5.1 – Angle Sums of a Polygon

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
• 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
1. Convex
2. Equilateral (same side lengths, congruent side lengths)
3. 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 prefixdia-, 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

• 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
• 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!

• 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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