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

Supplementary Angles

Complementary Angles

Complementary/Supplementary Angles

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

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

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

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°.

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°.

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

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.

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

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.

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

Pyramids Right pyramid – lateral faces are congruent isosceles triangles.

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

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.

Regular Polyhedra Cube Tetrahedron Octahedron

Regular Polyhedra Dodecahedron Icosahedron

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.

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.

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