Answer:
x⁶ +30x⁵ +375x⁴ +2500x³ +9375x² +18750x +15625
Step-by-step explanation:
The expansion is the sum of C(6, k)·x^(6-k)·5^k for k=0–6, where ...
... C(6, k) = 6!/(k!(6-k)!)
For k = 0–6, C(6, k) = {1, 6, 15, 20, 15, 6, 1}
Then the expansion is ...
... x⁶ +6·5¹·x⁵ +15·5²·x⁴ +20·5³·x³ +15·5⁴·x² +6·5⁵·x +5⁶
Using binomial theorem to expand the power of a binomial (X+5) to the power 6 and the result is [tex]X^6 + 30 \times X^5 + 300 \times X^4 + 1500 \times X^3 + 3750 \times X^2 + 4500 \times X + 3125[/tex].
To expand the expression (X+5) to the power 6 using the binomial theorem, we can use the formula [tex](a + b)^n = nC_0 \times a^n + nC_1 \times a^{(n-1)} \times b^1 + nC_2 \times a^{(n-2)} \times b^2 + ... + nC_n * b^n[/tex].
In this case, a = X, b = 5, and n = 6.
Using the binomial coefficients, the expanded expression becomes:
[tex]X^6 + 6 \times X^5 \times 5 + 15 \timesX^4 \times 5^2 + 20 \times X^3 \times 5^3 + 15 \times X^2 \times 5^4 + 6 \times X \times 5^5 + 5^6[/tex]
Simplifying this expression gives
[tex]X^6 + 30 \times X^5 + 300 \times X^4 + 1500 \times X^3 + 3750 \times X^2 + 4500 \times X + 3125[/tex].
Which postulate or theorem proves ∆MNQ ≅ ∆PNQ ?
SSS Congruence Postulate
AAS Congruence Postulate
ASA Congruence Postulate
AAS Congruence Postulate proves that ∆MNQ ≅ ∆PNQ
What is congruency?When two triangles have shape and sizes then the triangles are said to be congruent.There are 5 types of congruencies. They are SAS, AAS, ASA , SSS , RHS.How to know which postulate or theorem proves ∆MNQ ≅ ∆PNQ ?Considering ΔMNQ and ΔPNQ
∠QMN = ∠QPN (given)
∠QNM = ∠ QNP ( both are right angles)
Side NQ is common
∴ ΔMNQ and ΔPNQ are congruent by AAS congruency since two angles and a side are considered to prove the congruency.
The side is not included between the angles and hence ASA congruency is not the answer.The correct answer is option B. AAS congruency postulates.
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please hurry thank you
Answer:
7.9 ft
Step-by-step explanation:
The hypotenuse is the side opposite of the right angle, hence the hypotenuse for this triangle is 11 ft.
Side [tex]x[/tex] is adjacent to the angle given. So we need adjacent and we have hypotenuse. Which trigonometric ratio relates adjacent and hypotenuse? It is COSINE.
Thus, we can write:
[tex]cos(A)=\frac{adjacent}{hypotenuse}\\cos(44)=\frac{x}{11}[/tex]
Cross multiplying, we can solve for [tex]x[/tex]:
[tex]cos(44)=\frac{x}{11}\\x=cos(44)*11\\x=7.91[/tex]
Rounding to nearest tenth, we have [tex]x=7.9[/tex] ft.
A number is doubled and then 1 is added to it. The answer is divided by 5 and then increased by 16. The final result is 18.
A)Develop an equation that shows the step above
Equation:___________________
B) Solve the equation to determine the value of the number.
Solve the equation. Show your work
Answer: The number is___________
So let's break down this sentence (Let n = unknown number):
"A number is doubled and then 1 is added to it"; Remember that double means multiplied by 2. With this sentence, we can determine that "2n + 1" is a part of the equation."The answer is divided by 5, and then increased by 16"; The "answer" they refer to is "2n + 1" from the prior sentence. Since this is divided by 5 and then added by 16, we can determine that [tex]\frac{2n+1}{5}+16[/tex] is a part of our equation."The final result is 18"; This means that the prior part of the equation is equal (=) to 18. With this info, our full equation is [tex]\frac{2n+1}{5}+16=18[/tex]B.Now, let's solve our prior equation found in A. To solve for the unknown number, n, we need to isolate the variable onto 1 side of the equation. Firstly subtract both sides by 16 to cancel out the + 16 on the left side:
[tex]\frac{2n+1}{5}=2[/tex]
Next, multiply both sides by 5 to cancel out the division on the left side:
[tex]2n+1=10[/tex]
Next, subtract both sides by 1 to cancel out the + 1:
[tex]2n=9[/tex]
Lastly, divide both sides by 2 to cancel out the multiplication:
[tex]n=\frac{9}{2}\ \textsf{OR}\ 4.5[/tex]
In short, the number is 9/2 or 4.5.
