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Explanation: A midsegment of a trapezoid is a segment that connects the midpoints of the two non-parallel sides of a trapezoid. This segment has two special properties. It is parallel to the bases of the trapezoid. The length of the midsegment is the average of the lengths of the two bases.

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https://www.youtube.com/watch?v=aUy-_WnEV30

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Let's draw ourselves a triangle. Let's say this side has length 6. Let's say this side right over here has length 10. And let's say that this side right over here has length x. And what I'm going to think about is how large or how small that value x can be. How large or small can this side be? So the first question is how small can it get? Well, if we want to make this small, we would just literally have to look at this angle right over here. So let me take a look at this angle and make it smaller. So let's try to make that angle as small as possible. So we have our 10 side. Actually let me do it down here. So you have your 10 side, the side of length 10, and I'm going to make this angle really, really, really small, approaching 0. If that angle becomes 0, we end up with a degenerate triangle. It essentially becomes one dimension. We lose our two-dimensionality there. But as we approach 0, this side starts to coincide or get closer and closer to the 10 side. And you could imagine the case where it actually coincides with it and you actually get the degenerate. So if want this point right over here to get as close as possible to that point over there, essentially minimizing your distance x, the closest way is if you make the angle the way equal to 0, all the way. So let's actually-- let me draw a progression. So now the angle is getting smaller. This is length 6. x is getting smaller. Then we keep making that angle smaller and smaller and smaller all the way until we get a degenerate triangle. So let me draw that pink side. So you have the side of length 10. Now the angle is essentially 0, this angle that we care about. So this side is length 6. And so what is the distance between this point and this point? And that distance is length x. So in the degenerate case, this length right over here is x. We know that 6 plus x is going to be equal to 10. So in this degenerate case, x is going to be equal to 4. So if you want this to be a real triangle, at x equals 4 you've got these points as close as possible. It's degenerated into a line, into a line segment. If you want this to be a triangle, x has to be greater than 4. Now let's think about it the other way. How large can x be? Well to think about larger and larger x's, we need to make this angle bigger. So let's try to do that. So let's draw my 10 side again. So this is my 10 side. I'm going to make that angle bigger and bigger. So now let me take my 6 side and put it like that. And so now our angle is getting bigger and bigger and bigger. It's approaching 180 degrees. At 180 degrees, our triangle once again will be turned into a line segment. It'll become a degenerate triangle. So let me draw the side of length x, try to draw it straight. So we're trying to maximize the distance between that point and that point. So this is side of length x and let's go all the way to the degenerate case. In the degenerate case, at 180 degrees, the side of length 6 forms a straight line with the side of length 10. And this is how you can get this point and that point as far apart as possible. Well, in this situation, what is the distance between that point and that point, which is the distance which is going to be our x? Well in this situation, x is going to be 6 plus 10 is 16. If x is 16, we have a degenerate triangle. If we don't want a degenerate triangle, if we want to have two dimensions to the triangle, then x is going to have to be less than 16. Now the whole principle that we're working on right over here is called the triangle inequality theorem and it's a pretty basic idea. That any one side of a triangle has to be less, if you don't want a degenerate triangle, than the sum of the other two sides. So length of a side has to be less than the sum of the lengths of other two sides. If you're willing to deal with degenerate triangles-- where you essentially form a line segment, you lose all your dimensionality, you turn to a one-dimensional figure-- then you could say less than or equal, but we're just going to stick to non-degenerate triangles. So the length of a side has to be less than the sum of the lengths of the other two sides. And just using this principle, we could have come up with the same exact conclusion. You could say, well look, x is one of the sides. It has to be less than the sum of the lengths of the other two sides. So it has to be less than 6 plus 10, or x has to be less than 16-- the exact same result we got by visualizing it like this. You want to say how large can x be? Well you could say, well, 10 has to be less than-- Or how small can x be? You have to say 10 has to be less than 6 plus x, the sum of the lengths of the other two sides. If you subtract 6 from both sides right over here, you get 4 is less than x, or x is greater than 4. So this is a, in some level, it's a kind of a basic idea, but it's something that you'll see definitely in geometry. And then you'll go far into other types of mathematics and you'll see other versions of what's essentially this triangle inequality theorem.

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Triangle inequality theorem (video) | Khan Academy

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Let's draw ourselves a triangle. Let's say this side has length 6. Let's say this side right over here has length 10. And let's say that this side right over here has length x. And what I'm going to think about is how large or how small that value x can be.

