the probability of not being the victim of the theft involving the bicycle, if the victim of the theft is randomly selected, is 0.961.
According to the given data, it is given that there was a 3.9% probability of theft involving a bicycle in 2002. Thus, the probability of not being the victim of the theft involving the bicycle can be calculated by the complement of the probability of being the victim of the theft involving the bicycle.
The formula for calculating the probability of the complement is:
P(A') = 1 - P(A)
Where P(A) represents the probability of the event A, and P(A') represents the probability of the complement of event A.
Thus, the probability of not being the victim of the theft involving the bicycle can be calculated as:
P(not being the victim of the theft involving the bicycle) = 1 - P(the victim of the theft involving the bicycle)
Now, substituting the value of P(the victim of the theft involving the bicycle) = 3.9% = 0.039 in the above formula, we get:
P(not being the victim of the theft involving the bicycle) = 1 - 0.039P(not being the victim of the theft involving the bicycle) = 0.961
Therefore, the probability that the randomly selected victim was not the victim of bicycle theft is 0.961 Thus, the probability of not being the victim of the theft involving the bicycle, if the victim of the theft is randomly selected, is 0.961.
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Joseph has a bag filled with 3 red, 3 green, 15 yellow, and 9 purple marbles. Determine P(not red) when choosing one marble from the bag. 10% 30% 50% 90%
Answer:
P(not red) = 90%
Step-by-step explanation:
There are 3 red, 3 green, 15 yellow, and 9 purple marbles in all:
3 + 3 + 15 + 9 = 30
You are trying to find the probability of not obtaining a red marble. There are 3 red marbles, and 30 marbles in all. This means that there are 27 marbles that are not red. Divide 27 with 30:
27/30 = 0.9
0.9 = 90/100 = 90%
90% is your answer.
P(not red) = 90%.
~
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Answer: 90%
Step-by-step explanation:
HURRY WILL MARK BRAINLIEST An egg carton of a dozen eggs consists of 1 randomly selected empty cardboard container and 12 randomly selected eggs. The weights of the empty cardboard containers have a mean of 20
grams and a standard deviation of 1.7 grams. The weight of a single egg has a mean of 68.33 grams and standard deviation of 2.23 grams. It is reasonable to assume independence between the weights of the empty cardboard containers and the weights of the eggs. It is also reasonable to assume independence among the weights of the 12 eggs that are randomly selected for a full carton. Let the random variable X be the weight of a randomly selected carton of a dozen eggs.
i) The mean of x is 68.33 grams.
ii) The standard deviation of x is 2.23 grams.
What is standard deviation?
A standard deviation or (σ ) is a measurement of the data's dispersion from the mean. A low standard deviation indicates that data are concentrated around the mean, whereas a high standard deviation shows that data are more dispersed.
If the standard deviation is close to zero, the data points are close to the mean; otherwise, if the standard deviation is high or low, the data points are, respectively, above or below the mean.
i) Properties of expected values establish that
[tex]$\mathrm{E}(W)=\mathrm{E}(P)+\mathrm{E}\left(X_1\right)+\ldots+\mathrm{E}\left(X_{12}\right)$[/tex]
Because all 12 eggs have the same mean weight, this becomes
[tex]$\mathrm{E}(W)=\mathrm{E}(P)+12 \times \mathrm{E}\left(X_i\right)$[/tex]
We were told that
[tex]$\mathrm{E}(W)=840$ and $\mathrm{E}(P)=20$, so we can solve[/tex]
[tex]$840=20+12 \times \mathrm{E}\left(X_i\right)$ to find $\mathrm{E}\left(X_i\right)=\frac{840-20}{12} \approx 68.33$ grams.[/tex]
ii) Because of independence, properties of variance establish that
[tex]$${Var}(W)={Var}(P)+{Var}\left(X_1\right)+{Var}\left(X_2\right)+\ldots+{Var}\left(X_{12}\right)$$[/tex]
Because all 12 eggs have the same variance of their weights, this becomes
[tex]${Var}(W)={Var}(P)+12 \times {Var}\left(X_i\right)$[/tex]
We were told that
[tex]${SD}(W)=7.9\ \text {and}\ {SD}(P)=1.7$ Therefore, ${Var}(W)=(7.9)^2=62.41$ and ${Var}(P)=(1.7)^2=2.89$[/tex]
We can solve
[tex]$ 62.41=2.89+12 \times {Var}\left(X_i\right)$[/tex]
To find
[tex]${Var}\left(X_i\right)=\frac{62.41-2.89}{12}=4.96$[/tex]
Thus, SD[tex](X_i\right))=\sqrt{(4.96)} \approx[/tex] 2.23 grams.
