Questions on Geometry: Parallelograms answered by real tutors!

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The vertices of a parallelogram are A(-4,-1),B(5,-6), C(11,-3) and D(2,2). Show that the diagonals bisect each other.
A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
A parallelogram has:
A- 2 sets of parallel sides
B- 2 sets of congruent sides
C- opposite angles congruent
D- consecutive angles supplementary
E- diagonals bisect each other
F- diagonals form 2 congruent triangles
Plot all 4 points on the coordinate plane (I would use graph paper).
Then connect each point to form your parallelogram.
Find the distance from point A to point C and then the distance from point B to point D using the distance formula for points.
If the distances are equal, then the diagonals bisect each other.
Can you take it from here?

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true!!
:
A rhombus is actually just a special type of parallelogram. Recall that in a parallelogram each pair of opposite sides are equal in length. With a rhombus, all four sides are the same length.It therefore has all the properties of a parallelogram.
:
If the diagonals formed anything but 90 degree angles it would only be a parallelogram because we would not have equal sides

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Two 12 mm sides of a rhombus form a 60 degree angle. Fine the area of the rhombus. Round your answer to the nearest tenth.
:
You can use trig to find the area here: A = side^2 * sine of an interior angle
;
A = 12^2 * sin(60)
A = 144 * .866
A = 124.7 sq/mm

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First I checked to make sure that the point (8,-7) is not
a solution to either equation. That means neither line
passes through that point. Now all I have to do is find
2 lines. Both have to pass through (8,-7), one line must
be parallel to and the other must be
parallel to
The general point-slope formula is
where is slope
-------------------
For line parallel to ,






--------------------
For line parallel to ,




--------------------
Now I'll plot all 4 lines to check solution:

It looks like a parallelogram, so OK

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a diagonal of a square lies on the line 3x-5y=14. one vertex is at (0,4). Find an equation of the line that contains the other diagonal.
--------------------------
The diagonals of a square are perpendicular, so the 2nd one will have a slope that's the negative inverse of 3x-5y=14.
The slope, m1, of 3x-5y=14 is 3/5, so m2 will be -5/3.
There are an infinite number to choose from, notice it says "an equation."
If we have the 2nd diagonal pass thru the point (4,0), then find the equation:
y-y1 = m2*(x-x1) where (x1,y1) is the point (4,0)
y = (-5/3)*(x-4)
y = (-5/3)x + 20/3

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ABCD has vertices A(-1,0), B(3,3), C(6,-1), and D(2,-4)
Let's see 3 important properties of a Square:
1) All sides are EQUAL.
2) The Diagonals are equal.3
3) Each Angle is equal to 90 degress.
.
To Prove the properties, let's draw the Square first with given vertices:

1) All SIDES ARE EQUAL.
To prove, we use distance Formula ---->
For :


For :


For :


For


.
Therefore, you can see all SIDES are EQUAL IN LENGTH,
It follows, as you see in the graph:

.
2) The Diagonals are EQUAL.
The diagonals here are referred to adn . And again we use Distance formula,
For :


For ;


Therefore:
It satisfy the 2nd property, and as we see in the graph:

.
3) Each Angle is equal to 90 DEGREES
In this property we use Equation for Right Traingles, the Pythagorean theorem.
Just where the formula we used above is derived from.
For Triangle :



, A Right Triangle!
For Triangle :


, A Right Triangle!
For Triangle


, A Right Triangle
For Triangle


, A Right Triangle!
ALL satisfies the Pythagorean theorem, therefore all angles are 90 degrees:
.
Conclusion: ABCD that has vertices A(-1,0), B(3,3), C(6,-1), and D(2,-4) is a SQUARE.
Thank you,
Jojo

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A(-1,0), B(3,3), C(6,-1), D(2,-4).
slopes:
AB=(3-0)/(3+1)=3/4
DC=(-4+1)/(2-6)=3/4 THUS THESE 2 LINES ARE PARALLEL.
slopes:
BC=(-1-3)/6-3)=-4/3=-4/3
AD=(-4-0)/(2+1)=-4/3 THESE 2 LINES ARE PARALLEL & PERPENDICULAR TO THE AB & DC LINES.
THUS THIS IS A SQUARE FIGURE.

