Week 6 Day 1 (Introduction to Geometry)

Geometry Introduction

Geometry p. 211- 251; Review Test p. 274 (q. 1-47), Glossary p. 280-284

 

Note that the Glossary for Geometry is on p. 280-284. I will cover the terminology briefly as it occurs throughout this chapter.

 

Plane geometry – deals with two-dimensional geometry, or flat shapes and figures

Flat, 2 dimensional figures (2d):

Solid geometry – deals with three-dimensional geometry, or shapes and figures in 3D

Point – the most fundamental geometrical concept, an imaginary dot with no length, width, or thickness, just a location in space; named by a letter or number

 

Angles p. 220

Angle – formed by two rays, lines, or line segments (sides) meeting at a single point (vertex), measured from 0° to 360°; named by a letter, number, or a combination of the names of the sides, e.g. Ð1, ÐA, or ÐBAC (or CAB) for the picture below:

 

Angles can be right (exactly 90°), acute (less than 90°), obtuse (greater than 90° but less than 180°), or straight (exactly 180°).

Angles can also be adjacent (sharing one side) or vertical (also known as opposite), and complementary (adding up to 90°) or supplementary (adding up to 180°).

Angle bisectors divide an angle exactly in half.

 

Lines p. 224

Lines are a group of points extending infinitely in both directions; they are named by two points with capital letters with a double-arrow on top, or by a single letter, usually in lower-case:

Line segments are finite pieces of lines, usually named by their two endpoints and a line without any arrows:

Rays are infinite pieces of lines that have one endpoint and continue infinitely in one direction, usually named by the endpoint first and another point along the infinite direction next and a line with one arrow:

Lines may intersect (cross each other), and this intersection may or may not be perpendicular (i.e., a 90° angle) in which case we would say that e.g., l is perpendicular to m or l ^ m; alternatively, lines can be parallel (continuing to infinity without touching) in which case we would say that e.g., l is parallel to m or l || m.

 Parallel lines intersected by a transversal - p. 226

When parallel lines are intersected by a third line at an angle (parallel lines cut by a transversal), they form a number of angles: adjacent (touching = supplementary), vertical (opposite = identical in measure), corresponding (similar but on the other line = identical measure), alternate interior (opposites between the parallels = identical measure) or exterior angles (opposites outside the parallels = identical measure), and consecutive interior angles (stacked on top of each other between the parallels = supplementary). Given the measure of any single angle in such a situation, you can determine the value of all the other angles.

For example, Ð1 is adjacent to Ð2 and Ð3, vertical to Ð4, corresponding to Ð5, alternate exterior to Ð8. Ð3 is adjacent to Ð1 and Ð4, vertical to Ð2, corresponding to Ð7, alternate interior to Ð6, and consecutive interior to Ð5. This means that Ð1 is equal in measure to Ð4, Ð5, and Ð8, and will sum to 180° with Ð2, Ð3, Ð6, or Ð7. If Ð1 is 130°, then so are Ð4, Ð5, and Ð8, and that means Ð2, Ð3, Ð6, and Ð7 are each 50°.