# data[].attributes.headline
Triangle Song: Types of Triangles For Kids Video - YouTube

# data[].attributes.subject
Triangle Song: Types of Triangles For Kids Video - YouTube

# data[].attributes.attachment_link.title
Triangle Song | Types of Triangles Song | Classifying Triangles

# data[].attributes.attachment_link.description
Equilateral triangle: All angles are the same (60 degrees)

# data[].attributes.headline
Slope, Distance, and Midpoint Formulas - YouTube

# data[].attributes.subject
Slope, Distance, and Midpoint Formulas - YouTube

# data[].attributes.headline
An introduction to geometry (Geometry, Points, Lines, Planes and Angles) – Mathplanet

# data[].attributes.subject
An introduction to geometry (Geometry, Points, Lines, Planes and Angles) – Mathplanet

# data[].attributes.attachment_link.title
An introduction to geometry (Geometry, Points, Lines, Planes and Angles) - Mathplanet

# data[].attributes.attachment_link.description
Geometry Points, Lines, Planes and Angles: Finding distances and midpoints Geometry Points, Lines, Planes and Angles: Measure and classify an angle Geometry Points, Lines, Planes and Angles: Polygons

# data[].attributes.headline
Three Undefined Terms: Point, Line, and Plane - Concept - Geometry Video by Brightstorm

# data[].attributes.subject
Three Undefined Terms: Point, Line, and Plane - Concept - Geometry Video by Brightstorm

# data[].attributes.attachment_link.readable_text
In Geometry, we have several undefined terms: point, line and plane. From these three undefined terms, all other terms in Geometry can be defined. In Geometry, we define a point as a location and no size. A line is defined as something that extends infinitely in either direction but has no width and is one dimensional while a plane extends infinitely in two dimensions.

There are three undefined terms in geometry. From these terms we define everything else. The first term is point. The second term is plane. And the third undefined term is the line. So let's go back and define these as much as we can. Now we're not really defining point, we're just describing it. A point has no size; it only has a location. And the way that we label it is with a capital letter. So we can call this Point P. A plane is a flat surface that has no thickness, and it will extend infinitely in every direction.
So one way to visualize what a plane could be is to think about a sheet of paper. Or if there are two differences between a sheet of paper and a plane, the first is this paper does not extend in every direction. Secondly, this paper actually has some thickness and a plane will not.
Now you can name a plane using a single capital letter, usually written in cursive, or by three non-collinear points. And collinear we'll talk about in a second here, but collinear means they're not on the same line. So let's say you had a point right here: Point A, Point B, and Point C. You could call this plane, Plane ABC.
Definition of coplanar: We actually can define this, is points, lines, or anything, segments, polygons in the same plane. So two things are coplanar if they are, just like we have in the picture here, in the same plane. So "co", you can think of it as a word for sharing. So you can think of coplanar as sharing the same plane. Now the third undefined term is a line. And a line is set of points or, the word that you might learn later is locus, extending in either direction infinitely. So a line is going to be all the points, and we can actually select two of them to name it. So we can call this Line AB.
Now when you're labeling a line, it's key to include at least two points. Or if you have some sort of smaller letter over here, we can call this Line L. But notice how I'm writing the arrows above my letters; I have arrows on either side. And these arrows tell you, the geometry student, that it extends infinitely in this direction. Now this arrow here extends infinitely in that direction.
You can have points be collinear, that is, they share the same line. So here we could have, C, D, and E are all collinear. And if you look at Point F here, I drew this in to draw a contrast. You can see that Point F is not on this line, so F is not collinear with C, D, and E. But I could say that E is collinear with C and D, D is collinear with C and E, and C is collinear with D and E.
So the three key terms that are not definable, but only describable, are the line, which is a set of points extending infinitely in one or the other direction; plane, which is a flat surface with no thickness; and the third undefined term is point and that has a location and no size.

# data[].attributes.attachment_link.title
Three Undefined Terms: Point, Line, and Plane - Concept - Geometry Video by Brightstorm

# data[].attributes.attachment_link.description
In Geometry, we have several undefined terms: point, line and plane. From these three undefined terms, all other terms in Geometry can be defined. In Geometry, we define a point as a location and no size. A line is defined as something that extends infinitely in either direction but has no width and is one dimensional while a plane extends infinitely in two dimensions.

