A collection of numbers, variables, and mathematical operations is referred to as an expression. The expression is -2/5- 4/3P and 2/5+4/3P
What is meant by 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 .A collection of numbers, variables, and mathematical operations is referred to as an expression.
Given,
-4/3p - 2/5
= -20p - 6/15
= 2(10 p +3)/15
A mathematical expression is a sentence that includes numbers, variables, or both. The equal symbol is never used in expressions. Here are a few phrases as examples. A mathematical statement stating the equality of two expressions is known as an equation.
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What are asymptotes example?
An asymptote is a line that a curve approaches but never touches. There are two types of asymptotes: horizontal and vertical. Few examples of asymptotes are f(x) = 1/x, f(x) = 1/x² + 1 and f(x) = 1/(x-3).
Here are some examples of functions with horizontal asymptotes:
The function f(x) = 1/x has a horizontal asymptote at y = 0.
f(x) = 1/x
As x gets larger and larger, the value of the function approaches 0, but never quite reaches it. This means that the horizontal line y = 0 is an asymptote for the function f(x) = 1/x.
The function f(x) = 1/x² + 1 has a horizontal asymptote at y = 1.
f(x) = 1/x² + 1
As x gets larger and larger, the value of the function approaches 1, but never quite reaches it. This means that the horizontal line y = 1 is an asymptote for the function f(x) = 1/x² + 1.
Here is an example of a function with a vertical asymptote:
The function f(x) = 1/(x-3) has a vertical asymptote at x = 3.
f(x) = 1/(x-3)
As x gets closer and closer to 3, the value of the function approaches infinity, but never quite reaches it. This means that the vertical line x = 3 is an asymptote for the function f(x) = 1/(x-3).
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Determine whether each figure has rotational symmetry.
If so, determine the angle of rotation.
Which grapgh represents this system of inequalitys
y≤x +4
2x+y ≤-4
Answer:B
Step-by-step explanation:
Which graph represents the solution set of the system of inequalities? {y>−2x−2y≤x+4 A system of 2 linear inequalities graphed on a coordinate plane. The horizontal x-axis ranges from negative 5 to 5 in increments of 1. The vertical y-axis ranges from negative 5 to 5 in increments of 1. A dashed line passes through begin ordered pair negative 0.5 comma negative 1 end ordered pair, begin ordered pair negative 1 comma 0 end ordered pair and begin ordered pair 0 comma negative 2 end ordered pair. A solid line passes through begin ordered pair negative 0.5 comma negative 1 end ordered pair, begin ordered pair 0 comma 1 end ordered pair and begin ordered pair negative 1 comma negative 3 end ordered pair. The region below the dashed line and above the solid line is shaded. A system of 2 linear inequalities graphed on a coordinate plane. The horizontal x-axis ranges from negative 5 to 5 in increments of 1. The vertical y-axis ranges from negative 5 to 5 in increments of 1. A da
How do you describe the graph of the solution set of a quadratic inequality?
A quadratic inequality of the form
y>a[tex]x^{2}[/tex]+bx+c
(or substitute <,≥ or ≤ for > ) represents a region of the plane bounded by a parabola .
A quadratic inequality of the form
y>a[tex]x^{2}[/tex]+bx+c
(or substitute <,≥ or ≤ for > ) represents a region of the plane bounded by a parabola .
To graph a quadratic inequality, start by graphing the parabola. Then fill in the region either above or below it, depending on the inequality.
If the inequality symbol is ≤ or ≥ , then the region includes the parabola, so it should be graphed with a solid line.
Otherwise, if the inequality symbol is < or > , the parabola should be drawn with a dotted line to indicate that the region does not include its boundary.
Example:
Graph the quadratic inequality.
y≤[tex]x^{2}[/tex]−x−12
The related equation is:
y=[tex]x^{2}[/tex]−x−12
First we notice that a , the coefficient of the x2 term, is equal to 1 . Since a is positive, the parabola points upward.
