1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Art [367]
3 years ago
8

3. Complete the proof below. Help please!!!

Mathematics
1 answer:
baherus [9]3 years ago
5 0

Answer:

The completed proof is presented as follows;

The two column proof is presented as follows;

Statements    {}                                               Reason

1. \overline {HI} ║ \overline {KL}, J is the midpoint of \overline {HL} {}         1. Given

2. ∠IHJ ≅ ∠JLK{}                                            2. Alternate angles are congruent

3. ∠IJH ≅ ∠KJL   {}                                         3. Vertically opposite angles

4.  \overline {HJ} ≅ \overline {JL}   {}                                              4. Definition of midpoint

5. ΔHIJ ≅ ΔLKJ  {}                                         5. By ASA rule of congruency

Step-by-step explanation:

Alternate angles formed by the crossing of the two parallel lines \overline {HI} and \overline {KL}, by the transversal \overline {HL} are equal

Vertically opposite angles formed by the crossing of two straight lines \overline {IK} and \overline {HL} are always equal

A midpoint divides a line into two equal halves

Angle-Side-Angle, ASA rule of congruency states that two triangles ΔHIJ and ΔLKJ, that have two congruent angles, ∠IHJ in ΔHIJ ≅ ∠JLK{} in ΔLKJ and ∠IJH in ΔHIJ ≅ ∠KJL in ΔLKJ, and that the included sides between the two congruent angles is also congruent \overline {HJ} ≅ \overline {JL}, then the two triangles are congruent, ΔHIJ ≅ ΔLKJ.

You might be interested in
PLEASE HELP: Lucille Richards earns $1,250.00 semimonthly and has the
rjkz [21]
The answer would be B. $387.50
4 0
3 years ago
A fruit basket contains seven apples, four bananas, and eight pears. If a piece of fruit is picked at random from the basket, wh
vekshin1

Answer:

1) 0.21

Step-by-step explanation:

This is because there are 4 bananas out of all 19 fruits. 4/19 is equal to 0.21052631578.

5 0
3 years ago
Find an integer x such that 0<=x<527 and x^37===3 mod 527
Greeley [361]
Since 527=17\times31, we have that

x^{37}\equiv3\mod{527}\implies\begin{cases}x^{37}\equiv3\mod{17}\\x^{37}\equiv3\mod{31}\end{cases}

By Fermat's little theorem, and the fact that 37=2(17)+3=1(31)+6, we know that

x^{37}\equiv(x^2)^{17}x^3\equiv x^5\mod{17}
x^{37}\equiv(x^1)^{31}x^6\equiv x^7\mod{31}

so we have

\begin{cases}x^5\equiv3\mod{17}\\x^7\equiv3\mod{31}\end{cases}

Consider the first case. By Fermat's little theorem, we know that

x^{17}\equiv x^{16}x\equiv x\mod{17}

so if we were to raise x^5 to the nth power such that

(x^5)^n\equiv x^{5n}\equiv x\mod{17}

we would need to choose n such that 5n\equiv1\mod{16} (because 16+1\equiv1\mod{16}). We can find such an n by applying the Euclidean algorithm:

16=3(5)+1
\implies1=16-3(5)
\implies16-3(5)\equiv-3(5)\equiv1\mod{16}

which makes -3\equiv13\mod{16} the inverse of 5 modulo 16, and so n=13.

Now,

x^5\equiv3\mod{17}
\implies (x^5)^{13}\equiv x^{65}\equiv x\equiv3^{13}\equiv(3^4)^2\times3^4\times3^1\mod{17}

3^1\equiv3\mod{17}
3^4\equiv81\equiv4(17)+13\equiv13\equiv-4\mod{17}
3^8\equiv(3^4)^2\equiv(-4)^2\mod{17}
\implies3^{13}\equiv(-4)^2\times(-4)\times3\equiv(-1)\times(-4)\times3\equiv12\mod{17}

Similarly, we can look for m such that 7m\equiv1\mod{30}. Apply the Euclidean algorithm:

30=4(7)+2
7=3(2)+1
\implies1=7-3(2)=7-3(30-4(7))=13(7)-3(30)
\implies13(7)-3(30)\equiv13(7)equiv1\mod{30}

so that m=13 is also the inverse of 7 modulo 30.

