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From Wikipedia

Line segment

In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment is either an edge (of that polygon) if they are adjacent vertices, or otherwise a diagonal. When the end points both lie on a curve such as a circle, a line segment is called a chord (of that curve).

Definition

If V\,\! is a vector space over \mathbb{R} or \mathbb{C}, and L\,\! is a subset of V,\,\! then L\,\! is a line segment if L\,\! can be parameterized as

L = \{ \mathbf{u}+t\mathbf{v} \mid t\in[0,1]\}

for some vectors \mathbf{u}, \mathbf{v} \in V\,\!, in which case the vectors \mathbf{u} and \mathbf{u+v} are called the end points of L.\,\!

Sometimes one needs to distinguish between "open" and "closed" line segments. Then one defines a closed line segment as above, and an open line segment as a subset L\,\! that can be parametrized as

L = \{ \mathbf{u}+t\mathbf{v} \mid t\in(0,1)\}

for some vectors \mathbf{u}, \mathbf{v} \in V\,\!.

An alternative, equivalent, definition is as follows: A (closed) line segment is a convex hull of two points.

In proofs

In geometry, it is sometimes defined that a point B is between two other points A and C, if the distance AB added to the distance BC is equal to the distance AC.

In an axiomatic treatment of Geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or else defined in terms of an isometry of a line (used as a coordinate system).

Segments play an important role in other theories. For example, a set is convex if the segment that joins any two points of the set is contained in the set. This is important because it transforms some of the analysis of convex sets to the analysis of a line segment.

Transversal line

In geometry, a transversal line is aline that passes through two or more other coplanar lines at different points.

In Euclidean geometry if lines a and b are parallel, and line t intersects lines a and b, then corresponding angles formed by line t and the parallel lines are congruent.

Alternate angles

A transversal line is a line that transverses other lines

Theorems

There are at least eight geometrical theorems concerning transversals. They are as follows:

Theorem 9 If a transversal intersects two parallel lines, then the alternate interior angles are congruent.

Theorem 10 If a transversal intersects two parallel lines, then the corresponding angles are congruent.

Theorem 11 If a transversal intersects two parallel lines, then the alternate exterior angles are congruent.

Theorem 12 If a transversal intersects two parallel lines, then the interior angles on the same side of the transversal are supplementary.

The next four theorems are converses of the previous four theorems.

Theorem 13 If a transversal intersects two lines so that alternate interior angles are congruent, then the lines are parallel.

Theorem 14 If a transversal intersects two lines so that corresponding angles are congruent, then the lines are parallel.

Theorem 15 If a transversal intersects two lines so that alternate exterior angles are congruent, then the lines are parallel.

Theorem 16 If a transversal intersects two lines so that interior angles on the same side of the transversal are supplementary, then the lines are parallel.

Proofs

Theorem 9

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Given: y ll z; transversal t intersects line y and z at A and B; O is the midpoint of lineSEGMENTAB.

Prove: angle 1 is congruent to angle 3; angle 2 is congruent to angle 4.

1. y ll z; transversal intersects lines y and z at A and B; O is the midpoint of line AB.

2. Through O, draw line CD perpendicular to z.(P. means Postulate)

3. Line CD is perpendicular to y.(T. means Theorem)

4. Angle ACO and BDO are right angles.

5. Î”ACO and BDO are right triangles.

6. Angle 5 is congruent to angle 6.

7. Line AO is congruent to line BO.

8. âˆ´ Î”ACO is congruent to Î”BDO.

9. âˆ´ Angle 1 is congruent to angle 3.

10. Angle 2 is supplementary to angle 1; angle 4 is supplementary to angle 3.

11. âˆ´ Angle 2 is congruent to angle 4.

Theorem 11

thumb|center|500px|Theorem 11 Formal Proof Image.Given: Transversal t intersects lines m and n; m ll n.

Prove: Angle 1 is congruent to angle 7.

1. Transversal t intersects lines m and n; m ll n.

2. Angle 3 is congruent to angle 5.

3. Angle 1 is congruent to angle 3; angle 5 is congruent to angle 7.

4. âˆ´ Angle 1 is congruent to angle 7.

Theorem 12

Given: Transversal t intersects lines m' and n; m ll n.

Prove: Angle 1 is supplementary to angle 2.

1. Transversal t intersects lines m and n ; m ll n.

2. Angle 2 is congruent to angle 3.

3. Angle 1 is supplementary to angle '3.

4. âˆ´ Angle 1 is supplementary to angle 2.