What is the fraction in decimal form?
21/6
Enter your answer in the box
Answer: Your answer would come out to be 3.5
Step-by-step explanation: it would be 3.5 due to the fact that you have to break these numbers down to the smallest one possible meaning simplify which would result in this equation 7 over 2
7 over 2 = 3 1 over 2 this equals 3.5
1# Find the missing variable (s). Round to the nearest tenth.
Answer:
7.9
Step-by-step explanation:
The mnemonic SOH CAH TOA reminds you of the relationship of the hypotenuse and the side opposite an angle:
... Sin = Opposite/Hypotenuse
For your geometry, this is ...
... sin(21°) = x/22
Multiplying by 22 gives ...
... x = 22·sin(21°) ≈ 7.9
The equation C = 5/9(F - 32) is used to convert between Fahrenheit temperatures and Celsius temperatures. The temperature in Augusta on a summer day is 33° C.
Rounded to the nearest whole number, what is the temperature in Fahrenheit?
A) 55°F
B) 68°F
C) 79°F
D) 91°F
Answer:
D) 91 °F
Step-by-step explanation:
C = 5/9(F - 32)
33 = 5/9 (F – 32) Multiply each side by 9
9×33 = 5(F- 32) Divide each side by 5
9×33/5 = F – 32 Add 32 to each side
F = 9×33/5 + 32
F = 59.4 + 32
F = 91 °F
Answer:
Its D
Step-by-step explanation:
USA TP
please help!!!
A,B,C and D are four towns.
B is 30km due East of A
C is 30km due North of A
D is 45km due South of A
calculate the bearing of D from B
Answer:
213.7°
Step-by-step explanation:
The bearing of B from D is angle ADB, which is ...
... ∠ADB = arctan(30/45) ≈ 33.7°
The bearing of D from B is that angle plus 180°, so is ...
... 180° +33.7° = 213.7°
_____
We use angle ADB because the reference direction for bearings is North.
The tangent of an angle (ADB) is the ratio of its opposite side (AB) to its adjacent side (AD).
You are tired at the end of the term and decide to borrow $500 to go on a trip to whatever land. You go to the bank and borrow the money at 11% for 2 years. A) find the interest you will pay on the loan. B) how much will you have to pay the bank at the end of the two years ?
Answer:
A)55
B)110
Step-by-step explanation:500 /11%=55
55*2=110
The interest is $110 and the amount paid to the bank is $610.
What is simple interest?Simple interest is calculated just on the loan's principal amount, whereas compound interest is calculated on both the principal and the cumulative interest.
It is given that you decide to borrow $500 to go on a trip to whatever land. You go to the bank and borrow the money at 11% for 2 years.
A. The simple interest is,
SI = ( Principle x rate x time ) / 100
SI = ( 500 x 11 x 2 ) / 100
SI = $110
B. The amount paid to the bank after 2 years is,
Amount = P + SI
Amount = $500 + $110 = $610
Therefore, the interest is $110 and the amount paid to the bank is $610.
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I need help fast if you can PLEASE HELP
Answer:
333°
Step-by-step explanation:
Minor arc ST is the difference between minor arc RT and minor arc RS:
... 153° -126° = 27°
Major arc SPT is the remainder of the circle:
... 360° -27° = 333°
Given the following figure, what is the measure of angle B?
Answer:
132°
Step-by-step explanation:
Sum of the interior angles of a polygon is given by the formula:
[tex](n-2)*180[/tex]
Where [tex]n[/tex] is the number of sides of the polygon.