And similarly,

x^7\equiv3\mod{31}[/ex] [tex]\implies (x^7)^{13}\equiv3^{13}\mod{31}

Decomposing the power of 3 in a similar fashion, we have

3^{13}\equiv(3^3)^4\times3\mod{31}

3\equiv3\mod{31}
3^3\equiv27\equiv-4\mod{31}
\implies3^{13}\equiv(-4)^4\times3\equiv256\times3\equiv(8(31)+8)\times3\equiv24\mod{31}

So we have two linear congruences,

\begin{cases}x\equiv12\mod{17}\\x\equiv24\mod{31}\end{cases}

and because \mathrm{gcd}\,(17,31)=1, we can use the Chinese remainder theorem to solve for x.

Suppose x=31+17. Then modulo 17, we have

x\equiv31\equiv14\mod{17}

but we want to obtain x\equiv12\mod{17}. So let's assume x=31y+17, so that modulo 17 this reduces to

x\equiv31y+17\equiv14y\equiv1\mod{17}

Using the Euclidean algorithm:

17=1(14)+3
14=4(3)+2
3=1(2)+1
\implies1=3-2=5(3)-14=5(17)-6(14)
\implies-6(14)\equiv11(14)\equiv1\mod{17}

we find that y=11 is the inverse of 14 modulo 17, and so multiplying by 12, we guarantee that we are left with 12 modulo 17:

x\equiv31(11)(12)+17\equiv12\mod{17}

To satisfy the second condition that x\equiv24\mod{31}, taking x modulo 31 gives

x\equiv31(11)(12)+17\equiv17\mod{31}

To get this remainder to be 24, we first multiply by the inverse of 17 modulo 31, then multiply by 24. So let's find z such that 17z\equiv1\mod{31}. Euclidean algorithm:

31=1(17)+14
17=1(14)+3

and so on - we've already done this. So z=11 is the inverse of 17 modulo 31. Now, we take

x\equiv31(11)(12)+17(11)(24)\equiv24\mod{31}

as required. This means the congruence x^{37}\equiv3\mod{527} is satisfied by

x=31(11)(12)+17(11)(24)=8580

We want 0\le x, so just subtract as many multples of 527 from 8580 until this occurs.

8580=16(527)+148\implies x=148
3 0
3 years ago
Help i dont how to do this
IrinaK [193]
What is it you are trying to do I can help if you just say it. 
4 0
3 years ago
In 2010 the population of alaska was about 710200.What ir this written in Word form
Mazyrski [523]

Answer:

Seven hundred and ten thousand two hundred.

Step-by-step explanation:

6 digits would mean it's 100k, so you just go from there. Write out the numbers in word form adding "hundred" and "thousand" as you must. This applies to all problems similar to this.

7 0
3 years ago
Other questions:
  • Larry gave the clerk $10 for purchase of a $3.25 note book and a $2.58 set of markers.How much q change should he receive
    13·2 answers
  • Conversions <br>30qt=__gal __qt
    5·1 answer
  • What is 1.16 rounded to the nearest tenths?
    5·2 answers
  • Solve the equation -(-2-n)
    10·1 answer
  • 54 tacos was sold at lunch. What is six times as many as the number of hotdogs that were sold. How many hotdogs were sold?
    5·1 answer
  • I need help i do not understand ​
    9·1 answer
  • Lin borrowed 950 for a new laptop. She will pay $25,80 each month for the next 48 months (2 years). Find the amount of
    13·1 answer
  • 3 lines are shown. A line with points C, A, F intersects a line with points E, A, B at point A. A line extends from point A to p
    7·1 answer
  • What is the answer to this question?<br> I am confused.
    9·1 answer
  • Can someone help with this please??
    6·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!