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The "Triangle Proportionality Theroem" states: If a line is parallel to one side of a triangle and intersects the other two sides in two distinct points, then it separates these sides into segments of proportional length.
AD/DB does not equal AE/CE
AD = 2
DB = 3
AE = 4
EC = 4
We now have two fractions:
2/3 and 4/4
QUESTION:
Are these two fractions the same?
No, right?
So, the answer to your question is NO: DE is NOT parallel to BC

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A parallelogram has two pairs of congruent (same sized angles). The four angle sizes add up to 360 degrees.
Angle A and C will be the same size as will angles B and D.
angle A + angle B + angle C + angle D = 360
2x+19 + 3x+21 + 2x+19 + 3x+21 = 360
10x + 80 = 360
10x = 280
x=28
So angle A and C are 2*28+19 = 75 degrees
and angles B and D are 3*28+21 = 105 degrees.

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Area of rectangle is:
area = height * width
.
Let H = height
and plugging in the values given by the problem we have:
10x = h(2x)
.
To find h, divide both sides by 2x:
(10x)/(2x) = h
5 = h (height)

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Draw a parallelogram and look at the opposite angles in the upper left and lower right of the parallelogram. Extend the sides of the parallelogram that run horizontally so they extend beyond the parallelogram. You can see that the opposite angles are now equal because the alternate interior angle to the angle in the upper left is the complementary angle to the one in the lower right of the parallelogram and they must all be equal.

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There are three solutions. The first is (-4,-1), because it must 
be as far to the left of and as far below (-2,2) as (1,-1) is of
(3,2).



The second is (6,-1), because it must be as far to the right 
of and as far below (3,2) as (1,-1) is of (-2,2).



The third is (0,5), because it must be 
1. as far right of (-2,2) as (-1,1) is left (3,2)
and
2. as far above (-2,2) as (1,-1) is below (3,2).



Edwin

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simple; start by stating that you construct a parallelogram with one right angle.
then the consecutive angle needs to be also right because consecutive angles in a parallelogram add up to 180 (property of a parallelogram) and then continue with this statement until u complete all angles and that way you showed hat any parallelogram with one right angle it will be a rectangle.

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Yes they can. If the two parallelograms have unequal heights, then they will have unequal areas.

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Let's see the Rectangle:

.
By Pyth Theorem:

>>>>> 16ft, Length
Then,
>>>>>>>62ft
See again the rectangle:

Thank you,
Jojo

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A rhombus has sides of length 6 cm. One of its diagonals is 10 cm long. Find the length of the other diagonal ?
The sides of a rhombus are the same measure.
One diagonal measures 10m. This means that it is 5m + 5m.
The two diagonals meet and form right angles.
We can use the Pythagorean Theorem to find one leg of one of the right triangles formed inside the rhombus.
5^2 + x^2 = 6^2
25 + x^2 = 36
x^2 = 36 - 25
x^2 = 11
Take the square root of both sides.
x = sqrt{11}.
The other diagonal will be sqrt{11} + sqrt{11) = 2sqrt{11}m
Did you follow?

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the equation of a line y=mx-1. what is the value of m if the line passes through the point (2,3)
-----------------
Sub 2 for x and 3 for y
3 = m*2 - 1
2m = 4
m = 2
----
y-y1 = m*(x-x1) (x1,y1) is the point (2,3)
y-3 = 2*(x-2)
y-3 = 2x-4
y = 2x-1 slope-intercept form
2x-y = 1 standard form

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i need help to understand and solve these 5 problems, thank u
1.the quadrilateral whose consecutive sides have measures 2, 3, 2, 3 is a(n)----parallelogram
=====================================
2.the quad. whose cons. sides have meas. 2 , 3, 2, 3, is a(n)__________.
parallelogram
======================================
3.the quad. with two distinct pairs of NON-opposite angles congruent is a(n)trapezoid.
=======================================
4.if cons. midpoints of a rectangle are joined, a(n)parallelogram is formed.
========================================
5. if at leaset one pair of opposite angles of a quad. are congruent and the other pair of angles are each bisected by a diagonal, the figure formed is a(n)mystery.
=========================================
Cheers,
Stan H.