# data[].attributes.attachment_link.description
Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.

# data[].attributes.headline
Proving Triangles are Congruent - MathHelp.com - Math Help - YouTube

# data[].attributes.subject
Proving Triangles are Congruent - MathHelp.com - Math Help - YouTube

# data[].attributes.attachment_link.title
Proving Triangles are Congruent - MathHelp.com - Math Help

# data[].attributes.headline
4.2 Triangle Congruence by SSS and SAS - How to Prove Triangles Congurent - YouTube

# data[].attributes.subject
4.2 Triangle Congruence by SSS and SAS - How to Prove Triangles Congurent - YouTube

# data[].attributes.attachment_link.title
4.2 Triangle Congruence by SSS and SAS - How to Prove Triangles Congurent

# data[].attributes.attachment_link.description
http://mrpilarski.wordpress.com/2009/11/16/triangle-congruence-by-sss-and-sas-how-to-prove-triangles-congruent/ Side Side Side Postulate If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. SAS Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

# data[].attributes.headline
Proving Congruence with SSS and SAS | WyzAnt Resources

# data[].attributes.subject
Proving Congruence with SSS and SAS | WyzAnt Resources

# data[].attributes.attachment_link.readable_text
We have learned that triangles are congruent if their corresponding sides and angles are congruent. However, there are excessive requirements that need to be met in order for this claim to hold. In this section, we will learn two postulates that prove triangles congruent with less information required. These postulates are useful because they only require three corresponding parts of triangles to be congruent (rather than six corresponding parts like with CPCTC). Let's take a look at the first postulate.

SSS Postulate (Side-Side-Side)

If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

As you can see, the SSS Postulate does not concern itself with angles at all. Rather, it only focuses only on corresponding, congruent sides of triangles in order to determine that two triangles are congruent. An illustration of this postulate is shown below.

We conclude that ?ABC??DEF because all three corresponding sides of the triangles are congruent.

Let's work through an exercise that requires the use of the SSS Postulate.

Exercise 1

Solution:

The only information that we are given that requires no extensive work is that segment JK is congruent to segment NK. We are given the fact thatA is a midpoint, but we will have to analyze this information to derive facts that will be useful to us.

In the two triangles shown above, we only have one pair of corresponding sides that are equal. However, we can say that AK is equal to itself by the Reflexive Property to give two more corresponding sides of the triangles that are congruent.

Finally, we must make something of the fact A is the midpoint of JN. By definition, the midpoint of a line segment lies in the exact middle of a segment, so we can conclude thatJA?NA.

After doing some work on our original diagram, we should have a figure that looks like this:

Now, we have three sides of a triangle that are congruent to three sides of another triangle, so by the SSS Postulate, we conclude that ?JAK??NAK. Our two column proof is shown below.

We involved no angles in the SSS Postulate, but there are postulates that do include angles. Let's take a look at one of these postulates now.

SAS Postulate (Side-Angle-Side)

If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.

A key component of this postulate (that is easy to get mistaken) is that the angle must be formed by the two pairs of congruent, corresponding sides of the triangles. If the angles are not formed by the two sides that are congruent and corresponding to the other triangle's parts, then we cannot use the SAS Postulate. We show a correct and incorrect use of this postulate below.

Incorrect:

The diagram above uses the SAS Postulate incorrectly because the angles that are congruent are not formed by the congruent sides of the triangle.

Correct:

The diagram above uses the SAS Postulate correctly. Notice that the angles that are congruent are formed by the corresponding sides of the triangle that are congruent.

Let's use the SAS Postulate to prove our claim in this next exercise.

Exercise 2

Solution:

For this solution, we will try to prove that the triangles are congruent by the SAS Postulate. We are initially given that segments AC and EC are congruent, and that segment BC is congruent to DC.

If we can find a way to prove that ?ACB and ?ECD are congruent, we will be able to prove that the triangles are congruent because we will have two corresponding sides that are congruent, as well as congruent included angles. Trying to prove congruence between any other angles would not allow us to apply the SAS Postulate.