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Generally: another word for height. For Triangles: a line segment leaving at right angles from a…

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http://mathworld.wolfram.com/AngleBisector.html

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Triangle Centers Overview - Math Open Reference

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An overview of the various centers of a triangle

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http://www.virtualnerd.com/geometry/triangle-relationships/inequalities-one-triangle/triangle-order-sides-example

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ordering angle measurements based on side lengths

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ordering angle measurements based on side lengths

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Ordering Angles and Ordering Sides of a Triangle Based on Measure

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3 Problems Involving Ordering Angles and Ordering Sides of a Triangle Based on Measure

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a segment whose endpoints are a vertex to the midpoint of the opposite side

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As you well know by now, being able to deduce key information from a limited set of facts is the basis of geometry. An important type of segment, ray, or line that can help us prove congruence is called an angle bisector. Understanding what angle bisectors are and how they affect triangle relationships is crucial as we continue our study of geometry. Let's investigate different types of bisectors and the theorems that accompany them.

Perpendicular Bisector

A perpendicular bisector is a special kind of segment, ray, or line that

(1) intersects a given segment at a 90° angle, and

(2) passes through the given segment's midpoint.

Segment CD is the perpendicular bisector to segment AB.

We derive two important theorems from the characteristics of perpendicular bisectors. We can use these theorems in our two-column geometric proofs, or we can just use them to help us in geometric computations.

Perpendicular Bisector Theorem

If a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

Converse also true: If a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of the segment.

These theorems essentially just show that there exist a locus of points (which form the perpendicular bisector) that are equidistant from the endpoints of a given segment which meet at the midpoint of the segment at a right angle. An illustration of this concept is shown below.

Points E, F, G, and H (along with an infinite amount of points) are equidistant from A and B. Together, they form the perpendicular bisector of segment AB.

The perpendicular bisectors of a triangle have a very special property. Let's investigate it right now.

Circumcenter Theorem

The perpendicular bisectors of the sides of a triangle intersect at a point called the circumcenter of the triangle, which is equidistant from the vertices of the triangle.

Point G is the circumcenter of ?ABC.

Angle Bisectors

Now, we will study a geometric concept that will help us prove congruence between two angles. Any segment, ray, or line that divides an angle into two congruent angles is called an angle bisector.

We will use the following angle bisector theorems to derive important information from relatively simple geometric figures.

Angle Bisector Theorem

If a point lies on the bisector of an angle, then it is equidistant from the sides of the angle.

Converse also true: If a point in the interior of an angle is equidistant from the sides of the angle, then it lies on the bisector of the angle.

The points along ray AD are equidistant from either side of the angle. Together, they form a line that is the angle bisector.

Similar to the perpendicular bisectors of a triangle, there is a common point at which the angle bisectors of a triangle meet. Let's look at the corresponding theorem.

Incenter Theorem

The angle bisectors of a triangle intersect at a point called the incenter of the triangle, which is equidistant from the sides of the triangle.

Point G is the incenter of ?ABC.

Summary

While similar in many respects, it will be important to distinguish between perpendicular bisectors and angle bisectors. We use perpendicular bisectors to create a right angle at the midpoint of a segment. Any point on the perpendicular bisector is equidistant from the endpoints of the given segment. The point at which the perpendicular bisectors of a triangle meet, or the circumcenter, is equidistant from the vertices of the triangle.

On the other hand, angle bisectors simply split one angle into two congruent angles. Points on angle bisectors are equidistant from the sides of the given angle. We also note that the points at which angle bisectors meet, or the incenter of a triangle, is equidistant from the sides of the triangle.

Let's work on some exercises that will allow us to put what we've learned about perpendicular bisectors and angle bisectors to practice.

Exercise 1

Solution:

The most important fact to notice is that BC is the perpendicular bisector ofAD because, although it is just one statement, we can derive much information about the figure from it. The fact that it is a perpendicular bisector implies that segmentDB is equal to segment AB since it passes through the midpoint of segmentAD. Therefore, we have

Subtracting 7x from both sides of the equation yields

So, we have x=6.

Exercise 2

Solution:

In order to solve for y, we have to use the information given by the Circumcenter Theorem. This theorem states that the circumcenter is equidistant from the vertices of the triangle. Thus, we know thatRN=AN=KN. For this part of the problem, we only need to solve fory with AN=KN. We have

Exercise 3

Find the value of x.

Solution:

The illustration shows that points A and B are equidistant from pointL. By the converse of the Angle Bisector Theorem, we know thatL must lie on the angle bisector of ?AYB. This means that?AYL=?BYL, so we can solve for x as shown below:

So, our answer is x=4.

Exercise 4

From the information we've been given, we know that ?PQS is congruent to?SQR because QS bisects the whole angle, ?PQR. We have been given the measure of the whole angle and the measure of?SQR, which is half of the entire angle (since the angle has been bisected). Therefore, we have

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As you well know by now, being able to deduce key information from a limited set of facts is the basis of geometry. An important type of segment, ray, or line that can help us prove congruence is called an angle bisector.