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If a parallelogram has at least one right angle, then the other 3 angles must also be right angles (in order for the sides to be parallel). So this means that the bedroom will be the shape of a rectangle.


Go ahead and draw a picture if you need a clear visual.

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in a parallelogram, the opposite angles are equal to each other.
the sum of the angles in a parallelogram is 360.
-----
if the parallelogram is labeled ABCD, then angle A is equal to angle C because they are opposite each other, and angle B is equal to angle D because they are opposite to each other.
-----
the labeling of the parallelogram would be:
A is lower left corner
B is upper left corner
C is upper right corner
D is lower right corner
-----
assume A and C are the smaller angles.
assume B and D are the larger angles.
the parallelogram will be tilting to the right so that corner B is to the right of corner A.
-----
problem states that the larger angle is 3 times the smaller angle plus 40 degrees.
since B is the larger angle and A is the smaller angle, that means that:
B = 3*A + 40
-----
since the sum of the angles in a parallelogram = 360 degrees then angle A + B + C + D = 360.
since angle A = angle C, and angle B = angle D, this can be rewritten as 2*A + 2*B = 360 degrees by substituting A for C and B for D.
-----
equation you start out with is:
2*A + 2*B = 360
-----
you know that B = 3*A + 40 so you can substitute in equation.
2*A + 2*B = 360 becomes:
2*A + 2*(3*A+40) = 360
simplify:
2*A + 2*3*A + 2*40 = 360
2*A + 6*A + 80 = 360
8*A + 80 = 360
subtract 80 from both sides of equation:
8*A = 360 - 80
8*A = 280
divide both sides of equation by 8:
A = 280 / 8
A = 35
-----
angle A is 35 degrees.
angle B is 3 * 35 + 40
angle B is 145 degrees.
-----
angle A and angle B = 180 degrees.
2 * angle A and 2 * angle B = 360 degrees.
this checks out with the sum of the angles of a parallelogram so the angles are good.
answer is:
smaller angle is 35 degrees.
larger angle is 145 degrees.
-----

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Any parallelogram that does not have 4 right angles is a non-rectangular parallelogram.
To be a parallelogram opposite sides must be parallel, nothing more is needed.

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let ABCD be your parallelogram
A is bottom left
B is top left
C is top right
D is bottom right.
-----
let ABCD lean to the right so that point B is slightly to the right of point A.
all you need is a little tilt to show that it's not a rectangle.
-----
draw diagonals AC and BD.
AC is the long diagonal and BD is the short diagonal.
-----
BC congruent to AD (opposite sides of a parallelogram are congruent)
-----
note:
AC and BD are diagonals of the parallelogram.
they are also transversals that intersect two parallel lines (BC and AD).
-----
angle ACB congruent to angle CAD (opposite internal angles caused by the intersection of a transversal with two parallel lines are congruent).
likewise, angle DBC congruent to angle BDA.
-----
you have triangles AED congruent to triangle CEB (ASA)
the ASA is formed by:
side BC congruent to side AD
angle ACB congruent to angle CAD
angle DBC congruent to angle BDA.
-----
CE is congruent to AE (corresponding parts of congruent triangles are congruent)
BE is congruent to DE (same reason).
-----
a sketch of the parallelogram can be found at this website:
www.geocities.com/gonzo89p
look for 169616 parallelogram
shouldn't be too hard to find.
it's the only one there.
-----

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Frank's house needs new carpets.
the living room is 12 feet long and 13 feet wide. The dining room is 15 feet long and 11 feet wide .How many square feet of carpet will be need?
------------------
For the LR: 12*13 = 156 sf
For the DR: 15*11 = 165 sf
156+165 = 321 sq feet.
Carpet is sold and priced by the sq yard, not the square feet.

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There seems to be an error on the problem as it is written if AB is the long side and it is 5 shorter than the short side.
The general idea is to draw the parallelogram first and label all the vertices. Define the length of the short side to be x inches long so write x alongside each of the two shorter sides. The longer side is 5 inches longer than a short side so it will be x + 5 inches long. Mark in the lengths of the two long sides.
To find the perimeter of the shape add up each of the sides. This will give x + (x + 5) + x + (x + 5) which you already know must equal 48.





