Introduction to Geometry What is Geometry Why do we use Geometry What is Geometry? Geometry is a branch of mathematics that concerns itself with the questions of shape, size, position of figures, and the properties of space. Geometry became a useful study of mathematics when Rene Descartes introduced coordinates on the Cartesian plane. This allowed shapes to be graphically represented and allowed algebra to graphically represent equations and functions. Recall how we graphed linear equations in Algebra I. The study of geometry rose from the need to understand the concepts of length, area, and volume. As the formulas to describe length, area, and volume via the Pythagorean theorem, circumference and area of a circle, area of a triangle, volume of a cylinder, sphere, and pyramid, inaccessible measurements were able to be calculated based on similarity.
Introduction to Geometry What is Geometry Why do we use Geometry This lead to the development of trigonometry in astronomy. What is the size of the moon or sun? When the use of coordinates became common place, analytic geometry and algebraic geometry began to look at three dimensional curves on the Cartesian plane. This study developed into Calculus. For those interested, Einstein used geometry to explain his theories of relativity which incorporates gravitation, force, electro-magnetism, and lots and lots of mathematics. I would encourage all of you to spend a little time reading Einstein s theories, even if you do not understand the math or science behind the theories.
Introduction to Geometry What will we study in Geometry Developing reasoning skills in Geometry Problem solving steps Coordinate Graphing Angles Parallels Triangles and their uses Quadrilaterals and the types of quadrilaterals Circles
Chapter 1.1 Patterns and Inductive Reasoning Develop Inductive Reasoning skills When you see dark, towering clouds approaching, would you think a storm is coming? Sure, your past experience with weather tells you that a thunderstorm is likely to happen. You have made a conclusion based on past experiences or recognized patterns. When you make these conclusions, you are using inductive reasoning. Using inductive reasoning, we can find the next three terms of the sequence 33, 39, 45, Each consecutive term is 6 more than the previous; therefore, the next three terms are 51, 57, and 63. What is the next shape? First, the pattern is Triangle-Triangle-Square-Square-Triangle-Triangle. Second, the pattern is Blue-Red-Blue-Red-Blue-Red. Answer: Blue Square
Chapter 1.1 Patterns and Inductive Reasoning Develop Inductive Reasoning skills A conclusion based on inductive reasoning is known as a conjecture. A conjecture is an educated guess, which can be true or sometimes false. How can a conjecture, which is based on inductive reasoning, be false? When we test a conjecture and find an example where the conjecture is false. This false example is known as a counterexample. A student studies a table of data and forms the following conjecture The product of two positive numbers is always greater than either factor. Is there a counterexample? Yes, if a factor is a positive fraction, then the conjecture is false. Bookwork: page 8; problems 15-38 Factors 2 8 16 5 15 75 20 38 760 Product 54 62 3348
Chapter 1.2 Points, Lines, and Planes Define and determine characteristics of points, lines, and planes. Has anyone ever seen Sunday Afternoon on the Island of La Grande Jatte by George Seurat in 1884-1886? The dot, known as a point in geometry is the basic unit of geometry. A point has no size. A point is named by a capital letter. A line is made up of an infinite number of points. Arrows at the end of a line indicate that the line extends without end. A line can be named by a single lowercase script letter or by two points on the line A Ḃ This line is named line AB or line BA, or line l. The symbol for line AB is AB. l Three points that lie on the same line are said to be collinear. Points that do not lie on the same line are noncollinear. A
Chapter 1.2 Points, Lines, and Planes Define and determine characteristics of points, lines, and planes. A ray has a definite starting point and extends into infinity in one direction. A ray is named using the endpoint first, then a second point on the ray. The symbol for a ray is CD. A line segment is part of a line with two endpoints. A line segment is named using the endpoints. The symbol for a line segment is EF. A plane is a flat surface that extends into infinity in all directions. At least three points make up a plane. G J A plane can be named with a single uppercase script letter H or by three noncollinear points. Plane GHJ or plane M. M Points that lie in the same plane are coplanar. Points that do not lie in the same plane are noncoplanar. Bookwork: page 16; problems 8-30. E C D F
Chapter 1.3 Postulates Identify and use postulates about points, lines, and planes. In order to use geometry, we must accept certain truths and build on these truths. The truths in geometry that are used to build geometric thought are called postulates. A postulate is a statement in geometry that is accepted to be true. Postulate 1-1 states that two points determine a unique line. There is only one line that contains points E and F. E F Postulate 1-2 states that two distinct lines intersect at one point. Lines l and m intersect at point T. Postulate 1-3 states that three noncollinear points determine a plane. There is only one plane that contains points G, H, and J. l G T H J m
Chapter 1.3 Postulates Identify and use postulates about points, lines, and planes. Postulate 1-4 states that if two planes intersect, then their intersection is a line. Use construction paper exercise to show the intersection. These postulates allow us to draw figures. Postulate 1-3 states that at least three points make up a plane. We could define a plane using four points. Bookwork: page 21, problems 9-30.