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What are the 3 ways to calculate slope?
There are several ways to calculate the slope of a line, but the most common ways are:
Rise over Run: This is the most basic and intuitive way to calculate slope. It involves taking the difference in the y-coordinates (rise) and dividing it by the difference in the x-coordinates (run) between two points on the line. The slope is represented by the letter "m" and can be calculated using the following formula: m = (y2 - y1) / (x2 - x1)
For example, if two points on a line are (3,6) and (6,9) the slope can be calculated as: m = (9-6) / (6-3) = 3/3 = 1
Point-Slope Form: This method uses the coordinates of a single point on the line and the slope of the line to find the equation of the line. The slope is represented by the letter "m" and the point by the coordinates (x1, y1). The equation of the line in point-slope form is: y - y1 = m(x - x1)
Slope-Intercept Form: This method expresses the equation of a line in the form of y = mx + b, where m is the slope and b is the y-intercept. To find the slope-intercept form, you need to know the slope and the y-intercept (the point where the line crosses the y-axis) of the line.
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A set of data has a normal distribution with a mean of 50 and a standard
deviation of 8. Find the percent of data within the interval less than 50.
A. 34%
B. 68%
C. 50%
D. 97.5%
If a set of data has a normal distribution with a mean of 50 and a standard deviation of 8. the percent of data within the interval less than 50 is; C. 50%.
How to find the percent of data?Based on the details given the percent of data within the interval less than 50 is not 34% or 68% or 97.5% but rather the percent of data is 50%.
The reason why it is 50% is that the mean is 50 in which the data has a normal distribution, 50 is the center of the distribution. Based on this half the data falls above the mean and half the data falls below the mean.
Therefore the correct option is C.
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Use intercepts to graph the line described by the equation −2x − 5y = −20.
The x - intercept and y - intercept of the linear function are 10 and 4
Graph of a Linear FunctionA graph of a linear function is a straight line. The line will always pass through the origin (0,0) and have a constant slope. The equation for a linear function is y = mx + c, where m is the slope of the line and c is the y-intercept.
In this problem, we can model the linear function into the slope-intercept form and the plot using a graphing calculator.
-2x - 5y = -20
-5y = -20 + 2x
5y = 20 - 2x
y = (20 - 2x) / 5
y = 4 - 2/5x
Plotting this on a graph using a graphing calculator
The y - intercept is at 4, and the x - intercept is at 10
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Match each graph with the situation it represents
A) Leaving home in a car and going to a store, shopping for a while, and then coming back home (the direction of the store is positive) figure 2.
B) Rolling a marble down the stairs (the bottom of the stairs is the origin) figure 6.
C) Rolling a marble towards a table (the direction from which the marble came was the negative direction). figure 1.
What is graph?The graphs are extremely helpful and widely used data structures. They are applied to a wide range of object relationships and, in reality, are applicable to almost everything.
The graphs represent a generalised structure that allows for the modelling of a very broad range of real-world situations, as we shall see in more detail. Trees are a subset of graphs, and lists are special cases of both trees and thus of graphs.
Since graphs are frequently used in practise, there has been a great deal of research into "graph theory," where there are many known problems for graphs and many well-known solutions to them.
