Download Chapter 1 - Essentials of Geometry

Essentials of Geometry
CHAPTER 1
Euclid of Alexandria
•Euclid
• born circa 300 B.C. in Greece
• Best known for book called, Elements
Euclid’s Elements
• Began book with 5 unproved axioms (postulates)
1. Given two points there is one straight line that joins them.
2. A straight line segment can be prolonged indefinitely.
3. A circle can be constructed when a point for its centre and a distance for its radius are given.
4. All right angles are equal.
5. If a straight line falling on two straight lines makes the interior angles on the same side less than two
right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are
less than the two right angles.
Euclid continued
1.1 – Identify Points, Lines, & Planes
• The words, point, line and plane are undefined terms
• Have no formal definition, but there is agreement about what they mean
• Point
• Has no dimension, represented by a dot
• Line
• Has one dimension, represented by a line with two arrowheads, but extends without end
• Plane
• Has two dimensions, represented by a shape but extends in all directions without end
Collinear & Coplaner
• Collinear points
• Points that lie on same line
• Coplaner points
• Points that lie on same plane
Defined Terms
In geometry, terms that can be described using known words such as point or line are called
defined terms.
Defined Terms
• Segment
• Line segment, or segment, consists of two endpoints
• Ray
• Consists of one endpoint and all points extending in one direction
Intersections
Two or more geometric figures intersect if they have one or more points in common
• Intersection
• Set of all points two figures have in common
• Two lines intersect at one point
• Two planes intersect at one line (infinitely many points)
1.2 – Use Segments & Congruence
In geometry, rules that are accepted without proof are called postulates or axioms.
• Postulate 1 – Ruler Postulate
• Points on a line can be matched one to one with the real numbers. The real number that corresponds to
a point is the coordinate of the point
• The distance between points A and B, written as AB, is the absolute value of the difference of the
coordinates of A and B.
Adding Segments
When three points are collinear, you can say that one point is between other two points.
• Postulate 2 – Segment Addition Postulate
• If B is between A and C, then AB + BC = AC.
• If AB + BC = AC, then B is between A and C.
Congruent Segments
• Congruent segments
• Line segments that have the same length
Picture
Lengths are equal.
Segments are
congruent
AB = CD
𝐴𝐵 ≅ 𝐶𝐷
“is equal to”
“is congruent to”
1.3 – Use Midpoint & Distance Formulas
• Midpoint of segment
• Point that divides segment into two congruent segments
• Segment bisector
• Point, ray, line segment, or plane that intersects segment at its midpoint
• Midpoints and segment bisectors bisect a segment
Coordinate Plane & Midpoints
You can use a coordinate plane (x-y plane) and coordinates of endpoints to find the coordinates
of the midpoint.
• Midpoint Formula
• The coordinates of the midpoint of a segment are the averages of the x-coordinates and of the ycoordinates of the endpoints
• If 𝐴(𝑥1 , 𝑦1 ) and 𝐵(𝑥2 , 𝑦2 ) are points in a coordinate plane, then the midpoint M of segment AB has
coordinates
𝑥1 +𝑥2 𝑦1 +𝑦2
,
2
2
Distance Formula
Formula for computing distance between two points in a coordinate plane
• Distance Formula
• If 𝐴(𝑥1 , 𝑦1 ) and 𝐵(𝑥2 , 𝑦2 ) are points in a coordinate plane, then the distance between A and B is
𝐴𝐵 =
(𝑥1 − 𝑥2 )2 +(𝑦1 − 𝑦2 )2
1.4 – Measure & Classify Angles
• Angle
• Two different rays with same endpoint
• Rays are called sides of angle
• Endpoint is called vertex of angle
Measuring Angles
•A protractor can be used to approximate the measure of an angle
•Angles are measures in degrees (°)
• Postulate 3 – Protractor Postulate
• Consider line OB and a point A on for side of line OB. The rays of the form OA and OB can be matched
one to one with the real numbers from 0 to 180.
• The measure of angle AOB is equal to the absolute value of the difference between the real numbers for
ray OA and ray OB.
Classifying Angles
• Angles can be classified as: acute, obtuse, right, or straight
Acute angle
0° < 𝑚∠𝐴 < 90°
Obtuse angle
Right angle
Straight angle
90° < 𝑚∠𝐴 < 180°
𝑚∠𝐴 = 90°
𝑚∠𝐴 = 180°
Adding Angles
• Postulate 4 – Angle Addition Postulate
• If P is in the interior of ∠𝑅𝑆𝑇, then the measure of ∠𝑅𝑆𝑇 is equal to the sum of the measure of ∠𝑅𝑆𝑃
and ∠𝑃𝑆𝑇.
• 𝑚∠𝑅𝑆𝑇 = 𝑚∠𝑅𝑆𝑃 + 𝑚∠𝑃𝑆𝑇
Congruent Angles
• Congruent angles
• Two angles are congruent if they have the same measure
Angle measure are equal.
Angles are congruent.
𝑚∠𝐴 = 𝑚∠𝐵
∠𝐴 ≅ ∠𝐵
“is equal to”
“is congruent to”
Angle Bisector
• Angle bisector
• Ray that divides an angle into two angles that are congruent (same measure)
• Steps for constructing angle bisector
1.
2.
3.
4.
5.
1.5 – Describe Angle Pair Relationships
• Complementary angles
• Two angles are complementary if sum of their angle measures is 90°
• Each angle is complement of other
• Supplementary angles
• Two angles are supplementary if sum of their angle measures is 180°
• Each angle is supplement of other
• Complementary and supplementary angles can be both adjacent angles or nonadjacent angles
Angle Pairs
• Linear pair
• Two adjacent angles whose noncommon sides are opposite rays
• Angles are supplementary
• Vertical angles
• Two angles whose sides form two pairs of opposite rays
1.6 – Classify Polygons
In geometry, a figure that lies in a plane is called a plane figure
• Polygon
• Closed plane figure with following properties
1.
Formed by three or more line segments called sides
2.
Each side intersects exactly two sides, one at each endpoint
3.
No two sides with a common endpoint are collinear
• Vertex of a polygon
• Endpoint of each side is called vertex (plural is vertices)
• Polygon can be named by listing vertices in consecutive order
Convex or Concave?
A polygon can either be convex or concave
• Convex polygon
• No line that contains a side of polygon contains a point in the interior of polygon
• Concave polygon
• Polygons that are nonconvex
• Sides of polygon extend to interior of polygon
Classifying Polygons
Polygons are named by the number of sides it has
Number of Sides
Type of Polygon
3
Triangle
4
Quadrilateral
5
Pentagon
6
Hexagon
7
Heptagon
8
Octagon
9
Nonagon
10
Decagon
12
Dodecagon
n
N-gon
Is it regular?
• Equilateral
• Polygon with all sides congruent
• Equiangular
• Polygon with all interior angles congruent
• Regular
• Convex polygon that is both equilateral & equiangular