The right side can be factored as:
y=(x+3)(x−4)
So the parabola has x -intercepts at −3 and 4 . The vertex must lie midway between these, so the x -coordinate of the vertex is 0.5 .
Plugging in this x -value, we get:
y=(0.5+3)(0.5−4)
y=(3.5)(−3.5)
y=−12.25
So, the vertex is at (0.5,−12.25) .
We now have enough information to graph the parabola. Remember to graph it with a solid line, since the inequality is "less than or equal to".
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What is null set in math grade 7?
A null set is a set that contains no elements in mathematics, that is it is empty. It is also called empty set or void set.
It is denoted by the Greek letter Ф (phi). It is expressed as {}.
A null set is formed by intersection of two disjoint sets. For example, a set of odd numbers, say A = {1, 3, 5, ...} and set of even numbers say B = {0, 2, 4, ...}. A set of number of both odd and even is a null set as these sets A and B doesnt have any element in common, that is they are disjoint set.
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How to solve 997 1 ⁄ 3?
The value of the given mixed fraction 997 1 / 3 is equal to 2992 / 3.
As given in the question,
Given expression is equal to,
997 1 / 3
Simplify the mixed fraction using following method in standard form :
Q R / D = [ ( D × Q ) + R ] / D
In the numerator multiply divisor (D) to the quotient (Q) and add it to the remainder (R).
Denominator of the expression remain same.
Here Q = 997 , R = 1 and D = 3
Substitute the value in the formula we get,
997 1 / 3
= [ ( 3 × 997 ) + 1 ] / 3
= ( 2991 + 1 ) / 3
= 2992 / 3
Therefore, the value of the given mixed fraction is equal to 2992 / 3.
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PLEASE I REALLY NEED HELP THE BRAINLIST ANSWER WILL GET MARKED!!!!
Students in a class were asked to tell their favorite color.
a. What percent said red, blue, or yellow?
__ % said red, blue, or yellow.
b. How many times more students said red than yellow? __ times more students said red than yellow.
c. Use two methods to find the percent of students who said green. green: __%
Which method do you prefer? Explain.
Answer: a = 70%
b 6.5 times
c green = 16%
2. ????? Very poorly worded question
2. a 54%
b. 46%
c. 70%
d. 27%
Step-by-step explanation:
Students in a class were asked to tell their favorite color.
a. What percent said red, blue, or yellow? __% said red, blue, or yellow.
red is .26
blue is .40
yellow is .04
---------------------
.70 = 70 %
b. How many times more students said red than yellow?
red .26
yellow is .04
.26/.04 = 6.5 times
c. What percent of students said green? __% said green.
The circle must total 100 %
Red + blue + yellow + purple + green = 100
26+ 40 + 4 + 14 + green = 100
84 + green = 100
Subtract 84 from each side
84-84 + green = 100-84
green = 16
2. Which two methods could be used to find the percent of students who said green?
We can either convert the percents to decimals or the decimals to percents
The total percents must equal 100%
The total decimal must equal 1
a. Find the sum of the percents given in the circle graph.
If they only want the ones written on the graph as percents
(14+40) = 54%
b. Find the sum of the percent forms of the numbers given in the circle graph. Subtract the result from 100%.
100 - 54% = 46%
c. Find the sum of the decimal forms of the numbers given in the circle graph. Subtract that sum from 1. Change the result to a percent.
.26+.04 = .3
1-.3 = .7
70%
d. Find the sum of half of the percents given in the circle graph. Subtract that total from 50%.
1/2 (46%) = 23%
50 -23 = 27%
FOR 100 POINTS AND BRAINLIEST HURRTRRRRYYY
The water level is a number that indicates whether the level is above or below normal using positive or negative values. After a heavy storm, the water level of a river is −3.7 feet. This level is 1/2 times 2.5 feet more than the previous water level.
What was the previous water level?
Drag and drop a value into the box to correctly complete the statement
answer choices-
-18.5
-9.9
-4.9
-4.3
The previous water level is -9.9 feet. The correct option is B.