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A jar contain n nickel and dime. One gym ha a $150 joining fee and cot $35 per month. Another gym ha no joining fee and cot $60 per month. When would Caey pay the ame amount to be a member of either gym? How much would he pay?
Casey would pay the same amount to be a member of either gym in 6 months and that amount would be 360 dollars, that is $360.
Let the number of months be x.
In the 1st gym there is a $150 joining fee and cost $35 per month. So after x months the total cost in dollars in first gym is
= 150 + 35x
In the 2nd gym there is no joining fee and cost $60 per month. So after x months the total cost in dollars in second gym is
= 60x
When Casey pay the same amount to be a member of either gym, the total cost of the gym would be equal
150 + 35x = 60x
25x = 150
Dividing by 25
x = 6
The amount she pay to be a member of either gym when there is same amount is
= 60×6
= 360
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A rectangular container can hold 6 gallons of water. A second larger rectangular
container has dimensions that are triple the smaller container. How much water can the
larger container hold?
So on solving the provided question, we can say that Area of rectangle, A = 3x*x => x = 5
What is rectangle?A rectangle in Euclidean plane geometry is a quadrilateral with four right angles. You might also describe it as follows: a quadrilateral that is equiangular, which indicates that all of its angles are equal. The parallelogram might also have a straight angle. Squares are rectangles with four equally sized sides. A quadrilateral of the shape of a rectangle has four 90-degree vertices and equal parallel sides. As a result, it is sometimes referred to as an equirectangular rectangle. Because its opposite sides are equal and parallel, a rectangle is also known as a parallelogram.
here, length is 3 times breadth
so let length = x
breadth = 3x
Area of rectangle, A = 3x*x
A = 3x^2
75 = 3x^2
x = 5
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You design a tree house using a coordinate plane in which the coordinates are measured in feet. Three vertices of the rectangular floor are (-3,-4), (6,-4), (-3,4) . What is the fourth vertex?
u have to use the inequity expresstion to anwser the perentieses first it would come out to be a persent then turn that to a fration by devieding by the first numer in the perenthies by sencond number the answer would be 3 over six but negitive
Find the measure of the indicated arc or angle.
The measure of the arc QP is QP = 66°.
What is inscribed angle?
In terms of geometry, an inscribed angle is the angle created when two chords intersect within a circle. It can also be described as the angle formed by two specified points on a circle at a given point on the circle. Two chords of the circle sharing an endpoint are equivalently used to define an inscribed angle.
Given:
m ∠ RPQ = 57°
We have to find the measure of the arc QP.
Since,
RQ = 2*m ∠RPQ
RQ = 2*(57°)
RQ = 114°
Let
RP = RQ + QP
180° = 114° + QP
QP = 180° - 114°
QP = 66°
Hence, the measure of the arc QP is QP = 66°.
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Application 1: A mall city had 5000 people in 2005 and 6200 in 2010. Find the value of k in theexponential model Q(t) = Qoekt
A mall city had 5000 population of people in 2005 and 6200 in 2010, then the value of k is,
k = Log( [tex]\frac{\frac{Q(t)}{5000} }{t*log(e)}[/tex] ) , k = Log ( [tex]\frac{\frac{Q(t)}{6200} }{t*log(e)}[/tex] )
The value of t corresponds to the last two digits in the year.
If t=0 represents the year 2000, then
For the year 2005,
t = 5
For the year 2010,
t = 10
First part of the problem.
Find the value of k using these values.
Q(t) = 5000
t = 5
Q(t) = Qo[tex]e^{kt}[/tex]
Q(t) = 5000[tex]e^{kt}[/tex]
k is an exponential value. When we solve for an exponential value, we must use logarithms.
