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;
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3х^2 +7x —6 = 0
salve for x write both solutions
Answer:
x=2/3,-3,x=0.6 repeating decimals,-3
Step-by-step explanation:
Determine how long it will take for $4000 to quadruple in a bank account earning 3.57% interest compounded continuously. Round to the nearest year
Answer:
12 months, so 1 year
Step-by-step explanation:
James has a sidewalk that runs through his flower garden. He needs to put more mulch in the flower garden. If one bag of mulch covers 4 square yards, how many bags of mulch will he need?
Answer:
See Explanation
Step-by-step explanation:
Given
The given proportion can be represented as:
[tex]1\ bag = 4yd^2[/tex]
Required
The additional number of mulch needed
The question has incomplete details, as the amount of mulch he has applied and the area of the flower garden are not given.
So, I will make some assumptions to solve this question.
Let the area of his flower garden be x.
So, we have:
[tex]1\ bag = 4yd^2[/tex]
[tex]y = x\ yd^2[/tex]
Cross multiply
[tex]4yd^2 * y = xyd^2 * 1\ bag[/tex]
Make y the subject
[tex]y = \frac{x\ yd^2 * 1\ bag}{4yd^2}[/tex]
[tex]y = \frac{x* 1\ bag}{4}[/tex]
[tex]y = \frac{x}{4}\ bag[/tex] --- This represents the total amount of mulch required for the flower garden
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Assumptions
Assume that [tex]x = 16[/tex],
So, we have:
[tex]y = \frac{x}{4}\ bag[/tex]
[tex]y = \frac{16}{4}[/tex]
[tex]y = 4[/tex]
i.e 4 bags are needed.
Assume he has applied 2 bags already.
The number of bags needed to apply will be:
[tex]y - 2[/tex]
[tex]y - 2 = 4 - 2[/tex]
[tex]y - 2 = 2[/tex]
2 more bags needed
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If x=12 and y=6x , what is the value of 8x2y ?
Answer:
13824 is the answer
Step-by-step explanation:
8*12*2*6*12
help please not to sure what it can be!
Answer:
Y=0.5X+5
Step-by-step explanation:
Hope this helps ;)
If DE is the midsegment of ABC what is the value of x AND y?
Answer:
3.) x = 4.5
4.) y = 7
Step-by-step explanation:
3 is an easier one than 4. By the triangle midsegment theorem, the midsegment of a triangle will be half of the base if it is parallel to it. AB is our base, so the equation is like this!
[tex]AB = 2(ED)\\9=2(ED)\\4.5=ED[/tex]
Now for 4, it actually is pretty easy. By the same triangle midsegment theorem, we automatically know that EC and EA are congruent, and CD and DB are also congruent. Meaning y equals 7.
But another way we can determine this (the long, kinda unecessary, but useful way) is through examining the relationship that triangles ABC and EDC have! We know that they are similar triangles by SAS or AA. Meaning, their equation looks something like this...
[tex]4.5(7+y) = 9y\\31.5 + 4.5y = 9y\\31.5=4.5y\\7=y[/tex]
Like I said earlier, 7 is equal to y, but I figured I'd prove it so we are on the same page. Nothing wrong with double-checking ;)
Have a nice day. Feel free to Ask questions. Spread The Love.
5 more than the product of x and 2
Answer:
2x+5
Step-by-step explanation:
Answer:
5(2x) => 10x
Step-by-step explanation:
ice cream cost $0.85 per ounce. Which equation best represents y, the total cost of x ounces of ice cream.
Answer:
I'm not sure about my answer but I think it is y=0.85x
Step-by-step explanation:
0.85 is the rate and the rate = the cost of ounces of ice cream which is why 0.85 has an x to it
The product of two numbers is 144, but their sum is -24. What are these two numbers?
Answer:
The two numbers are - 12 and - 12 .
Step-by-step explanation:
Let the two numbers be x and y then
By the question
x * y = 144
or x = 144 / y ....... I)
Again
x + y = - 24
Or 144/y + y = - 24
or, ( 144 + y²) / y = - 24
or, 144 + y² = - 24y
y² + 24y + 144 = 0
y² + ( 12 + 12 ) y + 144 =0
y² + 12y + 12y + 144 = 0
y(y + 12 ) + 12 ( y + 14) = 0
( y + 12 ) ( y + 12 ) = 0
Either
y + 12 =0
y = - 12
Now
x = 144 / - 12 = - 12
Therefore the two numbers are - 12 and - 12 .
Hope it will help :)❤
At Paco’s Pizza Shop, a large pizza has a diameter of 16 inches. A small pizza has a diameter of 12 inches. What is the area of the small pizza? What is the area of the large pizza? Round to the nearest tenth.
what type of association is there ??
hi i just need some points rq tyStep-by-step explanation:
A box contains 1,200 marbles. One-fifth are blue, two-thirds are red, and the rest are white. How many of the marbles are white?
Answer:
7/15
Step-by-step explanation:
2/3. 1/5 3x5 =15 then add 2+5=7 which gets u 7/15
There are rainy days, sunny days, and cloudy days.
- September and October had the same number of rainy days.
- There are 10 rainy days in total for both months.
- There were 3 more cloudy days in October then in September.
