**Key Terms**

- Concave – Having one or more indentations.
- In math, a polygon is concave if a line can be drawn that contains a side of the polygon and also contains points inside the polygon.

- Convex – Having no indents. No lines can be drawn between vertices that are inside of the polygon.
- Decagon – A polygon with 10 sides.
- Diagonal – A line segment that connects two nonconsecutive vertices of a polygon.
- Exterior Angles – Angles that are on the outside of a polygon and form linear pairs with interior angles of the polygon.
- polygon – A closed plane figure.
- Regular Polygon – A convex polygon in which all sides and angles are congruent.

**Review**

- The word
**polygon**comes for the Greek prefix*poly-,*meaning many, and the Greek root*gonia*, which means angle.- So, a polygon is a shape with many angles!

- A
*vertex*of a polygon is the point where two of its sides meet.- The plural of
*vertex*is*vertices*

- The plural of
- Triangles are 3 sided polygons

**Notes**

- Polygons
- Two-dimensional (plane)
- Closed (no openings in sides)
- All straight sides (segments)
- At least three sides
- At least three angles
- At least three vertices

- Name polygons in clockwise and counter-clockwise order.

- Concave and Convex Polygons
- Click the “play” button to watch the video, then go to Pg 2 in Apex Study 5.1 to try sorting the rest of the polygons.

- Concave polygons have dents, and you can draw a line through them when you trace a side.

- Regular Polygons
- Convex
- Equilateral (same side lengths, congruent side lengths)
- Equiangular (same interior angle measures and same exterior angle measures)

- Click on the link below to practice adjusting the number of sides or side lengths of regular polygons

- Click the “play” button to watch the video, then go to Pg 4 in Apex Study 5.1 to try match the names of common polygons to their side lengths.

- Diagonals
- The word
*diagonal*comes from the Greek prefix*dia-*, meaning “across,” and the Greek root*gonia*, which means “angle.” - A diagonal goes across a polygon from angle to angle
- Formula for Finding the Number of a Polygon with “n” Sides: n(n − 3) / 2
- Ex. 10 sided polygon (decagon): 10(10-3)/2 = 10(7)/2 = 70/2 = 35 diagonals!
- Check out this Diagonals tool on Math Is Fun: http://www.mathsisfun.com/geometry/polygons-diagonals.html

- The word

- If you draw all of the diagonals from ONE vertex, you can divide the polygon into triangles

- Each triangle’s interior angles add up to 180°.
- How many triangles do you see in each polygon above?
- To find the SUM of the Interior Angles of a Polygon, multiple the number of triangles by 180°, or use this formula:
- (n-2) * 180 = Sum of Interior Angles of a Polygon
- Why n-2? Well, each n is a side. So, using the polygons above, notice the first one has 4 sides. And, 4 – 2 = 2. Do you see 2 triangles? 2 * 180 = 360.
- In the second polygon, you count 5 sides. 5 – 2 = 3. Do you see 3 triangles? 3 * 180 = 540.
- In the third polygon, you count 8 sides. 8 – 2 = 6. Do you see 6 triangles? 6 * 180 = 1080.

- Interior Angles of a Polygon
- Are supplementary (180°) with each exterior angle
- Are INSIDE of the polygon

- Click on the link below to practice adjusting the number of sides or side lengths of regular polygons

- Sum of Interior Angle Formula (learned with Diagonals, above)

- This table will come in handy as a quick reference guide

- Regular Polygons have CONGRUENT Interior Angles
- To find the measure of ONE interior angle of a regular polygon, find the SUM of the Interior Angles, then divide by how many angles (vertices) there are!
- Ex. A Heptagon has 7 sides and the sum of interior angles is 900 (see chart above).
- Since it has 7 sides, it also has 7 angles.
- 900 divided by 7 = 128.57°.
- So, each angle of a regular heptagon is 128.57 degrees!

- Play the game on Apex Study 5.1, Pg 15 to get some practice!

- Exterior Angles of a Polygon
- Add up to 360°
- Are on the OUTSIDE of the polygon
- Form LINEAR PAIRS with Interior Angles of a polygon
- Are supplementary (180°) with each interior angle of the polygon
- There are 2 exterior angles for every vertex of a polygon
- They are NOT vertical from interior angles
- They are NEVER obtuse!

- Click the links below to practice changing the exterior angles of a polygon.
- Notice that they will always add up to 360°, even with concave polygons (which are negative values)
- Link: http://www.mathopenref.com/common/appletframe.html?applet=polygonextangles&wid=600&ht=350
- Link: https://www.geogebratube.org/student/m19413

- Formula for Finding Exterior Angles of a Polygon (written 2 ways, because multiplication and division are inverses)
- 360 / n = Exterior Angle Measure
- Exterior Angle Measure * n = 360
- Click “play” to see an example of how to find the Sum of Interior Angles of a polygon if you only know the measure of ONE exterior angle.

- Try the practice problems in Apex 5.1 on Pg 25.

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