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Parallel Lines – Definition With Examples
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In geometry, parallel lines can be defined as two lines in the same plane that are at equal distance from each other and never meet. They can be both horizontal and vertical.
We can see parallel lines examples in our daily life like a zebra crossing, the lines of notebooks, and on railway tracks around us.
In the figure below, line "AB" is parallel to the line "CD".
The perpendicular distance is always the same between two parallel lines.
Sides of various shapes are parallel to each other. In the rectangle given below, the single arrow lines are parallel to each other, and similarly, the double arrow lines are also parallel to each other.
Parallel lines are represented with a pair of vertical lines between the names of the lines, using the sign: ︳︳
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Use a ruler to make them neat!
Make sure their ends never meet!
A transversal is a line that intersects two parallel lines (or lines on a plane) at different intersecting points, forming angles.
Properties of Parallel Lines
Parallel lines can be easily identified using the following fundamental properties and characteristics:
They are always straight lines with an equal distance between each other.
They are coplanar lines.
They never intersect, no matter how far you try to extend them in any given direction.
If there is a transversal line that intersects two parallel lines at two different points, it will form 4 angles at each point.
The corresponding angles formed by the two parallel lines and a transversal are equal.
Pairs of internal angles on the same side of the crossing are supplementary.
The vertically opposite angles/apex angles are equal.
Alternate internal/interior angles are equal.
Alternate external/exterior angles are also equal.
Linear equations are generally described by the slope-intercept represented by the equation $y = mx + b$.
Where "m" is the slope, "b" is the y-intercept, and y and x are variables.
The value of "m" determines the slope and indicates the steep slope of the line.
Note that the slopes of the two parallel lines are always the same.
For example, if the slope of the straight line in the equation y $= 4x + 3$ is 4, then all lines parallel to $y = 4x + 3$ have the same slope, or 4.
Parallel lines have different y-intersections and have no points or angles in common.
Example 1: Find out which lines are parallel to each other in the given figure.
Solution: All the three lines with arrows passing through them are parallel to each other, which means:
a || b || c
Lines with the double arrows, i.e., line d and e are transversals of lines a, b, and c, but they are parallel to each other.
So, we can say that d || e
Example 2: Find whether the given lines intersected by a transversal in the figure are parallel or not.
Solution: The two lines are parallel as they meet one of the properties of parallel lines "when the alternate interior angles are equal, the lines are parallel".
$∠a$ is equal to $∠c$, and both of these are alternate interior angles.
This proves that the two lines are parallel.
Example 3: Are the lines intersected by the transversal in this figure parallel?
Solution: According to the given properties of parallel lines, the alternating, corresponding, and consecutive angles should be the same to form parallel lines.
In this case, $∠a$ is not equal to $∠d $
Thus, these two lines are not parallel.
Parallel Lines - Definition With Examples
Attend this quiz & Test your knowledge.
Which of the following terms does not describe a pair of parallel lines?
Noncoplanar
Equidistant
Lines running in the same direction
CorrectIncorrect
Correct answer is: Noncoplanar
According to the definition of parallel lines, they are coplanar lines that are evenly
spaced but never intersect. This indicates that parallel lines are not noncoplanar.
Which of the following real-world examples does not represent a pair of parallel lines?
Wipers of a car
Strings of a tennis racket net
Correct answer is: Clock hands
The wiper of a car, stairs, railings, and the strings of a racket all have the same orientation and do not intersect. They are all in the same plane (that is, the strings are in the same plane in the net).
However, the hands of the clock intersect at the center of the clock and are not represented by a pair of parallel lines.
A transversal intersects these two parallel lines. If the value for one angle is given, can you find the value of ∠X?
$120°$
$0°$
Correct answer is: $112°$
If both the lines are parallel, then according to one of the properties of parallel lines, the corresponding angles and alternating interior and exterior angles formed are equal.
In this case, as the alternate exterior angles will be equal,
$∠X = 112°$ as given.
What are the different types of parallel lines?
Parallel lines can be vertical, diagonal, and horizontal.
How are parallel lines used in coordinate geometry?
If the graphs of two linear equations of coordinate geometry are parallel, then the two equations have no common solution.
The slopes of two parallel lines are the same and always equal in coordinate geometry.
How can you prove that two lines are parallel?
You can use some geometric relationships to prove that two lines are parallel.
A transversal is a line that intersects two or more lines. When a line intersects a pair of parallel lines, a pair of different angles are formed.
These different types of angles are used to prove whether the two lines are parallel to each other according to the given properties of parallel lines.