The figure shows a 5 sided polygon (aka Pentagon). So the sum of all the angles is:
Sum = [tex](5-2)*180\\=3*180\\=540[/tex]
If we add up all the expressions/angles given and equate to 540, we can figure out [tex]x[/tex]. So:
[tex]84+13x+6+9x-16+92+12x=540\\34x+166=540\\34x=540-166\\34x=374\\x=\frac{374}{34}=11[/tex]
But angle B measures [tex]12x[/tex], so plugging in [tex]x=11[/tex], we have:
Angle B = [tex]12x=12(11)=132[/tex]°
Second answer is right.
(2.)are NOT opposite reciprocals
(1.)are NOT equal
(2.) are NOT opposite reciprocals
REALLY NEED HELP WITH BOTH PLEASE FOR GRADE‼️‼️‼️‼️‼️‼️‼️‼️‼️‼️‼️
The slope of AB is ...
... slope = (change in y)/(change in x) = (-6-(-3))/(8-2) = -3/6 = -1/2
The slope of AD is ...
... slope = (change in y)/(change in x) = (0 -(-3))/(8 -2) = 3/6 = 1/2
We note that the differences have the same magnitudes in both cases (y changes by 6; x changes by 3). This tells us the line segments are the same length. (Each side of the figure is equivalent to the hypotenuse of a right triangle with legs 3 and 6.)
The opposite reciprocal of a number is found by dividing -1 by that number. Thus the opposite reciprocal of the slope of AB is ...
... -1/(-1/2) = (-1/1)·(-2/1) = 2
This value is not the same as the slope of AD, so the slopes are NOT opposite reciprocals.
_____
Plotting the points, we see that we have a rhombus, not a rectangle (or square).
please help me asap!
Julie wants to make a handmade Valentines she buys 1/2 yard of red ribbon and 3/4 yard of pink if each Valentine needs 1/8 yards how many can she create please show you work will Mark rainiest and we'll add you to help suow work
Answer:
10 Valentines
Step-by-step explanation:
1/2 = 4/8
3/4 = 6/8
Julie has a total of 4/8 + 6/8 = 10/8 yards of ribbon. If color doesn't matter, she can make 10 Valentines.
_____
10/8 = 10 × 1/8
Suppose a fair six-sided is rolled three times in a row. The probability of rolling a number less than 3 three times in a row is __________ . Express the probability as a fraction.
Answer:
1/27
Step-by-step explanation:
The probabaility of rolling a number less than 3 is 2/6 = 1/3, since only two of the six numbers are less than 3.
Assuming the die rolls are independent, the probability of doing this 3 times is the product of the probabilities of doing it each time:
... (1/3)^3 = 1/27
If m ABC is 53 degrees then m EDF is ?
Answer:
74°
Step-by-step explanation:
The triangle is isosceles and the congruent base angles are 53°. The sum of angles in a triangle is 180°, so the apex angle (∠EDF) will be ...
... 180° - 2×53° = 74°
Answer:
74* is the answer
Step-by-step explanation:
#PLATOFAM
#PLATOLIVESMATTER
#PLATOFAMILY
Write and solve a polynomial equation for the situation described.
A rectangular two-story horse barn is
being designed for a farm. The upper floor will be used
for storing hay, and the lower floor will have horse stalls
that extend 5 feet from both of the longer walls. The barn’s
length is twice the barn’s width, and the lower floor’s
ceiling height is 6 feet less than the barn’s width. What
should the dimensions of the lower floor be if the space
not used for stalls is to have a volume of 1920 cubic feet?
We can let x represent the width of the barn in feet. Then the length of the barn is 2x. The portion of width not used for stalls is (x-10). The height of that is (x-6).
So, the product of length, width not used for stalls, and height must be 1920 ft³. The equation for that is ...
... 2x(x -10)(x -6) = 1920 . . . . the equation
Solution
I generally find it convenient to use a graphing calculator to find the solutions to higher-order equations. Rearranging this so we're looking for a zero, it becomes ...
... 2x(x -10)(x -6) -1920 = 0
The graph shows one real root, at x = 16.
The dimensions of the lower floor should be 16 feet wide, 32 feet long, and a ceiling height of 10 feet.
_____
Alternate solution
Dividing by 2, we have
... x(x-6)(x-10) = 960
If we look at the factors of 960, we find that it is the product of 16, 10, and 6, which would let us assign x=16.
