10.3 Angles in polygons

Links and useful resources

• gr7's Prealgebra Course Outline START HERE<<<
• AoPS Online Textbook
• IXL Grade 7 index
• IXL Grade 8 index
• AoPS Alcumus
• IXL prealgebra practice index
• OpenSTAX Prealgebra-1 textbook

Lesson-specific resource links

• AOPS angles in a polygon 1
• AOPS angles in a polygon 2

Concept summary and lesson

• interior angles
• exterior angles
• sum of interior angles
• sum of exterior angles
• convex vs concave
• obtuse, right, and acute triangles
• regular polygon

Polygons are composed of vertices (singular: vertex), and sides. Each pair of sides meets at a vertex, and they form an angle that is not a straight angle.

Definitions

• Interior angle: An angle measured between adjacent sides of a polygon on the inside of the polygon
• Exterior angle: An angle measured between adjacent sides of a polygon on the outside of the polygon
• convex polygon: A polygon whose interior angles are all less than 180. You can think of it as a polygon with no "dents" poking into its interior. If you stretched a rubber band around a convex polygon, it would touch all of the vertices.
• concave polygon: A polygon with at least one interior angle greater than 180. These have a dent. A rubber band stretched around a concave polygon would skip over at least one vertex.

Some angle properties

• the sum of interior angles of any triangle is always 180$\mathrm{°}$
• The sum of interior angles of any quadrilateral is always $360\mathrm{°}$
• In general, the sum of the interior angles of any n-sided polygon is $180(n-2)\mathrm{°}$
• The sum of exterior angles of any polygon is always $360\mathrm{°}$

Guided practice

• Show that the angles in a quadrilateral must always add up to 360$\mathrm{°}$
• Show that the exterior angles of any convex polygon add up to $360\mathrm{°}$
• https://artofproblemsolving.com/ebooks/prealgebra-ebook/par/100277

Homework

• #hw (math) 10.3 Angles in Polygons (p. 397) 2025-03-05