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Dr. Lessig proposes a Citizens Equality Act — legislation that will ensure voter equality.
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After weeks of campaigning, it's almost time to find out what the voters think about their choices in the 2018 Primaries, to be held tomorrow.
The Laurens County Elections Commission joined with the State Elections Commission in reminding voters about who can vote in a primary.
Note, while you may select which primary you want to vote in tomorrow, you can only select one. State law prohibits voters from voting in more than one party's primary on the same day.
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Voting places around Laurens County will be open tomorrow from 7:00 am until 7:00 pm for primary-day voting.


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How to produce a Gmail-style checkbox selection interface?
May 20, 2006 2:07 AM Subscribe
Does anyone know how it's possible to produce an interface for checkbox selection like the one in Gmail? Namely, where you can hold down shift and click the first and last items in a list, and it'll automatically select all the ones in between. Is it possible with simple JavaScript without having to use AJAX or other fancy-pants technology...
posted by tommorris to Computers & Internet (8 answers total)
This has nothing to do with AJAX. It is simple javascript. Two event listeners: one monitors for the shift key, the other is an onclick (or onrelease) handler attached to each checkbox. The system also keeps track of the last clicked position to determine which direction it should check (or uncheck) rows. If any rows are already checked that in the "path" of the checking direction, they maintain their status.
The two most important things to look at when determining whether a shift-selection will be "check" or "uncheck" are the direction (based on the last selection) and the checked/unchecked status of the element you're clicking on. If you have a group selected, then click on an unchecked box at the other end of your selection, they will all be checked (since the checked status of the last box is false). If the checked status of the last item were instead true, it would unselect the items.
There's actually a bug in their selection model, if you play around with it: it appears that the last checked box becomes the new "root" checking node, but they aren't checking for the status of the checkbox before the shift-selection process.
To illustrate: click about ten boxes in a row in the middle of the page. #1 is the top box, #10 is the bottom box. Then, starting at box #7, unclick some boxes in the middle working your way up (say, #6, #5, #4). Then hold down shift and click the box directly above your selection (the equivalent of box #0). Box #4 will also get checked, because that was the last box you selected (and thus, is the current root selection).
posted by Civil_Disobedient at 3:02 AM on May 20, 2006
And while I'm sure you just want some easy event listener code and don't care for a treatise on checkboxes, you need to understand that the basics of this are pretty simple. It's ironing out all the permutations of user behavior that will separate something like GMail from the 20-line hack-jobs that are rife with bugginess and never quite work right.
Anyway. Here is a nice key handler routine.
Best answer: Heh, I didn't really go as into depth into the logistics of the checkbox selecting/deselecting, but just wrote up a simple example to show that it was possible (and pretty simple JavaScript): here. The source is pretty self-explanatory. No "AJAX" necessary (barf).
posted by zixyer at 3:21 AM on May 20, 2006
Consider the value of such a thing though. You now have to make it clear to the user the option exists, as opposed to more common conventions like putting a checkbox image in the header row of a table (since it's hard to come up with a label for that column anyway) that allows users to select and unselect all. I mention this because I am a web developer and a Gmail user and had no idea what you're describing was an option.
posted by yerfatma at 6:03 AM on May 20, 2006
zixyer, that's a nice, quick example. Unfortunately it suffers exactly from what I was talking about. I know it sounds like semantics, particularly since your example is just a quick mockup. But it's these little details will drive a user crazy. For example: click all the boxes. Now, using your shift-select, deselect all the boxes.
You can't do it.
I didn't implement deselecting. Shift-clicking only selects boxes.
I don't know if you specificially wanted it in Javascript, but there is a Greasemonkey script that allows you do to so. Very basic and easy to implement.
posted by apple scruff at 11:04 AM on May 20, 2006
I don't know why zixyer didn't get a Best Answer after the work he or she put in.
Especially as the question just asks if it's possible. Yes, it's possible.
Just to clear up any possible remaining confusion about "AJAX" (a technology that, like "DHTML" before it, doesn't really exist): it's only AJAX if the JavaScript gets information from a server as part of the script, using XMLHTTPRequest. JavaScript changing something about a local page is just ... JavaScript.
posted by AmbroseChapel at 2:57 PM on May 20, 2006
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Meet the volunteer board members and professional staff who keep our organization running.
The Board of Directors of Tacoma Historical Society meets at 6:30 pm, typically on the first Tuesday of each month. Members and friends of Tacoma history are invited to attend and share their ideas and comments.
Our staff members manage the day-to-day operations of Tacoma Historical Society, insuring that our community of dedicated volunteers have the support they need to pursue the THS mission.


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