First, divide both sides of equation by 5000
[tex]\frac{Q(t)}{5000}[/tex] = [tex]e^{kt}[/tex]
log both sides of the equation.
log( [tex]\frac{Q(t)}{5000}[/tex] ) = log ([tex]e^{kt}[/tex] )
Next, bring the exponents down as the coefficients of the log.
log( [tex]\frac{Q(t)}{5000}[/tex] ) = kt * log(e)
Divide by t*Log(e) both sides of equation to obtain k.
Log( [tex]\frac{\frac{Q(t)}{5000} }{t*log(e)}[/tex] ) = k
Now, plug in the values.
Second part of problem. Find P using these values.
t = 10
Q(t) = Qo[tex]e^{kt}[/tex]
Q(t) = 6200[tex]e^{kt}[/tex]
k is an exponential value. When we solve for an exponential value, we must use logarithms.
First, divide both sides of equation by 6200
[tex]\frac{Q(t)}{6200}[/tex] = [tex]e^{kt}[/tex]
log both sides of the equation.
log( [tex]\frac{Q(t)}{6200}[/tex] ) = log ([tex]e^{kt}[/tex] )
Next, bring the exponents down as the coefficients of the log.
log( [tex]\frac{Q(t)}{6200}[/tex] ) = kt * log(e)
Divide by t*Log(e) both sides of equation to obtain k.
Log ( [tex]\frac{\frac{Q(t)}{6200} }{t*log(e)}[/tex] ) = k
Now, plug in the values.
Therefore,
A mall city had 5000 population of people in 2005 and 6200 in 2010, then the value of k is,
k = Log( [tex]\frac{\frac{Q(t)}{5000} }{t*log(e)}[/tex] ) , k = Log ( [tex]\frac{\frac{Q(t)}{6200} }{t*log(e)}[/tex] )
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3 cars are washed every 27 minutes at a car wash
Express the unit rate as a shaded fraction of the circle below
Answer: The unit rate is the number of cars washed per minute. To find the unit rate, we need to divide the number of cars washed by the number of minutes.
3 cars / 27 minutes = 1/9 car/minute
To express the unit rate as a shaded fraction of a circle, we can divide the circle into 9 equal parts and shade 1 of those parts to represent the unit rate of 1/9 car/minute.
It is not possible to provide a shaded fraction of a circle here as it is a text based interface.
Step-by-step explanation:
Makayla run 3/5 of a mile in 1/6 of an hour. How fat, in mile per hour, doe he run
therefore Makayla runs = 3/5+3/5+3/5+3/5+3/5+3/5=18/5 = 3.6 miles in one hour
Makayla run 3/5 of a mile in 1/6 of an hour.
How far, in mile per hour, doe he run
in one hour = 1/6+1/6+1/6+1/6+1/6+1/6=1
therefore he runs = 3/5+3/5+3/5+3/5+3/5+3/5=18/5
= 3.6
What is linear equation?
An algebraic equation of the form y=mx+b is referred to as a linear equation involving only a constant and a first-order (linear) term, where m is the slope and b is the y-intercept. Occasionally, the above is called a "linear equation of two variables," where y and x are the variables.
Driving separate vehicles on the same highway are Makayla and Addison,
another eample Makayla is 290 miles away from stadium,
Makayla driving speed is 40 miles per miles.
345 miles separate Addison from the stadium.
Addison driving speed is 51 miles per hour.
Let tt represent hours.
Therefore the equation is
Mm=290-40tt
Aa =345-51tt.
question
One month, two search engines were used to conduct a total of 3.52 billion online searches. Search engine A was used to conduct 0.66 billion more searches than search engine B. How many searches were conducted on each search engine?
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Is this correct? If not, please explain.
Yes your answer was correct.
Shelly made treat bags for the guests attending her birthday party on Saturday. She made 19 bags; one for every boy and girl guest attending the party. If she made 11 bags for girl guests, what is the ratio of total boy guests to girl guests attending her party?
Answer: If Shelly made 19 treat bags, and 11 of them were for girl guests, then the remaining bags were for boy guests.
