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High School Geometry: Help and Review13 chapters | 162 lessons

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Lesson Transcript

Instructor:
*Miriam Snare*

Miriam has taught middle- and high-school math for over 10 years and has a master's degree in Curriculum and Instruction.

This lesson explains the definition and properties of a quadrilateral. We will look at examples of two categories of quadrilaterals. At the end, you will be able to test your knowledge with a quiz.

A **quadrilateral** is a closed figure with four straight sides. You can make a quadrilateral by taking (or imagining) anything straight and thin you might have handy: pens, toothpicks, chopsticks, etc. A square is one type of a special quadrilateral.

However, just to make it interesting, try to use four things that are not all the same length. Place your objects so that each end of one touches the end of another. I used pencils to form my quadrilateral, as you can see here. I placed each pencil tip so that it touches the eraser of another pencil. That way, I get a **closed** figure, meaning there are no gaps between sides and no side extends past the end of another side. Now we have our quadrilateral with four straight sides.

Let's simplify the figure by replacing the pencils with segments:

Each point where two sides touch is called a **vertex**. We name each vertex with a capital letter. Let's name our quadrilateral with the four vertices: *P*, *N*, *C*, and *L*. It also has four sides: the segments *PN*, *NC*, *CL*, and *LP*. To name the whole quadrilateral, we choose any vertex as a starting point and list all of the vertices going around either clockwise or counterclockwise. There are many possible names, including *PNCL*, *LCNP*, or *CLPN*.

If you play around a little with the objects that form your quadrilateral, it might be possible that you get a shape that looks something like this:

This is still a quadrilateral because it follows the definition; it has four straight sides that form a closed figure. Again, we can simplify the figure with segments and name the vertices.

This quadrilateral has four vertices: *A*, *R*, *O*, and *W*. It has four sides: segments *AR*, *RO*, *OW*, and *WA*. The whole quadrilateral could be named *AWOR*, *ROWA*, or *AROW*.

The two quadrilaterals, *PNCL* and *AROW*, are examples of two different types of quadrilaterals. There are several other special quadrilaterals, such as parallelograms, trapezoids, and kites, but we won't get into the properties specific to each of those types in this lesson. We are only going to discuss two categories of quadrilaterals. Let's look at how to tell the difference between those two types in the next section.

Depending on how the sides of a quadrilateral are connected to one another, quadrilaterals can be divided into two categories: convex and concave. To tell the difference between a convex and a concave quadrilateral, draw (or imagine) segments between each pair of unconnected vertices. If both segments lie inside the quadrilateral, then the figure is **convex**. If one of the segments lies outside of the quadrilateral, then the figure is **concave**.

Quadrilateral *PNCL* is convex because segments *PC* and *LN* are inside the quadrilateral. Quadrilateral *AROW* is concave because segment *AO* is outside of the quadrilateral.

Now that we've discussed the arrangement of the sides, let's turn our attention to the angles that are formed in a quadrilateral. The angles inside of every quadrilateral add up to 360 degrees. In a convex quadrilateral, each of the angles will measure less than 180 degrees. In a concave quadrilateral, one of the angles will measure more than 180 degrees.

A **quadrilateral** is a closed figure with four straight sides. A **closed** figure means there are no gaps between sides, and no side extends past the end of another side. If both segments lie inside the quadrilateral, then the figure is **convex**. If one of the segments lies outside of the quadrilateral, then the figure is **concave**.

A specific quadrilateral is named by listing its vertices in either clockwise or counterclockwise direction around the figure. Each point where two sides touch is called a **vertex**, which is the singular of 'vertices.'

The four angles of every quadrilateral add up to 360 degrees. In a convex quadrilateral, each of the angles will measure less than 180 degrees. In a concave quadrilateral, one of the angles will measure more than 180 degrees.

- A quadrilateral is made of four straight sides that meet at four angles, called vertices
- A quadrilateral can be convex or concave
- The sum of the angles of a quadrilateral is 360 degrees

When you are done, you should be able to:

- Identify a quadrilateral
- Locate the sides and vertices of a quadrilateral
- Distinguish between a concave quadrilateral and a convex quadrilateral

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High School Geometry: Help and Review13 chapters | 162 lessons

- Parallelograms: Definition, Properties, and Proof Theorems 5:20
- Measuring the Area of a Parallelogram: Formula & Examples 4:02
- What Is a Rhombus? - Definition and Properties 4:24
- Measuring the Area of a Rhombus: Formula & Examples 6:30
- Kites in Geometry: Definition and Properties 4:21
- Rectangles: Definition, Properties & Construction 4:08
- Measuring the Area of a Rectangle: Formula & Examples 4:40
- Squares: Definition and Properties 6:52
- Trapezoids: Definition and Properties 4:24
- Measuring the Area of a Trapezoid 4:38
- Using Heron's Formula in Geometry 5:54
- What is a Quadrilateral? - Definition, Properties, Types & Examples 4:45
- Go to Quadrilaterals: Help and Review

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