What is an expression?Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented by mathematical symbols.
The numerical variables and operations denoted by the addition, subtraction, multiplication, and division signs are combined in the mathematical expression. They can also denote the operation order and other properties of the logical syntax.
Given that after a heavy storm, the water level of a river is −3.7 feet. This level is 1/2 times 2.5 feet more than the previous water level.
The previous water level is calculated as,
1/2 ( x + 2.5 ) = -3.7
x + 2.5 = -7.4
x = -7.4 - 2.5
x = -9.9
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Step-by-step explanation:
Let's solve this problem step by step. The current water level is -3.7 feet, which is 1/2 times 2.5 feet more than the previous water level. Let's call the previous water level "x".
So, we can write an equation to represent this relationship:
-3.7 = x + (1/2) * 2.5
Now, let's solve for x:
-3.7 = x + 1.25
x = -3.7 - 1.25
x = -4.95
So, the previous water level was **-4.95 feet**.
Out of the answer choices you provided, the closest value to the correct answer is **-4.9**.
PLEASE MARK AS BRAINLIEST
The question is in the picture.I’ll mark Brainliest!
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How do you solve 3 simultaneous equations using elimination?
Answer:
Write all the equations in standard form cleared of decimals or fractions.Choose a variable to eliminate; then choose any two of the three equations and eliminate the chosen variable.
Step-by-step explanation:
What do Total Productive Maintenance (TPM) practices ensure? Select an answer: O equipment availability is minimized
O machines do not break down so often O scheduled maintenance is not necessary O equipment downtime is maximized
Total Productive Maintenance (TPM) practices ensure D. equipment downtime is maximized
What is Total Productive Maintenance?According to TPM, operators are in charge of sanitizing, enhancing, and maintaining their workstations to guarantee quality and safety throughout production cycles. The objective of TPM is to maintain equipment in top working order to ensure that production processes go without hiccups or delays.
Through the optimization of equipment availability and performance, TPM seeks to increase overall productivity and quality. The 5S methodology, a method of workforce organization that aids in streamlining and standardizing facility procedures, is the foundation on which TPM is based.
Therefore, the correct option is D.
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Who invented 0 number?
There are several hypotheses regarding how 0 was created. The idea of utilising 0 as a numeric value is thought to have originated in India. According to legend, Aryabhatta invented zero by using it as a placeholder number in the fifth century.
However, the idea of zero was not always present. Although the idea of nothingness or emptiness existed, applying it numerically had a profound effect on mathematics. The principles for zero were laid forth by Brahmagupta in the seventh century.
In America, the Mayans developed the idea of zero as a placeholder in the 4th century. But though being skilled in mathematics, they never thought the idea of using it in equations.
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A bottle contains 7 8 of a pitcher of water. A cup can hold 3 16 of a pitcher of water. Eve wants to pour all the water from the bottle into some cups. How many cups does Eve need?
Answer:
at least 5
Step-by-step explanation:
You want to know the number of cups needed to contain 7/8 pitcher of water when each cup holds 3/16 pitcher.
CupsThe number of cups can be found by dividing the water quantity by the amount each cup holds:
(7/8 pitcher)/(3/16 pitcher/cup) = 14/3 cups = 4 2/3 cups
Eve needs at least 5 cups to hold the amount of water in the bottle.
__
Additional comment
If every cup is filled to capacity, then the 5th cup will be 2/3 full. The number of cups can be any number greater than or equal to 5.
If you want, you can write an inequality relating the capacity of the cups to the amount of water:
n(3/16) ≥ 7/8
n ≥ 4 2/3 . . . . . . . . . where n is a counting number
n ∈ {5, 6, 7, ...}
This more clearly shows that the number is not restricted to 5.
