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Basic Geometry Math 1-Unit 1A Geometry • The word geometry comes from Greek words meaning “to measure the Earth” • Basically, Geometry is the study of shapes and is one of the oldest branches of mathematics The Greeks and Euclid • Our modern understanding of geometry began with the Greeks over 2000 years ago. • The Greeks felt the need to go beyond merely knowing certain facts to being able to prove why they were true. • Around 350 B.C., Euclid of Alexandria wrote The Elements, in which he recorded systematically all that was known about Geometry at that time. Undefined Terms? • The terms points, lines, and planes are the foundations of geometry, but… • • point, line, and plane are all what we call undefined terms. • How can that be? • Well, any definition we could give them would depend on the definition of some other mathematical idea that these three terms help define. In other words, the definition would be circular! Point • Has no dimension • Usually represented by a small dot A The above is called point A. Note the point is represented with a capital letter. Line • Extend in one dimension. • Represented with straight line with two arrowheads to indicate that the line extends without end in two directions. This is Line l, (using the lower case script letter) or symbolically we call it AB l A B NOTICE: The arrowheads are in both directions on the symbol Plane •Extend in two dimensions. •Represented by a slanted 4 sided figure, but you must envision it extends without end, even though the representation has edges. A B M C This is Plane M or plane ABC (be sure to only use three of the points when naming a plane) Undefined Concepts • Collinear points are points that lie on the same line. l B C A Points A, B and C are collinear. Undefined Concepts • Coplanar points are points that lie on the same plane. A B C Points A, B and C are coplanar. Line Segment •Let’s look at the idea of a point in between two other points on a line. Here is line AB, or recall symbolically AB A B The line segment does not extend without end. It has endpoints, in this case A and B. The segment contains all the points on the line between A and B This is segment AB Notice the difference in the symbolic notation! Ray Let’s look at a ray: A is called the initial point A B Ray AB extends in one direction without end. •Symbolized by AB The initial point is always the first letter in naming a ray. Notice the difference in symbols from both a line and segment. Symbol alert! •Not all symbols are created equal! AB is the same as BA A B AB is the same as BA A B BUT… Symbol alert!! The ray is different! AB is not the same as Initial point 1st BA A B A B AB BA Notice that the initial point is listed first in the symbol. Also note that the symbolic ray always has the arrowhead on the right regardless of the direction of the ray. Opposite Rays •If C is between A and B, A C then CA and B CB are opposite rays. C is the common initial point for the rays! Angles •Rays are important because they help us define something very important in geometry…Angles! •An angle consists of two different rays that have the same initial point. The rays are sides of the angles. The initial point is called the vertex. vertex B sides A C Notation: We denote an angle with three points and symbol. The middle point is always the vertex. We can also name the angle with just the vertex point. This angle can be denoted as: BAC , CAB, or A Classifying Angles •Angles are classified as acute, right, obtuse, and straight, according to their measures. Angles have measures greater than 0° and less or equal to 180°. A Straight angle A A A Acute angle Right angle Obtuse angle 0°< m A < 90° m A = 90° 90°< m A < 180° m A = 180° Adjacent angles are “side by side” and share a common ray. 15º 45º These are examples of adjacent angles. 80º 45º 35º 55º 130º 85º 20º 50º These angles are NOT adjacent. 100º 50º 35º 35º 55º 45º Supplementary angles add up to 180º. 40º 120º 60º Adjacent and Supplementary Angles 140º Supplementary Angles but not Adjacent Complementary angles add up to 90º. 30º 40º 50º 60º Adjacent and Complementary Angles Complementary Angles but not Adjacent Naming Geometric Figures • Parallel: Two lines in the same plane that never intersect and have the same slope. • Represented by symbol “||” (line 1 || line2) Transversal- Is a line that intersects two lines in the same plane 1 Transversal 4 2 6 3 7 5 8 Vertical Angels- Are opposite angles that share a vertex Vertical angles are shown with the same color 1 4 2 6 3 7 5 8 When 2 lines intersect, they make vertical angles. 75º 105º 105º 75º Vertical angles are opposite one another. 75º 105º 105º 75º Vertical angles are opposite one another. 75º 105º 105º 75º Corresponding Angels- Are in the same location and on the same side of the transversal Corresponding angles are shown with the same color-They both are on top of the line cut by the transversal 1 4 2 6 3 7 5 8 Alternate Interior Angles- Are the angles on the inside of the two lines but on opposite sides of the transversal Alternate Interior angles are shown with the same color 1 4 2 6 3 7 5 8 Alternate Exterior Angles- Are the angles on the outside of the two lines but on opposite sides of the transversal Alternate Interior angles are shown with the same color 1 4 2 6 3 7 5 8 Same side interior Angles- Are the angles on the inside of the two lines and on the same side of the transversal Same side Interior angles are shown with the same color 1 4 2 6 3 7 5 8 Basic Terms & Definitions • A ray starts at a point (called the endpoint) and extends indefinitely in one direction. A B AB • A line segment is part of a line and has two endpoints. A B AB • An angle is formed by two rays with the same endpoint. side vertex side • An angle is measured in degrees. The angle formed by a circle has a measure of 360 degrees. • A right angle has a measure of 90 degrees. • A straight angle has a measure of 180 degrees. Practice Time! Directions: Identify each pair of angles as vertical, supplementary, complementary, or none of the above. #1 120º 60º #1 120º 60º Supplementary Angles #2 30º 60º #2 30º 60º Complementary Angles #3 75º 75º #3 Vertical Angles 75º 75º #4 40º 60º #4 40º 60º None of the above #5 60º 60º #5 60º 60º Vertical Angles #6 135º 45º #6 135º 45º Supplementary Angles #7 25º 65º #7 25º 65º Complementary Angles #8 90º 50º #8 90º 50º None of the above Directions: Determine the missing angle. #1 ?º 45º #1 135º 45º #2 ?º 65º #2 25º 65º #3 ?º 35º #3 35º 35º #4 ?º 50º #4 130º 50º #5 ?º 140º #5 140º 140º #6 ?º 40º #6 50º 40º More Terms Defined • Congruent – Two or more objects that are the exact same size and shape. – Symbol for congruence is ≅ • Bisect – Dividing and object into two congruent (equal) parts More Terms Defined • Midpoint – The point that divides two objects into two congruent parts • Distance – The amount of space between to objects or people More Terms Defined • Proof – The process used to demonstrate truth of a statement • Postulate – A statement that is assumed to be true without proof. Postulates are the basic structure from which theorems are derived • Theorem – Is a statement that can be demonstrated to be true