___
There are 28 divisors of 960. According to the Rational Root theorem, any of them might be a solution. There are arguments that can be made regarding reasonable dimensions that suggest x will be more than 12 (6 ft ceiling, 2 ft aisle). So, the number of trial-and-error possibilities is reduced somewhat.
The dimensions of the lower barn floor should be a width of 8 feet, a length of 16 feet, and a height of 2 feet, determined by setting up and solving a polynomial equation based on the given parameters.
Explanation:The problem can be solved using polynomial equations. First, let's denote the width of the lower barn floor as x. Given that the length of the barn is twice the width, the length is 2x. The height of the lower barn floor is 6 feet less than the width of the barn, so it's x - 6 feet.
The volume of a rectangular solid is given by length times width times height. So according to the information provided, we can set up the following polynomial equation: 2x * x * (x - 6) = 1920.
By simplifying the equation, we get 2x^3 - 12x^2 - 1920 = 0.
This equation can be solved using various methods such as the rational root theorem, synthetic division, or factoring. After solving, we find that the root x = 8 is the only physically meaningful solution (length width and height must be positive). Therefore, the width of the lower floor should be 8 feet, the length should be 2 * 8 = 16 feet, and the height should be 8 - 6 = 2 feet.
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Find a polynomial, which if substituted for M will make the equation an identity: (4c^2–7c^2+6)–M=0
Answer:
-3c^2 +6
Step-by-step explanation:
The equation will be an identity (0=0) when M is equal to the contents of the parentheses. That polynomial can be used directly, or it can be simplified by combining the c^2 terms.
The contents of parentheses simplify as follows:
... (4c^2 -7c^2 +6) = (c^2(4 -7) +6) = (-3c^2 +6)
Answer:
4c^4-7c^2+6
Step-by-step explanation:
Solve each of these systems using the substitution method. Make sure to check your answers !
2x + 3y = -4
x − 2y = 5
a. x = 1, y = -2
b. x = -1, y = 2
c. x = 1, y = 2
d. x = -2, y = -1
Answer:
a. x = 1, y = -2
Step-by-step explanation:
The given equations are 2x + 3y = -4 -------------(1)
x - 2y = 5 -----------------(2)
From the second equation, we get x = 2y + 5
Now plug in x = 2y + 5 in the first equation, we get
2(2y + 5) + 3y = - 4
4y + 10 + 3y = -4
7y + 10 = -4
7y = -14
y = -2
Now plug in y = -2 in x = 2y + 5 to find the x value.
x = 2(-2) + 5
x = -4 + 5
x = 1
Therefore, x = 1, y = -2
Thank you.
Ben has $5. In order to have money for a new tablet computer, Ben is going to sell his collection of stamps for $3 per stamp. Ben has decided to graph a line showing the relationship between the number of stamps he sells and the total amount of money he has. Which will be the y-intercept of Ben's graph?
Answer:
5
Step-by-step explanation:
It is 5 because no matter how many stamps he sells, he has $5.
Answer:
The y-intercept of Ben's graph is [tex]y = 5[/tex].
Step-by-step explanation:
The function for Ben's amount of money y, depending on the number of stamps sold x, is as follows:
[tex]y = ax + b[/tex]
In which a is the price of each stamp and b is his initial money.
Building the function:
Ben has $5. This means that if he sells no stamps, he is still going to have $5. So [tex]b = 5[/tex].
For each stamp sold, he gains $3. So if sells 1 stamp, he will have $8, if he sells 2, he will have $11. So [tex]a = 3[/tex].
The function is:
[tex]y = 3x + 5[/tex]
Which will be the y-intercept of Ben's graph?
This is the value of y when x = 0.
So
[tex]y = 3*0 + 5 = 5[/tex]
The y-intercept of Ben's graph is [tex]y = 5[/tex].
Camacho is buying a monster truck. The price of the truck is xx dollars, and he also has to pay a 13\%13% monster truck tax.
how much Camacho pays in total for the truck?
Answer:
[tex]1.13x[/tex]
Step-by-step explanation:
We have been given that Camacho is buying a monster truck. The price of the truck is x dollars, and he also has to pay a 13% monster truck tax. We are asked to find the total amount paid by Camacho for the monster truck.
The total cost of monster truck would be x plus 13% of x.