Solve the IVPs by the Laplace transform. If necessary, use partial fraction expansion as in Example 4 of the text: Show all details. 1. Y’ + 5. 2y = 19. 4 sin 2t, y(0) = 0 2. Y’ + 2y = 0, y(0) = 1. 5 3. Y" – y’ + 6y = 0, y(0) = 11, y’(0) = 28 4. Y" + 9y = I0e^(-t) y(0) = 0, Y’(0) = 0 5. Y" – 1/4y = 0, y(0) = 12, Y’(0) = 0 6. Y" - 6y’ + 5y = 29 cos 2t, y(0) 3. 2 Y’(0) = 6. 2 7. Y" + 7y' + 12y = 21e^3t, y(0) 3. 5 y’(0) = -10 8. Y" - 4y' + 4y = 0, y(0) 8. 1, y’(0) = 3. 9 9. Y" - 4y' + 3y = 6t - 8, y(0) = 0, y’(0) = 0 10. Y" + 0. 04y = 0. 02t^2 y(0)=-25, y’(0)=0
y(t)= 3/5[tex]e^{-5.2}[/tex]-3/5esin2t-9/5sin2t when the IVP's is y'+5.2y=19.4sin2t, and y(0)=0 by Laplace transform and partial fraction.
Given that,
We have to solve the IVP's by the Laplace transform.
If necessary we have to use partial fraction expansion for y'+5.2y=19.4sin2t, and y(0)=0
We know that,
y'+5.2y=19.4sin2t, and y(0)=0
L(y(t))= y(s)
L(y'(t)) = sy(s) and
L(sin2t) = s/s²+4 ------->equation(1)
Thus if we take Laplace transform on both sides of the given equation
L(y'+5.2y)=L(19.4sin2t)
sy(s)-y(0)+5.2y(s) = 19.4L(sin2t)
y(s)[s+5.2]=19.4[s/s²+4] (from equation(1))
y(s)= 19.4[s/(s²+4)(s+5.2)]
Breaking left hand side into partial precautions,
y(s) = 3/5(1/s+5.2)-3/5(s/s²+4)-9/5(s/s²+4)
Taking inverse Laplace transform on both sides,
So,
y(t)= 3/5[tex]e^{-5.2}[/tex]-3/5esin2t-9/5sin2t
Therefore, y(t)= 3/5[tex]e^{-5.2}[/tex]-3/5esin2t-9/5sin2t when the IVP's is y'+5.2y=19.4sin2t, and y(0)=0 by Laplace transform and partial fraction.
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Drag each intermediary to its correct definition. Disintermediation Reintermediation Intermediaries Cybermediation Agents, software, or businesses that provide a trading infrastructure to bring buyers and sellers together Occurs when a business sells directly to the customer online and cuts out the intermediary Steps are added to the value chain as new players find ways to add value to the business process The creation of new kinds of intermediaries that simply could not have existed before the advent of ebusiness
Answer:
its provide cretion nt exit e business to advet
How do you prove two triangles are similar using transformations?
To see if the two triangles are similar, you first have to get them both in the same direction. You do this by rotating (turning) one shape to align with the other. Such a transformation is called a rotation.
Geometry transformations are movements of two-dimensional shapes in two dimensions, or within their plane. Like restricted game pieces on a game board, you can move two-dimensional shapes in only three ways:
Rotation -- Shapes are rotated or turned around an axisReflection -- Shapes are flipped across an imaginary line to make mirror imagesTranslation -- Shapes are slid across the planeTransform one of the triangles such that the orientation of the triangles is the same.
Identify the corresponding sides of the triangles and write each set of corresponding sides as a ratio. If the ratios of the side lengths are the same for all 3 sides, then the triangles are similar.
Hence, by performing two rigid motions, we can prove the congruence between these two triangles. Now once the triangles are superimposed, we can identify the corresponding parts by checking which side and angle of the initial triangle is superimposed on the transformed triangle.
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Solve pls brainliest
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The acceleration of segment d is m/s2. rank segments a, b, and c from least acceleration to greatest acceleration. least: greatest:
Calculating acceleration involves dividing the change in velocity by the change in time.. The equation is a = (v-u)/t, where a is acceleration, v is the final velocity, u is the initial velocity, and t is the time.
