Parallel & perpendicular lines | Basic geometry (video) | Khan Academy
If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on two interior right angles with the third line, that they will never converge—that is as I wanted, and the drawing would show that these lines still converged. If two straight lines are perpendicular to the same line, do they prove that we say that a line is parallel to itself as parallel lines never meet?. and create straight line graphs to show the relationship between two variables Parallel lines are a fixed distance apart and will never meet, no matter how.
So we can just rule this first choice out. I can stop reading, this started with a statement that we don't know, based on the information they gave us. So let's look at the second choice.
If two rays are rotated by the same amount, the angle between them will not change, so phi must be equal to theta.
So this is interesting, so let's just slow down and think about what they're saying. If ray OA and OC are each rotated degrees, so if you take ray OA, this right over here, if you rotate it degrees, it's gonna go all the way around and point in the other direction, it's going to become, it's going to map to ray OB.
So I definitely believe that. And so this first statement is true. And when people say respectively, they're saying in the same order. So I'm feeling good about that first sentence. If two rays are rotated by the same amount, the angle between them will not change.
Yeah, I could, yeah. Especially if they are rotated around, yeah, I'll go with that. So if we rotate both of these rays by degrees, then we've essentially mapped to OB and OD. And so the measure of those angles are going to be the same.
Proofs with transformations (video) | Khan Academy
So phi must be equal to theta. So I actually like this second statement a lot. So let's see this last statement. Rotations preserve lengths and angles. AB is congruent to CD. Actually, we don't know whether segment AB is congruent to CD, they never told us that. We don't know how far apart these things are. So we know that phi is equal to theta. So this statement right over here is just suspect.
And so actually I don't like that one. Line ST, we put the arrows on each end of that top bar to say that this is a line, not just a line segment. Line ST is parallel to line UV.
Parallel (geometry) - Wikipedia
And I think that's the only set of parallel lines in this diagram. Now let's think about perpendicular lines. Perpendicular lines are lines that intersect at a degree angle.
So, for example, line ST is perpendicular to line CD. So line ST is perpendicular to line CD.
Parallel lines from equation
And we know that they intersect at a right angle or at a degree angle because they gave us this little box here which literally means that the measure of this angle is 90 degrees. By the exact same argument, line the UV is perpendicular to CD. Let me make sure I specified these as lines. Line UV is perpendicular to CD. In such a space, straight lines would behave as Schumacher indicated. But, if space were bounded and curved, straight lines would behave differently.
That is why I thought of the image of comparing triangles on a sphere with triangles on a plane, since the sphere is bounded and curved, while the plane is infinitely extended and flat.
Thus, the triangles on the sphere have curved sides, while the triangles on the plane have straight sides [See Figure 6]. You are looking for certainty and truth in the images, not in the concepts.
Why do you say the triangles on the sphere have curved sides? Do you remember a pedagogical discussion written by Jonathan Tennenbaum back inin which he posed the experiment of drawing an equator on a sphere and two great circles that intersect that equator? When you look at those great circles from the center of the sphere, they appear to be straight lines. In fact, the image you see appears to be two parallel straight lines, intersecting another straight line at 90 degrees [See Figure 7].
Figure 7 Great circles on a sphere. I do remember that experiment, and I remember being startled by the result. Then you surely remember that when you looked straight ahead to the equator, the lines looked parallel, but when you turned your eye toward the pole, the lines began to converge.
- Line (geometry)
- Parallel (geometry)
- Proofs with transformations
That is what I recall being so surprising. Well, Gauss posed a similar paradox to Schumacher. If space had a negative curvature, then the sum of the angles of a triangle would be less than degrees. The sides of such triangles depicted on a flat piece of paper would look like hypocycloids, although, as discussed last week, this is just a depiction. For exactly the reason stated by Gauss above. Visualizing this idea seems to contradict your assumptions, because when you think of triangles whose angles add up to less than degrees, you think of their depiction in Euclidean space.
As such, you can only think of such triangles as having curved sides. But, if space were not Euclidean, this would not be a contradiction. As Gauss told Schumacher: This will take some thinking. Then draw a third line C, that cuts the other two. The parallel postulate says that lines A and B will intersect if the angles they make with C are less than degrees. Conversely, if the angles are greater than degrees they will diverge. If the angles are equal to degrees they will never meet.
Parallel & perpendicular lines
Now think of how the two images we just constructed pose a contradiction with what you just imagined. In the one by Tennenbaum, the straight lines A and B would be the great circles that intersect the equator C at degrees. According to the parallel postulate, they should never meet, yet, in fact, they converge at the pole.
In the image posed by Gauss, the straight lines A and B intersect C at less than degrees [Figure 3]. According to the parallel postulate, they should intersect, but if space were negatively curved, these lines would converge but never intersect. How can it be that we have three sets of three lines, all configured in the same way, and yet get three entirely different results? In a certain sense, this is an inversion of the problem Kepler posed with respect to the geometries of Ptolemy, Brahem and Copernicus.
In that case, three entirely different configurations yielded the equivalent result. In the paradox posed here, three equivalent configurations yield entirely different results.
Nevertheless the same principle is demonstrated: Now go back and think about why you thought the parallel postulate of Euclid was true in the first place. Because when I drew the picture, the two lines always intersected, as long as I extended them far enough. In other words, you thought it was true, because it always worked.
Yes, and there was nothing to indicate it would be otherwise.