So the number of boy guests = 19 - 11 = 8
Then the ratio of boy guests to girl guests is 8:11
or you can simplify it by dividing 8 and 11 by greatest common factor which is 8/11 in this case.
or you can express it as 8/19 for boy guests and 11/19 for girl guests
Step-by-step explanation:
Scientists are preparing two satellites to be launched. The equation y=7000xy=7000x represents the number of miles, yy, that the satellite, Space Explorer A, flies in xx hours. The satellite, Space Explorer B, flies 84000 miles in 20 hours.
Use the dropdown menu and answer-blank below to form a true statement.
The rate of flying explorer A is more than the rate of explorer B.
What is proportionality?Proportionality is defined as between two or more sets of values, and how these values are related to each other in the sense are they directly proportional or inversely proportional to each other.
here,
Proportionality equation for explorer A,
y=7000x
Rate of flying explore A = 7000
Rate of flying explore B,
Rate = distance / time = 84000 / 20 = 42000
Thus, the rate of flying explorer A is more than the rate of explorer B.
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can I have help with this question
Answer:
The Last Answer
Step-by-step explanation:
Process of elimination
He wants AT LEAST which means that is will be a less than symbol with a line under it, the last one is the only one with that!
what is the domain explain any restrictions
A function's domain is the set of all possible inputs to the function. Domains can be limited if the function is rational and the denominator is 0 for some x value or values.
What are the 3 domain restrictions?
The square root function, log function, and reciprocal function are the three functions with limited domains. Because you cannot take square roots of negative numbers and produce real numbers, the square root function has a limited domain.
To find the domain restriction of an expression, take the expression's denominator. Put that denominator to zero. Solve the resulting equation for the denominator zeroes. The domain consists of all other x-values.
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Is dilated cervix normal?
It is normal for the cervix to dilate during labor. The cervix opens when there is dilatation. The cervix may begin to shrink or stretch (efface) and open as labor approaches (dilate).
As a result, the cervix is ready for the baby to enter the delivery canal (vagina). Each woman's cervix thins and opens at a different rate. The cervix may begin to slowly efface and dilate over a few weeks in certain women. On the other hand, a first-time mother frequently won't dilate until active labor begins.
Your doctor may use his or her fingers to feel the cervix toward the end of your pregnancy to see how much it has effaced and dilated. To do this, he or she will put on sterile gloves. Your cervix opens (dilates) as your uterus contracts during labor. They also assist in positioning the infant for birth.
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A salesclerk is creating a display of 14 polo shirts and 56 team shirts. Only one type of shirt can be in each row. What is the maximum number of shirts in each row so that all the rows are equal in length?.
The maximum number of shirts in each row is 14 shirts. The result is obtained by finding the greatest common factor.
How to find the greatest common factor (GFC)?The GFC can be found by using factor trees. We will get prime factorization to determine the GFC. Here are the steps.
Make factor trees by dividing the number by the smallest prime that can divide it.Determine prime factorization.Choose the common factors and multiply them.A salesclerk is displaying 14 polo shirts and 56 team shirts. In each row, there are only one type of shirt. Find the maximum number of shirts in each row so that the length of all rows are equal!
To make the length of all rows equal, we should find the GFC of two kinds of shirts. That would be the maximum number of shirts.
See the factor trees illustration in the attachment!
The prime factorization
14 = 2 × 756 = 2 × 2 × 2 × 7The common factors are 2 and 7.
The greatest common factor is
= 2 × 7
= 14
Hence, the shirts displayed would be a maximum of 14 shirts in each row.
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How do you find the ordered pair of a function?
To find the ordered pair (x, f(x)) of a function, you must first know the function's equation. The ordered pair is a combination of the input value (x) and the output value (f(x)) of the function.