Acceleration is the rate of change in velocity over time. To calculate acceleration, you need to know the initial velocity (u), the final velocity (v), and the time (t) of the motion. The formula for calculating acceleration is a = (v-u)/t. To use this formula, subtract the initial velocity from the final velocity and divide by the time. This will give you the acceleration. For example, if the initial velocity is 10 m/s, the final velocity is 20 m/s, and the time is 2s, then the acceleration is (20-10)/2 = 10 m/s2. This means that the velocity has increased by 10 m/s every second.If the initial velocity is 10 m/s, the final velocity is 20 m/s, and the time is 2s, then the acceleration is (20-10)/2 = 10 m/s2.
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Answer:
1st Q: -20
2nd Q: C,B,A
Step-by-step explanation:
Its correct
What does ∆ mean in math?
Answer:
As far as I know, in math, Δ means:
triangledelta (slope)Examples:
ΔABC ≅ ΔDEF
"triangle ABC is congruent to triangle DEF"Δ = [tex]\frac{1}{2}[/tex]
"delta equals 1 over 2"Delta is another word for "slope""the slope equals 1 over 2"i need help with these
1. Your shooting percentage was 60%.
2. Your friend made 24 shots.
3. Hi Lowe's shooting percentage was 71%.
4. Julie Shootsalott made 17 shots.
5. Swish Malone's shooting percentage was 89%.
6. Slammin' Sammy made 19 shots.
7. Lightning Lizzy's shooting percentage was 94%.
8. The team's shooting percentage was 65%.
1. To calculate the shooting percentage,
divide the number of shots made (i.e..12) by the total number of shots taken(i.e.20)
12/20 = 0.6
Multiply this by 100 to get the percentage, which is 60%.
2. To calculate no. of shots,
multiply the shooting percentage (i.e.60%) by the total number of shots taken (i.e.40)
0.6 x 40 = 24
3. To calculate Hi Lowe's shooting percentage,
divide the number of shots made (i.e.25) by the total number of photos taken (i.e.35).
25/35 = 0.714
Multiply this by 100 to get the percentage,
which is 71.4%
4. To calculate Julie Shootsalott's shots,
multiply the shooting percentage (34%) by the total number of shots taken (50).
0.34 x 50 = 17
5. Swish Malone's shooting percentage was 89.3%. To calculate this, divide the number of shots made (i.e.25) by the total number of photos taken (i.e.28).
25/28 = 0.893
Multiply this by 100 to get the percentage,
which is 89.3%
6. To calculate Slammin' Sammy shots,
multiply the shooting percentage (i.e.95%) by the total number of shots taken (i.e.20)
0.95 x 20 = 19
7. To calculate Lightning Lizzy's shooting percentage,
divide the number of shots made (i.e.34) by the total number of shots taken (i,e,36)
34/36 = 0.944
Multiply this by 100 to get the percentage,
which is 94.4%
8. To calculate The team's shooting percentage,
divide the number of shots made (i.e.44) by the total number of shots taken (i.e.68). 44/68 = 0.647
Multiply this by 100 to get the percentage,
which is 64.7%.
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Bones brothers & associates prepare individual tax returns. Over prior years, bones brothers has maintained careful records regarding the time to prepare a return. The mean time to prepare a return is 90 minutes, and the population standard deviation of this distribution is 15 minutes. Suppose 100 returns from this year are selected and analyzed regarding the preparation time. What is the probability that the mean time for the sample of 100 returns for this year is greater than 92?.
The probability that the mean time for the sample of 100 returns for this year is greater than 92 is 0.0764.
The term probability in math refers the possibility of occurring the event.
Here we have given that Bones brothers & associates prepare individual tax returns. Over prior years, bones brothers has maintained careful records regarding the time to prepare a return.
