平行線的角度 2 (英)
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- Let's do a couple of examples dealing with angles between
- parallel lines and transversals.
- So let's say that these two lines are a parallel, so I can
- a label them as being parallel.
- That tells us that they will never intersect; that they're
- sitting in the same plane.
- And let's say I have a transversal right here, which
- is just a line that will intersect both of those
- parallel lines, and I were to tell you that this angle right
- there is 60 degrees and then I were to ask you what is this
- angle right over there?
- You might say, oh, that's very difficult; that's
- on a different line.
- But you just have to remember, and the one thing I always
- remember, is that corresponding angles are always equivalent.
- And so if you look at this angle up here on this top line
- where the transversal intersects the top line, what
- is the corresponding angle to where the transversal
- intersects this bottom line?
- Well this is kind of the bottom right angle; you could see
- that there's one, two, three, four angles.
- So this is on the bottom and kind of to the
- right a little bit.
- Or maybe you could kind of view it as the southeast angle
- if we're thinking in directions that way.
- And so the corresponding angle is right over here.
- And they're going to be equivalent.
- So this right here is 60 degrees.
- Now if this angle is 60 degrees, what is the
- question mark angle?
- Well the question mark angle-- let's call it x --the question
- mark angle plus the 60 degree angle, they go halfway
- around the circle.
- They are supplementary; They will add up to 180 degrees.
- So we could write x plus 60 degrees is equal
- to 180 degrees.
- And if you subtract 60 from both sides of this equation you
- get x is equal to 120 degrees.
- And you could keep going.
- You could actually figure out every angle formed between
- the transversals and the parallel lines.
- If this is 120 degrees, then the angle opposite to
- it is also 120 degrees.
- If this angle is 60 degrees, then this one right here
- is also 60 degrees.
- If this is 60, then its opposite angle is 60 degrees.
- And then you could either say that, hey, this has to be
- supplementary to either this 60 degree or this 60 degree.
- Or you could say that this angle corresponds to this 120
- degrees, so it is also 120, and make the same exact argument.
- This angle is the same as this angle, so it
- is also 120 degrees.
- Let's do another one.
- Let's say I have two lines.
- So that's one line.
- Let me do that in purple and let me do the other line in a
- different shade of purple.
- Let me darken that other one a little bit more.
- So you have that purple line and the other one
- that's another line.
- That's blue or something like that.
- And then I have a line that intersects both of them; we
- draw that a little bit straighter.
- And let's say that this angle right here is 50 degrees.
- And let's say that I were also to tell you that this angle
- right here is 120 degrees.
- Now the question I want to ask here is, are these
- two lines parallel?
- Is this magenta line and this blue line parallel?
- So the way to think about is what would have happened
- if they were parallel?
- If they were parallel, then this and this would be
- corresponding angles, and so then this would be 50 degrees.
- This would have to be 50 degrees.
- We don't know, so maybe I should put a little asterisk
- there to say, we're not sure whether that's 50 degrees.
- Maybe put a question mark.
- This would be 50 degrees if they were parallel, but this
- and this would have to be supplementary; they would have
- to add up to 180 degrees.
- Actually, regardless of whether the lines are parallel, if I
- just take any line and I have something intersecting, if this
- angle is 50 and whatever this angle would be, they would have
- to add up to 180 degrees.
- But we see right here that this will not add up to 180 degrees.
- 50 plus 120 adds up to 170.
- So these lines aren't parallel.
- Another way you could have thought about it-- I guess this
- would have maybe been a more exact way to think about it
- --is if this is 120 degrees, this angle right here has to be
- supplementary to that; it has to add up to 180.
- So this angle-- do it in this screen --this angle right
- here has to be 60 degrees.
- Now this angle corresponds to that angle, but
- they're not equal.
- The corresponding angles are not equal, so these
- lines are not parallel.