1. Vertical Angles
Vertical angles are across from each other and are created by intersecting lines

2. Supplementary Angles

3. Complementary Angles

4. Complementary/Supplementary Angles

5. Transversals Interior angles Exterior angles Alternate interior angles
Alternate exterior angles
Corresponding angles

6. In the figure, m || n and p is a transversal.
Example
In the figure, m || n and p is a transversal.
Explain why m∠1 + m∠6 = 180°.
Vertical angles have the same measure.
E.g., ∠3 & ∠6; ∠1 & ∠5; ∠4 & ∠8; ∠2 & ∠7

7. Measures of the Interior Angles of a Polygon
The sum of the measures of the interior angles of any convex polygon with n sides is (n – 2)180°.
The measure of a single interior angle of a regular
n-gon is

8. Measures of the Exterior Angles of a n-gon
The sum of the measures of the exterior angles of a convex n-gon is 360°.

9. Example
a. Find the measure of each interior angle of a regular decagon.
b. Find the number of sides of a regular polygon each of whose interior angles has measure 175°.

10. Example
Lines m and n are parallel, find the measures of the numbered angles.

11. Simple Closed Surfaces
Sphere – the set of all points at a given distance from a given point (the center)
Solid – the set of all points on a simple closed surface together with all interior points
Polyhedron – a simple, closed surface made up of polygonal regions (faces). The vertices of the polygonal regions are the vertices of the polyhedron. The sides of the polygonal regions are called the edges of the polyhedron.

12. Polyhedron Simple, closed surface made up of polygonal regions.
Polyhedra:
Not polyhedra:

13. Prisms vs. Pyramids
A prism is a polyhedron in which two congruent faces lie in parallel planes and the other faces are bounded by parallelograms.
A pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face.

14. Prisms Right prism – lateral faces are perpendicular to the bases.
Oblique prism – lateral faces are not perpendicular to the bases.

15. Pyramids
Right pyramid – lateral faces are congruent isosceles triangles.

16. Truncated Polyhedron
If one or more corners of a polyhedron is removed by an intersecting plane or planes, the polyhedron is a truncated polyhedron.

17. Regular Polyhedra
Convex polyhedron – a polyhedron in which a segment connecting any two points in the interior of the polyhedron is in the interior of the polyhedron.
Regular polyhedron – a convex polyhedron whose faces are congruent regular polygonal regions such that the number of edges that meet at each vertex is the same for all the vertices of the polyhedron.

18. Regular Polyhedra
Cube
Tetrahedron
Octahedron

19. Regular Polyhedra
Dodecahedron
Icosahedron

20. Cylinders and Cones
Cylinder – a simple, closed surface that is not a polyhedron; formed as a segment AB parallel to a given line  traces a planar curve other than a polygon.

21. Cylinders and Cones
Cone – the union of the line segments connecting a point P with each point of a simple, closed curve, the simple, closed curve, and the interior of the curve.

22. Nets
These patterns can be used to construct the five regular polyhedra.