And we need to find the probability that the mean time for the sample of 100 returns for this year is greater than 92
As per the given question, we know the following values,
=> mean,μ=90
=> population standard deviation,σ=14
=> sample size ,n=100
Then the standard error is 1.4
Therefore, the z value is obtained by the z table is 0.9236.
Then the resulting probability is calculated.
=> 1 - 0.9236
=> 0.0764
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What is the value of x in the sequence 20 10 10 15 30 75 x?
The value of x in the given sequence is found as 225.
Explain the term sequence?A list containing numbers in a specific order is called a sequence. A term is the name given to each number in such a sequence. A sequence has a place for each phrase (first, second, third and so on). The length of a sequence is the number of arranged elements (potentially infinite).20/2 = 10 (0.5)
10x10 = 1 (1)
10x1.5 = 15 (1.5)
15x2 = 30 (2)
30x2.5 = 75 (2.5)
There is an increase of 0.5 in each term and then multiplies it with the previous term.
So,
{ 2.5 + 0.5 = 3 }
75x3 = 225 (3)
Thus, the value of the x in the given sequence is found as 225.
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What is the nature of the roots of x2 6x 9 0?
The value of the Discriminant is 0 so we have equal roots = b/2a = 6/2 = 3
The given equation is a quadratic equation.
x^2+6x+9 = 0.
To tell the nature of the roots we have to understand how to find the root of a quadratic equation.
The roots of a quadratic equation are the values of the variables that fulfil the equation. In other words, if f(α) = 0, then x = α is a root of the quadratic equation f(x).
The term Discriminant is used in quadratic equations.
The discriminant of a quadratic equation is the expression [tex](b^2 - 4ac)[/tex] in the quadratic formula. A quadratic equation's discriminant shows the nature of roots.
Discriminant important thing is that it is used to find the nature of roots.
x^2+6x+9 = 0. a =1, b = 6, c=9
discriminant d = b^2 - 4*a*c = 36-4*1*9 = 36-36 = 0
So the value of the discriminant is 0 so we have equal roots = b/2a = 6/2 = 3
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A model rocket is shot upward at a rate of 230 ft/sec. The height of the rocket, in feet, after t seconds
is given by the formula h=16t² + 230t. Answer the following question and round to the nearest tenth where applicable. Be sure to show all of your algebraic work and formulas.
When will the rocket reach its highest point?
Answer:
Step-by-step explanation:
given the formula
[tex]h(t)=16t^{2}+230t[/tex]
Find the axis of symmetry
[tex]t=\frac{-b}{2a}[/tex]
Notice the function is of the form
[tex]h(t)=at^{2}+bt[/tex]
where
[tex]a=16, b=230[/tex]
To find the time at which the rocket reached its highest point, use the formula for the axis of symmetry. Substitute 16 for a and 230 for b. This gives you,
[tex]t=\frac{-(230)}{2(16)}[/tex]
simplify the arithmetic expression..
[tex]t=\frac{-230}{32}[/tex]
[tex]t[/tex]≈-7.2 seconds
I THINK the answer should be -7.2 seconds but I do not know how this would really play into the problem being that seconds should be positive, not negative... hope this helps!
How many years would it take a 6000$ loan at 5% to earn 12000$ in interest?
Answer: 15 years
Step-by-step explanation:
6000 x 1.05^x = 12000
1.05^x=2
log(1.05^x)=log(2)(Take log of both sides)
x*(log(1.05))=log(2)
x= log(2)
log(1.05)
x=14.206699
15 years
What is the probability that a 3 will be drawn out of a deck of 52 cards?
Answer:
1/13
Step-by-step explanation:
There are 4 cards that are a "3" in a deck of cards
3 of hearts, 3 of spades, 3 of clubs, 3 of diamonds
P( 3) = number of cards that are a "3" / total number of cards
P(3) = 4/52 = 1/13
Answer:
[tex]\frac{1}{13}[/tex] (one thirteenth)
1/13
Step-by-step explanation:
There's 4 of the card 3 in a deck. (3 of hearts, 3 of diamonds, 3 of clubs, 3 of spades.)