[tex]\text{The total cost of monster truck}=x+\frac{13}{100}x[/tex]
[tex]\text{The total cost of monster truck}=x+0.13x[/tex]
[tex]\text{The total cost of monster truck}=1.13x[/tex]
Therefore, Camacho paid [tex]1.13x[/tex] dollars in total for the truck.
Final answer:
To find the total cost of the monster truck, convert the 13\% tax to decimal form (0.13), multiply it by the price of the truck (xx dollars), then add the result to the original price to get the total cost, which is 1.13xx.
Explanation:
To determine the total amount Camacho will pay for the monster truck, we need to calculate the sales tax and add it to the original price of the truck. The sales tax is 13\%, which can be converted to a decimal for calculation purposes. First, convert 13\% to a decimal by dividing by 100, which gives us 0.13. Then multiply 0.13 by the price of the truck (xx dollars) to find the amount of tax Camacho will have to pay.
After calculating the tax, you can find the total cost by adding the tax to the original price of the truck.
Step 1: Convert 13\% to a decimal = 0.13
Step 2: Multiply the price of the truck (xx dollars) by 0.13 to find the tax = 0.13 * xx
Step 3: Add the tax to the original price to find the total cost: xx + (0.13 * xx).
Total cost = xx + 0.13xx = 1.13xx
please help me asap!
7. Law of cosines
a² = b² + c² - 2bc cos A
where a is the side opposite to ∠A and b and c are adjacent sides to ∠A.
Here, a=BC= 3.8
b=CA =4.8
c = AB = 3.2
Substituting the values in the formula,
3.8² = 4.8² + 3.2² - 2(4.8)(3.2) cos A
30.72 cos A =23.04 + 10.24 -14.44=18.84
cos A = 18.84/30.72 = 0.6133
∴ cos A =0.6133
8. It is a right angled triangle.
Pythagoras theorem
hypotenuse² = Perpendicular² + base²
Here, hypotenuse = 6.7, base = 5.4
Perpendicular² = 6.7² -5.4²=44.89 - 29.16 =15.73
Perpendicular = √15.73 = 3.966 = 3.97
∴ The length of the missing side is 3.97.
9. Given: Side of new cube = 5 * Side of original cube
Let the side of the original cube measure 1 unit.
So, volume of original cube = side³ = 1
The side of new cube = 5 * 1 =5 units
So, volume of new cube = side³ = 5³ =125
The volume of the bigger cube is 125 times larger than the original cube.
10. Given : Height of the cone = 7 in
Radius of the base of the cone = 2 in
Volume of cone = (1/3) πr²h
=(1/3)π(2²)(7)
=29.32 cubic inches
∴ Volume of the cone is 29.32 cubic inches.
11. The given equation is:
x² -18x +10 =0
(x² -18x + 9²) +10 -9² =0
(x-9)²-71 =0
(x-9)² =71
x-9 =√71
x = 9 + √71
∴ x = 9+√71
The formula for the area A of a trapezoid is A=1/2(b1+b2)h, where b1 and b2 represents the lengths of the bases and h represents the height. Which equation can be used to find the height of a trapezoid?
A. h=2A-b1-b2
B. h=2A/b1+b2
C. h=A(b1+b2)/2
D. h=A-2/b1-b2
Answer:
B. h=2A/(b1+b2) . . . . parentheses are needed
Step-by-step explanation:
Multiply the equation by the inverse of the coefficient of h. That coefficient is (b1 +b2)/2, so its inverse is 2/(b1 +b2). Parentheses are needed.
... 2A/(b1 +b2) = h
_____
Comment on the answer choices
As written, none of them are correct. The denominator can only be written without parentheses if the division bar is horizontal, and so serves also as a grouping symbol.
Please help if you can
Step-by-step explanation:
sqrt of 81 is 9 so rational
Sqrt of 89 is irrational
Sqrt of 121 is 11 so rational
Sqrt of 131 is irrational.
What is the apparent solution to the system of equations?
y=12x+2y=2x−1
y=12x+2y=2x−1
Graph the system of equations using the Line tool. Plot a point at the apparent solution to the system using the Point tool.
Answer:
(x, y) = (-0.3, -1.6)
Step-by-step explanation:
The graph below shows these equations graphed. Their point of intersection is (-0.3, -1.6).
_____
Analytical solution
You can equate the expressions for y to get ...