The way in which we can prove that ?ACB and ?ECD are congruent is by applying the Vertical Angles Theorem. This theorem states that vertical angles are congruent, so we know that?ACB and ?ECD have the same measure. Our figure show look like this:

Now we have two pairs of corresponding, congruent sides, as well as congruent included angles. Applying the SAS Postulate proves that ?ABC??EDC. The two-column geometric proof for our argument is shown below.

# data[].attributes.attachment_link.title
Proving Congruence with SSS and SAS | Wyzant Resources

# data[].attributes.attachment_link.description
We have learned that triangles are congruent if their corresponding sides and angles are congruent. However, there are excessive requirements that need to be met in order for this claim to hold. In this section, we will learn two postulates that prove triangles congruent with less information required.

# data[].attributes.headline
Practice with Proving Triangles Congruent

# data[].attributes.subject
Practice with Proving Triangles Congruent

# data[].attributes.headline
IXL - Proving triangles congruent by SSS, SAS, ASA, and AAS (Geometry practice)

# data[].attributes.subject
IXL - Proving triangles congruent by SSS, SAS, ASA, and AAS (Geometry practice)

# data[].attributes.attachment_link.title
Practicing Geometry math: 'Proving triangles congruent by SSS, SAS, ASA, and AAS'

# data[].attributes.attachment_link.description
Fun math practice! Improve your skills with free problems in 'Proving triangles congruent by SSS, SAS, ASA, and AAS' and thousands of other practice lessons.

# data[].attributes.headline
Surface Area and Volume Formulas for Figures

# data[].attributes.subject
Surface Area and Volume Formulas for Figures

# data[].attributes.attachment_link.readable_text
In math (especially geometry) and science, you will often need to calculate the surface area, volume, or perimeter of a variety of shapes. Whether it's a sphere or a circle, a rectangle or a cube, a pyramid or a triangle, each shape has specific formulas that you must follow to get the correct measurements.

We're going to examine the formulas you will need to figure out the surface area and volume of three-dimensional shapes as well as the area and perimeter of two-dimensional shapes. You can study this lesson to learn each formula, then keep it around for a quick reference next time you need it. The good news is that each formula uses many of the same basic measurements, so learning each new one gets a little easier.

A cone is a pyramid with a circular base that has sloping sides which meet at a central point. In order to calculate its surface area or volume, you must know the radius of the base and the length of the side.

If you do not know it, you can find the side length ( s) using the radius ( r) and the cone's height ( h).

With that, you can then find the total surface area, which is the sum of the area of the base and area of the side.

To find the volume of a sphere, you only need the radius and the height.

A pyramid with a square base and faces made of equilateral triangles is relatively easy to work with.

You will need to know the measurement for one length of the base ( b). The height ( h) is the distance from the base to the center point of the pyramid. The side ( s) is the length of one face of the pyramid, from the base to the top point.

Another way to calculate this is to use the perimeter ( P) and the area ( A) of the base shape. This can be used on a pyramid that has a rectangular rather than a square base.

When you switch from a pyramid to an isosceles triangular prism, you must also factor in the length ( l) of the shape. Remember the abbreviations for base ( b), height ( h), and side ( s) because they are needed for these calculations.

Yet, a prism can be any stack of shapes. If you have to determine the area or volume of an odd prism, you can rely on the area ( A) and the perimeter ( P) of the base shape. Many times, this formula will use the height of the prism, or depth ( d), rather than the length ( l), though you may see either abbreviation.

An ellipse is also called an oval and it is, essentially, an elongated circle. The distances from the center point to the side are not constant, which does make the formula for finding its area a little tricky.

To use this formula, you must know:

The sum of these two points does remain constant. That is why we can use the following formula to calculate the area of any ellipse.

On occasion, you may see this formula written with (radius 1 or semiminor axis) and (radius 2 or semimajor axis) rather than a and b.

The triangle is one of the simplest shapes and calculating the perimeter of this three-sided form is rather easy. You will need to know the lengths of all three sides ( a, b, c) to measure the full perimeter.