This means that the short sides are 9.5 inches long and the longer sides are 5 inches longer ie 14.5 inches. Check to make sure that the sides give a total perimeter of 48 inches.

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Compounded semi-annually. P dollars is invested at annual interest rate r for 1
year. If the interest is compounded semi-annually, then the polynomial p(1+r/2)^2
represents the value of the investment after 1 year.
:
Rewrite this expression without parenthesis.
A = p(1+r/2)^2
:
FOIL (1+(r/2))*(1+(r/2)) to get rid of the exponent
A = p *(1 + + )
:
A = p (1 + r + ); cancel 2
:
Multiply what's inside the brackets by p
A = p + pr +
:
Evaluate the polynomial if P= $200 and r =10%.
Substitute for p and r in the above equation
:
A = 200 + 200*.10 +
:
A = 200 + 20 +
:
A = 220 +
:
A = 220 + .50
:
A = 220.50
:
:
Check solution on a calc: enter 200*(1+(.1/2))^2 = 220.5

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for abcd to be a parallelogram, opposite sides have to be equal and parallel to each other.

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+=plus
-=minus
*=multiply
/=divide
^=power.
-----------------------
h*b=27
3b*b=27
3b^2=27
b^2=27/3
b^2=9
b=sqrt9
b=3 in. is the base.
3*3=9 in. for the height.
Proof:
3*7=27
----------------------
2*3+2*9=6+18=24 in. is the perimeter.

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Hi, Hope I can help,
.
I would mean alot to me if someone could help me out on this problem
I did find a smiliar question to mine, but could really understand how i would get the answer.
Consider the points P(10,2),Q(1,11), and R(-8,2) on a coordinate plane. Where must the point S be located so that the quadrilateral PQRS is a square?
i found the slope of PQ i got -1
I found the slope of QR i got 1
then what do we do after gettting the slope
please help
.
First we need to plot the three points on a graph
.
Here are the 3 points
.
(P)(10,2) = red
(Q)(1,11) = green
(R)(-8,2) = blue
.

.
We will need to find two equations,
.
1.equation of the line from the unknown point(x,y) to (-8,2)
2.equation of the line from the unknown point(x,y) to (10,2)
.
The line from the unknown point (x,y) to (-8,2) is parallel to the line from (10,2) to (1,11)(Line PQ)(redish brown line)(line has slope of (-1))
.
Since we are trying to find the equation of the line (contains unknown point(x,y) and (-8,2)), the line we are trying to find is parallel to the line with the slope of (-1), since parallel lines have the same slope, we know our unknown line has the slope of (-1), we can replace "m" with (-1) in our slope intercept equation, , where "m" is the slope, "b" is the y intercept
.
= = , since we have a point on the line (-8,2), we can replace "x" and "y" with (-8,2)(x,y)
.
= =
.
To solve for "b" we will move "8" to the left side
.
= = = =
.
"b" = (-6), we can replace "b" with (-6) in our equation
.
= =
.
We can check by replacing "x" and "y" with (-8,2)(x,y)
.
(-8,2) = = = (True)
.
One of our two equation we have to solve is (Line RS),
.

.
we will now solve for the second unknown line equation
.
2. equation of the line that contains the unknown point(x,y) and (10,2)
.
This line is parallel to the line that contains (-8,2) and (1,11)(Line QR)(green line)(line that has slope of (1)).
.
The line that contains the unknown point and (10,2) is parallel to the line , (They both have the same slope)
.
The line that contains the unknown point and (10,2) has a slope of "1", lets replace "m" with "1" in our slope intercept equation
.
= =
.
Since the line contains (10,2) we can replace "x" and "y" with (10,2)(x,y)
.
= = , we will move "10" to the left side to solve "b"
.
= = = = , we can replace "b" with (-8) in our equation
.
= , We can check by replacing "x" and "y" with (10,2)
.
(10,2), = = = (True)
.
(Line QS) is our second equation we had to solve
.

.
We found our two unknown equations(our two unknown line equations)
.
(Line QS)
.
(Line RS)
.
We can now solve for both "x" and "y" to get our unknown point, this is the way I solve systems of equations
.
First solve for a letter(doesn't matter which one)(since we don't have to do anything to solve for "y"(it is solved already for us), we will put the two answers that "y" is together into an equation(since "y" is one number, both of the answers equal each other)
.
Here are our two answers
.

.

.
Lets put them together
.
= , we will move (-x) to the left side
.
= = , We will move (-8) to the right side to solve "x" even more
.
= =
.
We will divide each side by "2" to get "x"
.
= =
.
"x" = 1, we can replace "x" with "1" in one of the two equations to get "y"
.

.

.
We will use the first equation
.
= = =
.
"y" = (-7), we can check our answers by replacing "x" and "y" in our two equations
.
"x" = "1"
"y" = (-7)
.
= = = (True)
.
= = = ( True)
.
"x" = "1"
"y" = (-7)
.
Since points are given as (x,y), our unknown point(Point "S") is (1,-7)
.

.
You can find this answer by a shortcut
.
Since this is a square, the diagonals are the same, you could measure the distance between (-8,2) and (10,2)(18 squares), you would just go from (1,11) and move down 18 squares, and you would find the other point
.
Unknown point( Point "S") = (1,-7)
.
Hope I helped, Levi

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Get the that is perpendicular to the and multiplied by the base= _____ sq units
In getting the you can resort to Pyth. Theorem since that perpendicular line to the will form a triangle.
Thank you,
Jojo

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Given: , , , &
Substitution:


---->

------------------------------> length of the other base
Thank you,
Jojo

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10x (25x-30)
Distribute:
--------------------> working eqn
Divdide the whole eqn by 250;




Check it out via working eqn:




Thank you,
Jojo

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1) ABCD is a parallelogram if AB = 2 AD and P is the midpoint of AB.prove that angle CPD = 90.
SOLUTION:
In isosceles triangle APD,
< APD = < ADP = (180 - < A)/2 = 90 - < A/2
In isosceles triangle PCB,
< BPC = < BCP = (180 - < B)/2 = 90 - < B/2
Next express < DPC in terms of < A and < B.
< DPC = 180 - < APD - < BPC = 180 - (90 - < A/2) - (90 - < B/2) = < A/2+ < B/2
= (< A + < B)/2
Note that in any parallelogram the sum of any two adjacent angles is 180. so:
< A + < B = 180
Therefore
< DPC = (< A + < B)/2 = 180/2 = 90
ALTERNATIVE SOLUTION:
First prove that
PD bisects < D
PC bisect < C
(Let Q be the midpoint of CD. As triangle APD and QPD are congruent, so < ADP = < QDP. For the same reason < BCP = < QCP)
Next
< PDQ = < D/2
< PCQ = < C/2
Thus < PDQ + < PCQ = < D/2 + < C/2 = (< D + < C)/2 = 180/2 = 90.
********************************************************************************
2) If the diagonal PR and QS of a parallelogram PQRS are equal, prove that PQRS is a rectangle.
SOLUTION:
We need to prove its angles are 90 degree.
First prove that triangle PQR and triangle QRS are congruent.(Reason: PQ = SR, QR = QR and PR = QS)
So < Q = < R
Next note that < Q + < R = 180.
As < Q = < R, < Q = < R = 90.
So PQRS is a rectangle.
********************************************************************************
PQRS is a parallelogram. PS is produced to M so that SM = SR and MR is produced to meet PQ produced to N. Prove that QN = QR.
SOLUTION:
We need to prove triangle QNR is an isosceles triangle.
First show that < N = < SRM and < QRN = < M ( ? )
Next show that < SRM = < M ( ? )
Conclusion: < N = < QRN ( ? ), So triangle QNR is isosceles.

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2*THE ORIGINAL SQUARE PERIMETER IS THE PERIMETER OF THE 10 cm & 6 cm EXTENDED SIDES OF THE SQUARE TO FORM A RECTANGLE.
2*4X=2(X+10)+2(X+6)
8X=2X+20+2X+12
8X-4X=32
4X=32
X=32/4
X=8 FOR THE ORIGINAL SIDES OF THE SQUARE (AB & AD)
PROOF:
2*4*8=2(8+10)+2(8+6)
64=2*18+2*14
64=36+28
64=64

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Since you provide no more information than the lengths of the sides of the parallelogram, we have to assume that the parallelogram in your problem is a rectangle, in which case, the area is given by: A = L*W where L = 9cm and W = 3cm.
So the area is: A = 9*3 = 27 sq.cm.

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there is not enough information to find the diagonals

you need one of the corner angles or the height

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We know that a parallelogram has 2 opposite sides with EQUAL lengths and 2 opposite bases also with equal lengths.
To find the opposite sides with given,
-------> A
--------> B
In B we get the value of "x" --->
Then substitute this in A,
=
=
=
= ----------> , Length of each side
Thank you,
Jojo

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We start with the formula, -----> basic eqn.




2 values:
----------> don't use, "negative"
------------> the one to used = height
---> = base
Go back basic eqn to check,


Thank you,
Jojo

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Let the larger angles be A and the smaller angles be B.
You know that in any quadrilateral, the sum of the interior angles is 360 degrees, so you can write:
2A+2B = 360 Simplify this to:
A+B = 180
From the problem statement, you have that A = 3B-20, right?
Substitute this into the first equation and solve for B.
(3B-20)+B = 180
4B-20 = 180
4B = 200
B = 50 degrees.
A = 3B-20
A = 3(50)-20
A = 150-20
A = 130 degrees.

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the diagonals of a rhombus are perpendicular bisectors of each other
__ so they form 4 congruent right triangles, with legs that are half the length of the diagonals
__ and a side of the rhombus as the hypotenuse

in this case the triangles are 9-12-? __ hint: think 3-4-5

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I have a parallelogram (rv lot)45 feet by 30 feet with angle of 60 degrees..question is...will my rv of 40 feet by 10 feet fit on lot without crossing boundries?
-------------------------
Draw the picture of the rv in the parallelogram.
The back of the rv is the base of an equilateral
triangle with 60 degree angles and sides of 10 ft.
------------
The triangle with the 120 degree angle has sides
of 20 ft and 45 ft.
Determine whether the 3rd side is >=40 ft. by
using the Law of Cosines
-----------
x^2 = 20^2 + 45^2 -2*20*45*cos120 = 3325
x = 57.66 ft.
-----------------
Yes your 40 foot long rv will fit in the lot.
=================
Cheers,
Stan H.

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I think that's a trick question. Sure it will ... if you park it at a 60 degree angle.
If you have to park it 'straight', then we have to do a little math.
Draw your parallelogram, 45 feet and 30 feet with 60 degree angle between the 30 and 45 sides.
Drop a line from one end of the a 45 foot side down. Make it perpendicular to the 30 foot base line.
Now take 45 *cos(60) to see how long the side of the resulting right triangle is. Take 45*sin(60) to see how long the other side is. Then make the call about whether the rv can fit

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(7-1)(4-1)=18

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The interior angles of the parallelogram equal 360 degrees.
Also, the oppposite angles are equal.
Angle A = Angle C
Angle B = Angle D
The sum of the angles is 360.
Substitute for B and C.
Simplify.
Divide by 2.
Substitute for A and D.
Simplify.
Solve for x.
Solve for A and C.
Angle A = Angle C = 155 degrees
Angle B = Angle D = 25 degrees

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There is no answer the way you have named the angles. When you use three points to name an angle, the convention is that vertex point be in the middle of the three point designation. No matter how you arrange the names of the points, J, K, L, and M, you cannot simultaneously have an angle LMK and KJM. So, you need to figure out where the naming error lies. But I will tell you this: The sum of the interior angles of any quadrilateral is 360. In a rhombus, or any other parallelogram for that matter, angles opposite each other are equal, and angles adjacent to each other are supplementary, that is the sum of their measures is 180. So if the angle whose measure you seek is opposite the 36 degree angle, it must also be 36. Otherwise it is 180 - 36.

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Put the line into the form y = mx + b


Now it is in the form y = mx + b.
So what is m?
. The slope of this line is 1.
The slope of any line parallel to this line is also 1, since parallel lines have the same slope.