Chapter 1.4 Conditional Statements and Converses Write if-then statements and converse statements. If a number is an even number, then the number is divisible by two. This is an example of an if-then statement. If-then statements join two statements that state a condition. If-then statements are also called conditional statements. Conditional statements have two parts: the hypothesis, if a number is an even number, and the conclusion, then the number is divisible by two. How can we determine if a conditional statement is true? If a figure is a triangle, then it has three angles? Is this true? Yes! One way to test a conditional statement is to state the converse. By testing the conclusion. The converse of a conditional statement is formed by exchanging the hypothesis and the conclusion. If a figure has three angles, then the figure is a triangle. Is this true? Maybe! It could have four angles.
Chapter 1.4 Conditional Statements and Converses Write if-then statements and converse statements. State the hypothesis and conclusion in the following statement. If it is raining, then we will read a book. Write two other statements that mean the same as, if two lines are parallel, then they never intersect. All parallel lines never intersect. Lines never intersect if they are parallel. Write the converse of the following statement. If today is Saturday, then there is no school. Is this always true? Bookwork: page 27, problems 10-27 and emphasis on problem 31.
l Geometry Chapter 1.6 Problem Solving Develop and use the four step plan to solve problems. Determine the formulas for perimeter and area. A formula is an equation that shows the relationship of the certain quantities identified in the equation. Like equations, formulas allow us to fine unknown information from known information. Lets look at a couple of useful formulas. When a rancher rides the fence line looking for breaks, the rancher is travelling along the perimeter.. The perimeter of an object or figure, is the distance around the object or figure. Perimeter can be expressed as the sum of the sides of a figure. Perimeter of a rectangle, P = l + w + l + w, or P = 2l + 2w w
Chapter 1.6 Problem Solving Develop and use the four step plan to solve problems. Determine the formulas for perimeter and area. The area of a figure is the number of square units needed to cover the figure s surface. The area of a rectangle, A = l w Notice by definition, the figure must be divided into square units. Also, the unit chosen will determine the magnitude of the area measurement, cm, ft, in, etc. This means the calculated number must have squared units or unit 2 because feet times feet is equal to feet 2. Count the number of squares for the area of this rectangle, 20. And four times five equals twenty.
Chapter 1.6 Problem Solving Develop and use the four step plan to solve problems. Determine the formulas for perimeter and area. A parallelogram is a figure that has two pairs of opposite sides that are parallel. We will show later that opposite sides are congruent or equal in length. The area of a parallelogram, A = b h; where b is the base length and h is the height. h b
Chapter 1.6 Problem Solving Develop and use the four step plan to solve problems. Determine the formulas for perimeter and area. Some math problems can be solved using a formula, like perimeter and area. Other problems can be solved using patterns or models. All problems need to use a four-step plan. Step 1: Explore the problem. Step 2: Plan the solution. Step 3: Solve the problem. Step 4: Examine the solution. Lets look at example 4 on page 37. Bookwork: page 39, problems 12-32.