To find out the triangle's area, you will need only the length of the base ( b) and the height ( h), which is measured from the base to the peak of the triangle. This formula works for any triangle, no matter if the sides are equal or not.

Similar to a sphere, you will need to know the radius ( r) of a circle to find out its diameter ( d) and circumference ( c). Keep in mind that a circle is an ellipse that has an equal distance from the center point to every side (the radius), so it does not matter where on the edge you measure to.

These two measurements are used in a formula to calculate the circle's area. It's also important to remember that the ratio between a circle's circumference and its diameter is equal to pi ( π).

The parallelogram has two sets of opposite sides that run parallel to one another. The shape is a quadrangle, so it has four sides: two sides of one length ( a) and two sides of another length ( b).

To find out the perimeter of any parallelogram, use this simple formula:

When you need to find the area of a parallelogram, you will need the height ( h). This is the distance between two parallel sides. The base ( b) is also required and this is the length of one of the sides.

Keep in mind that the b in the area formula is not the same as the b in the perimeter formula. You can use any of the sides-which were paired as a and b when calculating perimeter-though most often we use a side that is perpendicular to the height.

The rectangle is also a quadrangle. Unlike the parallelogram, the interior angles are always equal to 90 degrees. Also, the sides opposite one another will always measure the same length.

To use the formulas for perimeter and area, you will need to measure the rectangle's length ( l) and its width ( w).

The square is even easier than the rectangle because it is a rectangle with four equal sides. That means you only need to know the length of one side ( s) in order to find its perimeter and area.

The trapezoid is a quadrangle that can look like a challenge, but it's actually quite easy. For this shape, only two sides are parallel to one another, though all four sides can be of different lengths. This means that you will need to know the length of each side ( a, b₁, b₂, c) to find a trapezoid's perimeter.

To find the area of a trapezoid, you will also need the height ( h). This is the distance between the two parallel sides.

A six-sided polygon with equal sides is a regular hexagon. The length of each side is equal to the radius ( r). While it may seem like a complicated shape, calculating the perimeter is a simple matter of multiplying the radius by the six sides.

Figuring out the area of a hexagon is a little more difficult and you will have to memorize this formula:

A regular octagon is similar to a hexagon, though this polygon has eight equal sides. To find the perimeter and area of this shape, you will need the length of one side ( a).

# data[].attributes.attachment_link.title
Surface Area and Volume Formulas for Geometric Shapes

# data[].attributes.attachment_link.description
In math (especially geometry) and science, you will often need to calculate the surface area, volume, or perimeter of a variety of shapes. Whether it's a sphere or a circle, a rectangle or a cube, a pyramid or a triangle, each shape has specific formulas that you must follow to get the correct measurements.

# data[].attributes.attachment_link.description
For a polygon (or any other closed curve, such as a circle) - perimeter is the distance around the outside. For a 3D shape, volume is the amount of space displaced.

# data[].attributes.attachment_link.readable_text
After any of those transformations (turn, flip or slide), the shape still has the same size, area, angles and line lengths.

When one shape can become another using only Turns, Flips and/or Slides, then the two shapes are Congruent

When a Resize is neeeded for one shape to become another (we may also Turn, Flip and/or Slide) the two shapes are

In other words:

When one shape becomes another ...

The shapes are ...

... using only Rotate, Reflect and/or Translate

... using a Resize (may also Rotate, Reflect
and/or Translate)

# data[].attributes.attachment_link.description
Learn about the Four Transformations: Rotation, Reflection, Translation and Resizing

# data[].attributes.headline
Intersection of Parabola and Line - 2 solutions! - YouTube

# data[].attributes.subject
Intersection of Parabola and Line - 2 solutions! - YouTube

# data[].attributes.attachment_link.title
Intersection of Parabola and Line - 2 solutions!

# data[].attributes.attachment_link.description
Also called linear-quadratic systems. How to find the two points at which a parabola and line intersect.

# data[].attributes.headline
Right Triangle Calculator with detailed explanation

# data[].attributes.subject
Right Triangle Calculator with detailed explanation

# data[].attributes.attachment_link.title
Right Triangle Calculator with detailed explanation

# data[].attributes.attachment_link.description
Calculator that shows work to solve right triangle problems.