This means 4 of 52 cards are a 3, which makes the fraction [tex]\frac{4}{52}[/tex]. This is the probability.
This fraction can be simplified to [tex]\frac{1}{13}[/tex] by dividing the top and bottom by four. This is the final answer, since it is in the simplest form.
Please help me quick it is geometry
What is the solution of simultaneous equations 5x 3y 8 and 3x Y 2?
The solution of simultaneous equations 5x-3y=8 and 3x+Y=2 is (1,-1)
A linear equation is an equation in which each term has at max one degree. Linear equations in variables x and y can be written in the form y=mx+c
Linear equation with two variables, when graphed on the cartesian plane with axes of those variables, give a straight line.
The given equations are:
5x-3y=8 ------(1)
3x+y=2 -------(2)
By solving the equations we get the points
Multiply equation(1) by 3 and equation (2) by 5 then we get
15x-9y=24
15x+5y=10
By solving the above equations we get x and y values.
-14y=14
y=-1
substitute y=-1 in equation(1) we get x value.
5x-3(-1)=8
5x=5
x=1
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Solve 2x²-8=0
A. X = 4
B. X = 2
C. x = 4, -4
D.x = 2, -2
Answer:
D. x=2, -2
Step-by-step explanation:
I think you're asking for the correct answer to the problem because all of the values for x won't work for the equation.
So all we have to do is simplify the equation to isolate x.
[tex]2x^2-8=0[/tex] (original problem)
[tex]2x^2=8[/tex] (add 8 to both sides)
[tex]x^2=4[/tex] (divide both sides by 2)
[tex]x=2, x=-2[/tex] (find the square root of both sides)
Note that x = ±2 and note just 2 because both [tex]2^2[/tex] and [tex](-2)^2[/tex] [tex]=4[/tex].
Hope this helped :D
What is the meaning of AAA theorem?
If three angles of one triangle are congruent to three angles of another triangle then by the AAA similarity theorem, the two triangles are similar. Actually, you need only two angles of one triangle being congruent to two angle of the second triangle.
How to prove AAA Theorem?
Specifying three angles A, B, and C does not uniquely define a triangle, but any two triangles with the same angles are similar. Specifying two angles of a triangle automatically gives the third since the sum of angles in a triangle sums to 180 degrees (pi radians), i.e.,
C = π - A - B .
Prove:
Triangles ABC and DEF such that ∠A = ∠D; ∠B = ∠E; ∠C = ∠F
Δ ABC ~ ΔDEF
Construction : We mark point P on the line DE and Q on the line DF such that AB = DP and AC = DQ, we join PQ.
There are three cases :
Case ( i ) : AB = DE, thus P coincides with E.
⇒ AB = DE, BC = EF and AC = DF
Consequently, Q coincides with F.
AB BC CA
---- = ------ = ------
DE EF FA
Since the corresponding angles are equal, we conclude that Δ ABC ~ Δ DEF.
Case( ii ) : AB < DE. Then P lies in DE
In triangles ABC and DPQ,
Statements ------ Reasons
1) AB = DP- 1) By construction
2) ∠A = ∠D -2) Given
3) AC = DQ -3) By construction
4) ΔABC ≅ ΔDPQ -4) By SAS postulate
5) ∠B = ∠DPQ 5) CPCTC
6) ∠B = ∠E- 6) Given
7) ∠E = ∠DPQ- 7) By transitive property
( from above)
8) PQ || EF -8) If two corresponding angles are congruent then the lines are parallel
9) DP/DE = DQ/DF -9) By basic proportionality theorem
10) AB/DE = BC/EF- 10) By construction
11) AB/DE = AC/DF- 11) By substitution property
12) Δ ABC ~ Δ DEF - 12) By SAS postulate
Learn more about AAA theorem here;
https://brainly.com/question/30108144
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