From Encyclopedia

Lines, Parallel and Perpendicular Lines, Parallel and Perpendicular

In mathematics, the term "straight line" is one of the few terms that is left undefined. However, most people are comfortable with this undefined concept, which can be modeled by a pencil, a stiff wire, the edge of a ruler, or even an uncooked piece of spaghetti. Mathematicians sometimes think of a line as a point moving forever through space. Lines can be curved or straight, but in this entry, only straight lines are considered. A line, in the language of mathematics, has only one dimensionâ€”lengthâ€”and has no end. It stretches on forever in both directions, so that its length cannot be measured. When a line is modeled with a piece of spaghetti, a line segment is actually being represented. The model of a line segment has thickness (or width), while the idea that it modelsâ€”a mathematical lineâ€”does not. So a mathematical line is a notion in one's mind, rather than a real object one can touch and feel, just as the notion of "two" is an idea in one's mindâ€”that quality and meaning that is shared by two apples, two trucks, and the symbols //, 2, â˜ºâ˜º, and ii. Think of two straight lines in a plane (another undefined term in geometry ). Someone can model this idea, imperfectly, by two pencils or two pieces of spaghetti lying on a desktop. Now, mentally or on a desktop, push these lines around, still keeping them on the plane, and see the different ways two lines can be arranged. If these two lines meet or cross, they have one point in common. In the language of mathematics, the two lines intersect at one point, their point of intersection. If two lines are moved so that they coincide, or become one line, then they have all of their points in common. What other arrangements are possible for two lines in a plane? One can place them so that they do not coincide (that is, one can see that they are two separate lines), and yet they do not cross, and will never cross, no matter how far they are extended. Two lines in the same plane, which have no point in common and will never meet, are called parallel lines. If one draws a grid, or coordinate system, on the plane, she can see that two parallel lines have the same slope, or steepness. Are there any parallel lines in nature, or in the human-made world? There are many models of parallel lines in the world we build: railroad tracks, the opposite sides of a picture frame, the lines at the corners of a room, fence posts. In nature, parallel lines are not quite so common, and the models are only approximate: tracks of an animal in the snow, tree trunks in a forest, rays of sunlight. The only other possible arrangement for two lines in the plane is also modeled by a picture frame, or a piece of poster board. Two sides of a rectangle that are not parallel are perpendicular . Perpendicular lines meet, or intersect, at right angles, that is, the four angles formed are all equal. The first pair of lines in part (a) of the figure below meet to form four equal angles; they are perpendicular. The second pair in part (b) forms two larger angles and two smaller ones; they are not perpendicular. Perpendicular lines occur everywhere in buildings and in other constructions. Like parallel lines, they are less common in nature. On a coordinate system, two perpendicular lines (unless one of them is horizontal) have slopes that multiply to a product of -1; for example, if a line has a slope of 3, any line perpendicular to it will have a slope of -â…“. see also Lines, Skew; Slope. Lucia McKay Anderson, Raymond W. Romping Through Mathematics. New York: Alfred A. Knopf, 1961. Juster, Norton. The Dot and the Line: A Romance in Lower Mathematics. New York: Random House, 1963. Konkle, Gail S. Shapes and Perception: An Intuitive Approach to Geometry. Boston: Prindle, Weber and Schmidt, Inc., 1974.

Question:write the word that describes the lines or line segments. 1.the strings on a guitar 2.the marks left by a skidding car 3.sidewalks on opposite sides of a street 4.the segments that make up a + sign 5.the wires suspended between telephone poles 6.the hands of a clock at 9:00 P.M. 7the trunks of grown trees in a forest. please help. i dont understand this. :(

Answers:1) Parallel 2) Parallel 3) Parallel 4) Perpendicular 5) Parallel 6) Perpendicular 7) Parallel If the instructions are written as you wrote them, then it means pick the word that describes the relationship of the lines in each example. Like with the guitar strings or the sidewalk, these lines run in the same direction so they are parallel. Since a + sign has opposite lines, it is perpendicular, same with 9:00 because the hour hand is horizontal and the minute hand is vertical. Make sense?

Question:How do I draw a right angle, ray, line segment, line, endpoint?

Question:I have to take pictures of things in everyday life that represent these geometry subjects. They are: Acute Angle Adjacent Angle Angle Bisector Collinear Concave Polygon Congruent Convex Polygon Line Line Segment Linear pair Midpoint Obtuse Angle Parallel lines Perpendicular Lines Plane Point Ray Right Angle Slope Vertical Angles If you could help me by giving examples of what in the real world could represent any of these, that would be great. I have a couple but I just want to make sure that I am good on them. Thank you. =]

Answers:Acute Angle : find something triangular. All triangles have at least one acute angle Adjacent Angle : take that same triangular thing and any 2 angles are adjacent. Angle Bisector : Collinear find something with a straight line and any 3 (or more) things along that line are collinear Concave Polygon Congruent: get two things that are identical in size Convex Polygon Line : use a yardstick and mention it goes on forever. Route 66 goes across the country, but it is not straight Line Segment : use a ruler Linear pair Midpoint: the number 6 on a ruler Obtuse Angle; get a triangle again Parallel lines; find a box and two of the edges will be parallel Perpendicular Lines ; use the same box and use two edges that are perpendicular Plane ; a piece of paper could represent a plane surface, understand it goes on in all directions Point; has no dimensions, but for school purposes, a dot Ray: the graph of the absolute value of x Right Angle: anything that meets at a right angle. Use the box again Slope: a hill Vertical Angles: a map where 2 roads cross good luck

Question:A student says that he is confused between the difference of a line, line segment and a ray. How do you respond?

Answers:This is usually confusing because in geometry you are changing the definition of a line that kids have known their entire lives. Acknowledge that change. A "line segment" is what they have known as a "line." No arrows. A "line" is an infinite line, which is impossible to draw but can be represented by arrows at both ends. A "ray" is 1/2 a line. It is also infinite, but only in 1 direction. Drawn with an arrow at one end. But, the trick is to keep coming of with ways to explain it and not repeating the same thing over and over.