... 12x +2 = 2x -1
... 10x = -3 . . . . subtract (2x+2)
... x = -0.3 . . . . . divide by 10
... y = 2(-.3)-1 = -1.6 . . . . use the 2nd equation to find y.
GUYS HELP PLEASE!!!! I NEED THIS ASAP!!!!!!!
Answer:
slope = .4
She ran .4 laps per min.
She ran 12 laps in 30 minutes
Step-by-step explanation:
The slope is the rate of change. It can be found by using the formula
slope = (y2-y1)/(x2-x1)
The given points are (11, 4.4) (5.5 ,2.2)
slope = (2.2-4.4(/ (5.5-11)
= -2.2/-5.5
.4
The slope represents .4 laps per min since y is in laps and x is in minutes.
If she ran 30 minutes, we multiply the time by the slope to calculate the laps.
30 * .4 = 12
She ran 12 laps.
A) The slope is the change in Y over the change in X.
Using the 2 points given on the graph (11,4.4) and (5.5, 2.2)
The first number in the parenthesis are the X value and the 2nd numbers are the Y values.
Slope = (4.4 - 2.2) / (11-5.5)
= 2.2 / 5.5
= 0.4
B) From the slope, she ran 0.4 laps per minute.
C) To find the total laps, multiple laps per minute by total minutes:
30 minutes x 0.4 laps per minute = 12 laps.
1/3(y+7)=(y-1) y= what?
Answer:
y=5
Step-by-step explanation:
1/3(y+7)=(y-1)
I will multiply by 3 on each side to get rid of the fractions
3*1/3(y+7)=(y-1)*3
y+7 = 3*(y-1)
Distribute the 3
y+7 = 3y -3
Subtract y from each side
y-y+7 = 3y-y-3
7 = 2y-3
Add 3 to each side
7+3 = 2y-3+3
10=2y
Divide each side by 2
10/2 = 2y/2
5 =y
Bryan purchases a new boat. The expression below represents the value of the boat after x years. 22000(0.88)^(X+2)
Which statement is true?
The price of the boat decreases by 88% every year.
The price of the boat increases by 88% every year.
The price of the boat decreases by 12% every year.
The price of the boat increases by 12% every year.
None of the above. All the answers refer to price. The given equation refers to value. We imagine price may stay the same or actually increase year-to-year. The problem statement gives insufficient information to conclude anything about price.
Math is about attention to detail. The equation tells you ...
... The value of the boat decreases by 12% every year.
Step-by-step explanation:The value of the boat is multiplied by 0.88 each time X increases by 1. That means the value is 88% of what it was the year before, a decrease of 12%.
In trapezoid ABCD with bases AB and DC , diagonals intersect at point O. Find the length of diagonal BD , if BO=6 cm and: AO/OC = 3/1
Answer:
8 cm
Step-by-step explanation:
... BO/OD = AO/OC = 3/1
Written another way, this is ...
... OD : BO = 1 : 3
Now, BD = OD + BO, so we have
... BD : BO = (OD +BO) : BO = (1 +3) : 3 = 4 : 3
That is, BD = 4/3 × BO
... BD = 4/3 × 6 cm = 8 cm
To find the length of diagonal BD in trapezoid ABCD, we set up an equation using the fact that the diagonals bisect each other. We solve for x and then substitute it into the equation for BD.
Explanation:To find the length of diagonal BD in trapezoid ABCD, we can use the fact that the diagonals of a trapezoid bisect each other. Let AO = 3x and OC = x. Since the diagonals intersect at point O, we can set up the equation (3x + 6) / x = 3 / 1. Solving for x, we find that x = 1.5. Therefore, BD = 2 * x + 6 = 2 * 1.5 + 6 = 9 cm.
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Ally and Beth are at the grocery store. Ally buys a container of milk for $4 and 4.75 pounds of apples. Beth buys a box of cereal for $3 and 5.55 pounds of apples. If the girls spend the same amount, how much does 1 pound of apples cost?
Answer:
$1.25
Step-by-step explanation:
The sum of the measures of the angles shown is 63° . Which choice is the measure of angle y ?
Answer:
63
Step-by-step explanation:
Answer:
25 degrees.
Step-by-step If one of the angles is 38 degrees then in order for the sum to be 63, the answer is 25