of elephant is 14000 pounds.
And, the drop in weight is 2100 pounds.
Now, the percent value of the drop in weight is calculated as follows
Percent drop = Drop in weight / Original weight × 100
⇒ 2100/14000 × 100
⇒ 15%
Hence, the percent of decrease in weight is given as 15%.
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felipe graphs the linear relationship passing through the points 5,11 and -3,-9
slope -20/-8
line y = -5/2x - 3/2
it is NOT proportional because it doesnt pass thru 0,0
Shaina is 55 kg. What is Shaina's weight in grams
Answer: 55000 grams
Step-by-step explanation:
The function g is defined as follows for the domain given.
g(x)=2x-2, domain = {-3, -2, 1, 5)
Write the range of g using set notation. Then graph g.
Answer:
1. range = {-8,-6,0,8}
2. My graph is attached to this response as a photo.
Step-by-step explanation:
1. ⭐What is the "domain" of a function?
Domain is another word for the x-values⭐What is the "range" of a function?
Range is another word for the y-values⭐What is the "set notation"?
Set notation is a way you can list numbers, such as x-values and y-values. When writing in set notation, you must list the numbers from least to greatest.Therefore, we have to find the corresponding y-values for the given x-values, and then write said y-values in set notation.
To find the corresponding y-values for the given x-values, substitute each given x-value into g(x) and solve for g(x).
✦ When x = -3
[tex]g(x) = 2x-2[/tex]
[tex]g(-3) = 2(-3) -2[/tex]
[tex]g(-3) = -6 -2[/tex]
[tex]g(x) = -8[/tex]
✧ y = -8
________________________________________________________
✦ When x = -2
[tex]g(-2) = 2(-2) -2[/tex]
[tex]g(-2) = -4-2[/tex]
[tex]g(-2) = -6[/tex]
✧ y = -6
________________________________________________________
✦ When x = 1
[tex]g(1) = 2(1) -2[/tex]
[tex]g(1) = 2-2[/tex]
[tex]g(1) = 0[/tex]
✧ y = 0
________________________________________________________
✦ When x = 5
[tex]g(5) = 2(5) -2[/tex]
[tex]g(5) = 10 -2[/tex]
[tex]g(5) = 8[/tex]
✧ y = 8
________________________________________________________
Next, we have to write the y-values in set notation. Make sure to write it from least to greatest.
range = {-8, -6, 0, 8}
An amusement park charges an admission fee of 40 dollars per person. The cost, C (in dollars), of admission for a group of p people is given by the following.
C=40 p
What is the cost of admission for a group of 4 people?
_____dollars
The cost of admission for a group of 4 people is $160
How to determine the cost of admission for a group of 4 people?From the question, we have the following parameters that can be used in our computation:
C = 40 p
Where p is the number of people
For a group of 4 people, we have
p = 4
Substitute p = 4 in the equation C = 40 p
So, we have the following representation
C(4) = 40 * 4
Evaluate the products
C(4) = 160
hence, the cost is $160
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URGENT!! ILL GIVE BRAINLIEST!!!! AND 100 POINTS!!!
The length of diameter of the base of the cone will be 112 cm.
What is meant by an equation?A mathematical expression with an equals sign is referred to as an equation. Algebra is widely used in equations. When performing calculations but unsure of the precise amount, algebra is used.
A formula that expresses the connection between two expressions on each side of a sign. Typically, it has a single variable and an equal sign.
Equations are mathematical expressions with two algebras flanking the equal (=) sign on either side. This relationship is illustrated by the left and right expressions being equal to one another. Left-hand side equals right-hand side is a basic, straightforward equation.
By using Pythagoras's theorem,
106² = 90² + r²
simplifying the above equation, we get
r² = 106² - 90²
r = √3136
radius = 56 cm.
Then, the diameter will be,
Diameter = 2 × radius
substitute the values in the above equation,
D = 2(56)
D = 112 cm.
Therefore, the correct answer is option b. 112 cm.
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The following questions are about a spherical balloon that is being filled with air such that its radius is increasing at a constant rate of 2 cm/sec.
Part A:
How fast is the surface area increasing when the radius of the sphere is 10 cm? Round to the nearest thousandths.
Part B:
How fast is the volume increasing when the radius is 10 cm? Round to the nearest thousandths.
The surface area increases when the radius of the sphere is 160π cm^2/s
Given that dt/dr =10 cm/s and r=15 cm
Find out dt/ds=?
We know that the surface area of sphere S=4πr^2
dt/ds=4π(2r)
dt/dr=8π(2)(10)
=160π cm^2/s.
Let r, V, and S be respectively the radius, volume, and surface area of the spherical balloon at time t.
∴V=4/3 πr^3 and S=4πr^2
Rate of change of surface area with respect to t
⇒ dt/dS=2
⇒ d/dt=(4πr^2)=2
⇒8πr dt/dr =24
⇒ dr/dt=1/4πr.... (i)
Rate of change of volume of the balloon with respect to t
=dV/dt
=d/dt(4/3πr^3)
= 4/3π⋅3r^2dr/dt
=4πr ^2⋅ dr/dt
=4πr^2.1/4πr ....[From (i)]
r=10
Hence, the volume of the spherical balloon is increasing at the rate of 10cm^3/sec.
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Which statement is missing in the proof?
The correct answer is m√5 + m√2 = 180°.
What are parallel lines?Parallel lines are those in a plane that are regularly spaced apart from one another. There is no crossing of parallel lines.
Given, k and p are parallel lines.
In an expression or equation, one value can be substituted for another and the result will still be the same. This is known as the substitution property of equality. If the values of x and y are the same, then x and y can be substituted for one another.
And we have,
m√1 + m√2 = 180°
And m√5 = m√1
So, m√5 + m√2 = 180°
Therefore, in the proof, the missing term is m√5 + m√2 = 180°.
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