Chapter 1 Essentials of Geometry

1.1 Identifying Points, Lines, and Planes Geometry: Study of land or Earth measurements Study of a set of points Includes undefined terms, definitions, postulates (axioms), theorems, properties, and logic

Undefined terms Point- a location, no dimension, occupies no space Name: A

Line- one dimension, has no thickness, extends without end, contains an infinite number of points, use 2 points to name a line or a lower case letter Name: m, AB, BA, BC, CA Through any 2 points, there is exactly one line

Plane- 2 dimensions, extends without end in all directions, has length and width but no depth Name: ABF, EBC, plane T Through any 3 points there is exactly one plane.

Collinear- points that lie on the same line

Coplanar- points that lie in the same plane

Segment- consists of the endpoints and all points between the endpoints. Endpoint- a point at the end of a segment or the starting point of a ray. Name: AE or EA, BC, BD

Ray- consists of a part of a line that starts at a point and extends infinitely in one direction. Name: AB, DC

Opposite rays If point C lies on AB between A and B, then CA and CB are opposite rays.

Intersection Geometric figures with one or more points in common The set of all points that figures have in common

1.2 Use Segments and Congruence Postulate or Axiom- a rule that is accepted without proof, “ If - then statements” Postulates: (p. 96) The intersection of two lines is a _______________. The intersection of two planes is a ______________. Through any two points there is exactly one _______. Through any three noncollinear points there is exactly one ________________. If two points lie in a plane, then the line containing them _______________________.

Ruler Postulate The points on a line can be matched one to one with the real numbers. Coordinate- the real number that corresponds to a point Distance- the absolute value of the difference of the coordinates

Segment Addition Postulate If B is between A and C on a line, then AB + BC = AC If AB + BC = AC, then B is between A and C.

1.3 Use Midpoint and Distance Formula Midpoint- the point that divides the segment into two congruent segments Segment bisector- a point, ray, line, line segment or plane that intersects a segment at its midpoint Midpoint formula :

Other Formulas Distance formula: Slope formula: Equation of a line:

Example The midpoint of AB is M(5,8). One endpoint is A(2,-3). Find the coordinate of B. Find the distance of AB. Write the equation of the line segment in slope-intercept form.

1.4 Measure and Classify Angles Angle- a figure formed by 2 rays with a common endpoint, the union of 2 rays with a common endpoint Side (of an angle)- rays Vertex- the common endpoint of the rays that form the angle Interior/ Exterior of an angle- Name:

Protractor Postulate Consider OB and a point A on one side of OB. The ray that forms OA can be matched one to one with real numbers from 0 to 180.

Classifying Angles Acute- 0 o < x < 90 o Right - x = 90 o Obtuse- 90 o < x < 180 o Straight- x = 180 o

Angle Addition Postulate If P is in the interior of <RST, then m<RST = m<RSP + m<PST

Congruent angles- angles with the same measure Angle bisector- a line, segment, or ray that divides an angle into 2 congruent angles

Example Find m<ABC and m<CBD.

1.5 Describe Angle Pair Relationships Complementary angles- 2 angles whose measures sum to 90 o  (complement) Supplementary angles- 2 angles whose measures sum to 180 o  (supplement) Adjacent angles- 2 angles that share a common vertex and side but have not common interior points

Linear Pair- 2 angles formed when the endpoint of a ray falls on a line, non common sides are opposite rays Vertical angles- 2 angles formed by 2 intersecting lines

Examples Name a pair of complementary angles supplementary angles adjacent angles

1.6 Classify Polygons Polygon- a closed plane figure formed by coplanar segments such that ▫Each segment intersects exactly 2 other segments one at each endpoint and ▫No 2 points with a common endpoint are collinear Sides- coplanar segments Vertex- point where 2 segments’ endpoints intersect Diagonal- a segment that joins 2 nonconsecutive vertices of a polygon Name:

Convex or Concave Convex- no line that contains a side of the polygon contains a point in the interior of the polygon Concave- (or nonconvex) a polygon that is not convex Classifying

Equilateral, equiangular, or regular Equilateral- all sides are congruent Equiangular- all angles on the interior are congruent Regular- a convex polygon that is equilateral and equiangular

Classify by sides TriangleOcta- QuadrilateralNona- PentagonDeca- HexaDodeca- Heptan-gon N-gon n= the number of sides of the polygon

Examples Classify the polygon ab

cd

A billiard rack has the shape of an equilateral triangle. Find the length of a side.

1.7 Find Perimeter, Circumference, and Area Perimeter- the distance around a figure, measured in linear units Circumference- the distance around a circle Area- the amount of surface covered by a figure, measured in units 2

Formulas Square P = 4sA = s 2 RectangleP = 2l + 2wA = lw TriangleP = a + b + cA = ½·bh CircleC = ∏ d or 2 ∏ r A = ∏ r 2

Converting Units 100 in 2 to ft 2 1.2yd 3 to in 3 o.39m 2 to mm 2

Examples Find the perimeter of ∆ABC. Leave in simple radical form. A(2,5) B(4,1) C(8,3)

The base of a triangle is 14cm. The area is 42cm 2. Find the height.

You can mow 600 sq. ft of grass in 10 min. How long will it take to mow a rectangular field that is 60 ft. wide and 80 ft. long?