How many sunny days were there in September?
Answer:
3 more sunny days in September then in October
Solve for the unknown value
QUICKKK!!!!! PLEASEEEEEEEEEE
Answer:
10.29 (rounded = 10.30)
Step-by-step explanation:
X = a^2 + b^2 = c^2
X = 9^2 + 5^2 = c^2
X = (81) + (25) = c^2
X = 106 = c^2
X = √106 = c^2
X = 10.295630141 = c^2
X = 10.03 = c^2 (10.03 is 10.295630141 rounded to the nearest tenth)
Select ALL the expressions that are equivalent to (x^2)(square root of x).
Answer:
x^(2.5) ("x to the power 5/2")
Step-by-step explanation:
(x^2)(√x) can also be written as x·x·x^(1/2). We just add the exponents together: (x^2)(√x) = x^1·x^1·x^(1/2) = x^(2.5) ("x to the power 5/2").
Next time, please share the answer choices.
Candice is collecting donations from her neighbors for the local animal shelter. She started by adding some of her own money into the collection jar. After collecting d more dollars in donations, she will have a total of 10+d dollars in the jar. Today, she collected $36 more in donations.
How much money does Candice have in her collection jar? plz help
plz help me 2 plz 2 plz 2
Step-by-step explanation:
P(-4,7)
Q(-1,-1)
R(-8,-1)
Select two equations of lines that are perpendicular
to the line whose equation is 2x + 14y = 70.
Answer: I know C it’s one answer
Step-by-step explanation:
A recent survey of 800 seventh graders from across the state shows 4 out there of 10 like to play paper football. How many students said they like to play paper football?
Answer:
320
Step-by-step explanation:
800 = 10
x = 4
800 x 4 = 10 x
3200= 10x
x = 320
Which statement about the data represented in the graph appears to be de
A)
100% of employees hold a master's degree.
B)
69% of employees hold a bachelor's degree.
44% of employees hold a bachelor's degree.
D)
31% of employees hold a PhD.
Answer:
A. 100% of employees hold a master's degree.
Step-by-step explanation:
Sana nakatulong
could someone help me with this problem?
Answer:
The question already got answered: https://brainly.com/question/23062055
Answer:
1/2
Step-by-step explanation:
The slope of the line is going down 1/2 at a time
Joyce wants to mix granola
Answer:
good for him or her can u help me with my math question
Step-by-step explanation:
I got the rest done and now my brain Isn't cooperating, h e l p
1 x= 2
2 x= 16
3 x= 12
Heart or brainly me!
the square below has sides that are 3 1/2 inches long. what is the perimeter of the square? write an addition equation to solve.
Help me Tysm pls ;Dty again
Answer:
its a
Step-by-step explanation: i took the test
Callie brushes her teeth three times each day. She leaves the water running. Is it reasonable to say that she uses 2 cups of water in one day? Explain
Answer: She uses more than 2 cups of water per day.
Step-by-step explanation:
It takes 2 minutes to brush your teeth and if you times that by 3 you get 6 so she spends 6 minuntes per day brushing her teeth and water runs fast. so if your leaving the water on for 6 mintues your filling up much more than 2 cups
Point B has coordinates (4,1). The x-coordinate of point A is -4. The distance between point A and point B is 10 units. What are the possible coordinates of point A?
Given:
Point B has coordinates (4,1).
The x-coordinate of point A is -4.
The distance between point A and point B is 10 units.
To find:
The possible coordinates of point A.
Solution:
Let the y-coordinate of point A be y. Then the two points are A(-4,y) and B(4,1).
Distance formula:
[tex]D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
The distance between point A and point B is 10 units.
[tex]\sqrt{(4-(-4))^2+(1-y)^2}=10[/tex]
Taking square on both sides, we get
[tex](8)^2+(1-y)^2=100[/tex]
[tex](1-y)^2=100-64[/tex]
[tex](1-y)^2=36[/tex]
Taking square root on both sides, we get
[tex](1-y)=\pm \sqrt{36}[/tex]
[tex]-y=\pm 6-1[/tex]
[tex]y=1\mp 6[/tex]
[tex]y=1-6[/tex] and [tex]y=1+6[/tex]
[tex]y=-5[/tex] and [tex]y=7[/tex]
Therefore, the possible coordinates of point A are either (-4,-5) or (-4,7).
Simplify the following trigonometric expression. cosθtan²θcotθ
This question requires the understanding of trigonometric identities. I'll list the ones that are needed to answer this question:
cotθ = 1 / tanθ
tanθ = sinθ / cosθ
Look at the latter identity. We can rewrite it as such:
tanθ * cosθ = sinθ
We are given the expression:
cosθ * tan²θ * cotθ
Referring to the first trigonometric identity above, substitute cotθ:
cosθ * tan²θ * (1 / tanθ)
Multiply tan²θ and the fraction:
cosθ * (tan²θ / tanθ)
cosθ * tanθ
Refer to the second trigonometric identity:
cosθ * tanθ
sinθ
cosθ * tan²θ * cotθ = sinθ
If you have any questions about how I got to the answer, feel free to ask
- breezyツ
slope between the points (5, 1) and (7,
-9)
Answer:
Slope : 5x
Step-by-step explanation: