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Question 902233: find the value of each variable and the measure of each angle.
give the equations used to find the measure.
there are two intersecting lines.
the measure of the angle on the left is (4y)
the measure of the angle on the top is (2x)
the measure of the angle on the right is (x+y+10)
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 2 intersecting lines form vertical angles and adjacent angles.
the angles opposite each other are the vertical angles.
the angles next to each other are the adjacent angles.
how can you tell the difference?
the angles that are on the opposite side of both lines are the vertical angles.
the angles that are on the opposite side of only one of the lines and on the same side of the other line are the adjacent angles.
your angles are:
one on the left is equal to 4y
one on the right is equal to x + y + 10
one on the top is equal to 2x
the one on the left and the one on the right are the vertical angles.
they are on opposite sides of both lines.
the one on the top is a supplementary angle to the one on the left and also a supplementary angle to the one on the right.
that's because the angles are on the opposite side of only one of the lines while it is on the same side of the other line.
the equations formed are:
2x + 4y = 180 (adjacent therefore supplementary)
2x + x + y + 10 = 180 (adjacent therefore supplementary)
4y = x + y + 10 (vertical therefore equal)
we only need to solve two of these equations simultaneously to get the value of x and y that we need.
we'll choose 2x + 4y = 180 and 4y = x + y + 10.
2x + 4y = 180 doesn't need to be simplified.
4y = x + y + 10 needs to be simplified and converted to the same form as the first equation as follows:
subtract x and subtract y from both sides of the equation to get:
4y - x - y = 10
simplify to get:
3y - x = 10
rearrange the terms to get:
-x + 3y = 10
the two equations to solve simultaneously are:
2x + 4y = 180
-x + 3y = 10
we'll multiply both sides of the second equation by 2 to get:
2x + 4y = 180
-2x + 6y = 20
we'll add the two equations together to get:
10y = 200
we'll divide both sides of this equation by 10 to get:
y = 20
replace y with 20 in the first equation to get:
2x + 4y = 180 becomes:
2x + 80 = 180
subtract 80 from both sides of the equation to get:
2x = 100
divide both sides of the equation by 2 to get:
x = 50
your solutions are:
x = 50
y = 20
let's see if they work.
the adjacent angles of 2x and 4y should add up to be equal to 180 degrees because they are supplementary7.
you get 2*50 + 4*20 = 100 + 80 = 180
since they add up to 180, the solution of x = 50 and y = 20 looks good so far.
the vertical angles of 4y and x+y+10 should be equal to each other.
you get 4*20 = 20 + 50 + 10 becomes 80 = 80.
this confirms the values of x and y are good in this equation as well.
you can go one step further since 2x + x + y + 10 are also supplementary and their sum should be equal to 180.
you get 2*50 + 50 + 20 + 10 = 100 + 80 = 180.
this confirms the values are good in this equation as well.
looks like you're good all around, so your solution is:
x = 50
y = 20
the measure of the angle on the left is (4y) which is equal to 80 degrees.
the measure of the angle on the top is (2x) which is equal to 100 degrees.
the measure of the angle on the right is (x+y+10) which is equal to 80 degrees.
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