Here are the steps to find the ordered pair of a function:
Identify the input value (x) for which you want to find the corresponding output value (f(x)). Substitute the input value (x) into the function's equation. For example, if the function is f(x) = 2x + 3, and you want to find the ordered pair for x = 4, you would substitute 4 into the equation:f(4) = 2(4) + 3 = 8 + 3 = 11
The output value (f(x)) is the result of the equation when you substitute the input value (x). In this example, f(4) = 11The ordered pair for this input value (x) is (4, 11). The first value in the ordered pair is the input value (x), and the second value is the output value (f(x)).It's important to note that the ordered pair will be different for every input value (x) you substitute into the function's equation.
For instance, if you want the ordered pair of f(x) = 2x + 3 for x = 5 you will have (5,13) as an ordered pair.
It's also worth mentioning that in some cases you might have functions like[tex]f(x) = (x-2)^2 + 1[/tex], you can use the same process but you will be solving the equation rather than just substituting it.
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How do you use section formula?
We use Section formula for internal division: P(x, y) = x = (m₁x₂ + m₂x₁)/(m₁+ m₂) and y = (m₁y₂ + m₂y₁)/(m₁+ m₂).
Section formula for external division: P(x, y) = x = ((m₁x₂ - m₂x₁)/(m₁+ m₂) , y = (m₁y₂ - m₂y₁)/(m₁+ m₂).
What is the section formula?
In coordinate geometry, the Section formula is used to determine the internal or external ratio at which a line segment is divided by a point. It is used to determine a triangle's centroid, incenter, and excenter. It is employed in physics to identify equilibrium locations, system centers of mass, etc.
Here,
We have to find the use of the section formula.
we concluded that the section formula is used for internal division: P(x, y) = x = (m₁x₂ + m₂x₁)/(m₁+ m₂) and y = (m₁y₂ + m₂y₁)/(m₁+ m₂).
Section formula for external division: P(x, y) = x = ((m₁x₂ - m₂x₁)/(m₁+ m₂) , y = (m₁y₂ - m₂y₁)/(m₁+ m₂).
Hence, the section formula is x = (m₁x₂ + m₂x₁)/(m₁+ m₂) and y = (m₁y₂ + m₂y₁)/(m₁+ m₂).
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Do sides 7 cm 24 cm 25 cm form a right angled triangle?
Yes, the sides 7 cm 24 cm 25 cm form a right angled triangle.
The right angled triangle contains hypotenuse, which is opposite the 90° angle. Other two sides are base and perpendicular. The side of right angled triangle follows Pythagoras theorem.
It says, hypotenuse² = perpendicular² + base²
We know 25 cm will be hypotenuse as it is the longest side. Keeping the values now -
25² = 24² + 7²
Taking square of all the values
625 = 576 + 49
Performing addition on Right Hand Side of the equation
625 = 625
As the square of hypotenuse is equal to sum of square of other two sides, it is right angled triangle.
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Jim scores 8 more points than
Kevin in the basketball game. If
altogether they scored 22 points
how many points did Jim score?
Answer: 14
Step-by-step explanation: Because if altogether they have 22 and Kevin already has 8 then you just add 14 more
The average height of a giraffe is about 2 x 10¹ feet. The average height of an ant is about 4 x 10-³ feet.
Based on this information, about how many times taller is a giraffe than an ant?
Answer:
1. Convert to decimal notationThe exponent is -3, making it 10 to the power of negative 3. When an exponent is negative, the solution is a number less than the origin or base number. To find our answer, we move the decimal to the left 3 times: 4.0
2. Final result
0.004
What are altitudes of a triangle class 7?
Answer:
Step-by-step explanation:
The altitude of a triangle is the perpendicular drawn from the vertex of the triangle to the opposite side.
At 10:00 a. M. , Han and Tyler both tarted running toward each other from oppoite end of a 10-mile path along a river. Han run at a pace of 12 minute per mile. Tyler run at a pace of 15 minute per mile. How far doe Han run after a half hour? After an hour?
2.5 miles Han run after a half hour
it will take 1[tex]\frac{1}{9}[/tex] to meet both
Given that
At 10:00 a. M. , Han and Tyler both tarted running toward each other from opposite end of a 10-mile path along a river.
