Wednesday, September 10, 2008

Geometry - Chapter 1 - Essentials of Geometry

Chapter 1 - Essentials of Geometry

1-1 Undefined Terms:
A. Undefined terms:
their meaning is accepted without definition
1. Point: definition - is that which has no part, a dot.
2. Set: is a collection of objects
3. Line - is breathless length;
4. Straight line: is a line that lies evenly with the points on itself.
5. Plane: is a set of points that form a flat surface extending indefinitely in all directions.
1-2 The real numbers and their Properties
Every real number corresponds to a point on a number line and every point on the number line corresponds to a real number.

1-3 Definitions, lines, and line segments

A. Definition: is a statement of the meaning of the term
B. Collinear set of points: is a set of points all of which lie on the same straight line.
C. Noncollinear set of points: is a set of three or more points that DO NOT all lie on the same straight line.

D. Distance between two points - every point on a line corresponds to a real number called its coordinate. To find the distance between any two points, find the absolute value of the difference between the coordinates of the two points.

If point A is at -2 and point D is at 1, what is the distance between A and D?

take the absolute value of (-2 - 1) = the absolute value of (-3) = 3

E. Order of points on a line
1. Betweenness:
B is between A and C if and only if A, B, and C are distinct collinear points and AB + BC = AC
also, this is the partition postulate: the part + the part = the whole

2. Line Segment: or segment, is a set of points consisting of two endpoints on a line, called endpoints, and all of the points on the line between the endpoints.

1-5 Rays and Angles:
Definition: two points, A and B, are on one side of a point P if A, B, and P are collinear and P is not between A and B.

Half-line: every point on a line divides the line into 2 opposite set of points called half-lines.

Definition: A ray consists of a point (endpoint) on a line and all points on one side of the point.

Definition: Opposite rays: are two rays of the same line with a common endpoint and no other point in common.

Point B is the vertex, BA is a side of the angle and BC is the other side of the angle.
The measure of an angle is the number of degrees in the angle.
A straight angle is an angle that is the union of opposite rays.
Its degree measure is 180 degrees
Acute Angles - is an angle whose degree measure is greater than 0 degrees and less than 90 degres.
Right angles - is an angle whose degree measure is 90 degrees.
Obtuse angle - is an angle whose degree measure is greater than 90 degrees and less than 180 degrees.
Note:If you bisect a straight angle, you get two 90 degree angles or 2 right anglesIf you bisect an obtuse angle, you get two acute angles.
1-6 More angle definitions
Congruent angles are angles that have the same measure.
Definition: A bisector of an angle is a ray whose endpoint is the vertex of the angle and that divides that angle into two congruent angles.
Given angle ABC with angle bisector BD, we can conclude that angle ABD is congruent to angle CBD.
If angle ABC measures 70 degrees, what is the measure of angle ABD? 35 degrees
Example: Given angle QRS with angle bisector RT, if the measure of angle QRS = 10x, and the measure of angle SRT = 3x + 30, what is the measure of angle QRS?
we know: measure of angle QRT = measure of angle SRT + measure of angle QRS
and since the angle QRS is bisected by ray RT, then angle QRT = angle SRT
so 3x + 30 = measure of angle SRT,
3x + 30 + 3x + 30 = 10x
6x + 60 = 10x
60 = 4x
15 = x
measure of angle QRS = 10x = 10 (15) = 150 degrees
1-7 Triangles:
I) Vocabulary:
A) Base angles - the two angles adjacent to the base of an isosceles triangle.
B) Vertex angles - the angle opposite the base of an isosceles triangle.
II) Theorems:
A) Base Angles Theorem: if two sides of a triangle are congruent, then the angles opposite them are congruent.
B) Converse of the Base Angles Theorem: if two angles of a triangle are congruent, the the sides opposite them are congruent.
A) If a triangle is equilateral, then it is equiangular.
B) If a triangle is equiangular, then it is equilateral.
I) Vocabulary:
A) Triangle - is a figure formed by three segments joining three noncollinear points.
B) Classification of triangles:
1) By sides:
a) Equilateral Triangle - has 3 congruent sides
b) Isosceles Triangle - has at least 2 congruent sides
c) Scalene Triangle - has no congruent sides
2) By angles:
a) Acute Triangle - has 3 acute angles
b) Equiangular Triangle - has 3 congruent angles that measure 60 degrees each
c) Right Triangle - has one right angle and 2 acute angles
d) Obtuse Triangle - has one obtuse angle and 2 acute angles
C) Vertex - each of the three points joining the sides of a triangle (plural - vertices)
D) Adjacent Sides - in a triangle, two sides sharing a common vertex.
E) Right triangles have 2 sides that form the right angle called the legs. The side opposite the right angle is the hypotenuse of the triangle.
F) Interior angles - when the sides of a triangle are extended, other angles are formed. the three original angles are the interior angles.
G) Exterior angles - the angles that are adjacent to the interior angles.
H) Theorem:
1) Triangle sum theorem - the sum of the measures of the interior angles of a triangle is 180 degrees.
2) Exterior Angle Theorem - the measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.
I) Corollary to a theorem - is a statement that can be proved easily using the theorem.
Example: The acute angles of a right triangle are complementary.

Example: given Triangle ABC with side BC extended through point D, if angle A = 65 degrees and angle ACD = 2x + 10 and angle B = x, solve for x.
angle A + angle B = angle ACD65 + x = 2x + 1055 = x