Han run at a pace of 12 minute per mile.
Tyler run at a pace of 15 minute per mile.
We can say that he is capable of running at a pace of 5 mph. (Since speed is equal to 1/pace, and 12 minutes equal 1 mile and 60 minutes equal 5 miles),
Similar to this, Tyler would run at a speed of 4 miles per hour if he ran a mile at a pace of 15 minutes.
distance = speed * time
The distance covered by Han in half an hour = 1/2 × 5 = 2.5 miles
the time at which Han and tyler will meet,
T= Total distance/ ( the sum of their speed)
T = 10/5+4 = 10/9
T = 1[tex]\frac{1}{9}[/tex] hours is greater than 1 hour
so, they will not meet within 1 hour
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Find the sum of the first 25 terms. 2, 8, 14, 20
For the mathematical series, the total of the first 25 terms is 1850.
What is arithmetic sequence?Arithmetic progression or arithmetic sequence is a numerical series in which the difference between subsequent terms is constant. For example, the sequence 5, 7, 9, 11, 13, 15.. An arithmetic sequence is a series of integers that are ordered and share a common difference between each succeeding term. In the arithmetic sequence 3, 9, 15, 21, 27, for example, the common difference is 6. An arithmetic progression is another name for an arithmetic sequence.
Here,
This is an arithmetic sequence as each term has a common difference to the previous term. Any arithmetic sequence can be expressed as:
a(n)=a+d(n-1), a=initial term, d=common difference, n=term number.
We know that a=2 and d=(20-14)=(14-8)=(8-2)=6 so
a(25)=2+6(25-1)
a(25)=146
The sum of an arithmetic sequence is the average of the first and last terms times the number of terms, in this case:
s(25)=25(2+146)/2
s(25)=1850
The sum of the first 25 terms is 1850 for the arithmetic sequence.
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Aa similarity postulate
a) Yes, the given triangle is Angle - Angle similarity posulate.
b) Yes , 2 triangles are similar using AA similarity posulate.
c) If the triangles are follow the AA similarity posulate then they are congruent to each other also.
What is triangle?
A triangle is a polygon with three vertices and three sides. The angles of the triangle are formed by the connection of the three sides end to end at a point. The triangle's three angles add up to 180 degrees in total.
What is AA similarity posulate?
According to the AA Similarity Postulate and Theorem, each triangle must have two matching angles that are congruent in order to demonstrate that two triangles are similar.
Think about the triangles which are given in the question;
ΔABC with scale factor of [tex]\frac{AB}{DE}[/tex] to create ΔA'B'C'.
ΔABC is dilated about point P with a scale factor of [tex]\frac{AB}{DE}[/tex] to create ΔA'B'C'.
After a dilation, corresponding angles are congruent, thus
∠B ≅ ∠B'
Therefore, ∠B' ≅ ∠E.
Similarly, ∠A ≅ ∠A'
therefore, ∠A' ≅ ∠F . Due to the scaling factor being [tex]\frac{AB}{DE}[/tex]
A'B' = [tex]\frac{AB}{DE}[/tex] · AB = DE
So, [tex]\overline{A'B'}[/tex] ≅ [tex]\overline{F'E'}[/tex].
Now, we have to prove ΔABC [tex]\sim[/tex] ΔDEF.
We know that,
∠B' ≅ ∠E , ∠A' ≅ ∠F and [tex]\overline{A'B'}[/tex] ≅ [tex]\overline{F'E'}[/tex].
This means that,
ΔA'B'C' ≅ ΔDEF by ASA rule.
Consequently, a series of stiff transformations must occur that will convey;
ΔDEF to ΔA'B'C' then there will be a dilatation.
ΔA'B'C' to ΔABC.
About the initial two triangles, it was all that was known was two sets of parallel angles. You have therefore demonstrated that AA is a need for triangle similarity.
Learn more about AA similarity posulate click here;
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