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Geometry 1 Unit 1: Basics of Geometry

Geometry 1 Unit 1: Basics of Geometry

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Geometry 1 Unit 1: Basics of Geometry

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  1. Geometry 1 Unit 1: Basics of Geometry

  2. Geometry 1 Unit 1 1.1 Patterns and Inductive Reasoning

  3. Describe how to sketch the fourth figure in the pattern. Then sketch the fourth figure. Each circle is divided into twice as many equal regions as the figure number. Sketch the fourth figure by dividing a circle into eighths. Shade the section just above the horizontal segment at the left. EXAMPLE 1 Describe a visual pattern SOLUTION

  4. Sketch the fifth figure in the pattern in example 1. ANSWER for Examples 1 and 2 GUIDED PRACTICE

  5. Notice that each number in the pattern is three times the previous number. ANSWER Continue the pattern. The next three numbers are –567, –1701, and –5103. EXAMPLE 2 Describe a number pattern Describe the pattern in the numbers –7, –21, –63, –189,… and write the next three numbers in the pattern.

  6. 2. Describe the pattern in the numbers 5.01, 5.03, 5.05, 5.07,… Write the next three numbers in the pattern. Notice that each number in the pattern is increasing by 0.02. 5.01 5.03 5.05 5.09 5.11 5.13 5.07 +0.02 +0.02 +0.02 +0.02 +0.02 +0.02 ANSWER Continue the pattern. The next three numbers are 5.09, 5.11 and 5.13 for Examples 1 and 2 GUIDED PRACTICE

  7. Patterns and Inductive Reasoning • Conjecture • An unproven statement that is based on observations. • Inductive Reasoning • The process of looking for patterns and making conjectures.

  8. EXAMPLE 3 Make a conjecture Given five students, make a conjecture about the number of different handshakes that can take place. SOLUTION Make a table and look for a pattern. Notice the pattern in how the number of connections increases. You can use the pattern to make a conjecture.

  9. ANSWER Conjecture: Five students can shake hands in 6 + 4, or 10 different ways. EXAMPLE 3 Make a conjecture

  10. = 4 3 = 8 3 = 113 = 17 3 ANSWER Conjecture: The sum of any three consecutive integers is three times the second number. EXAMPLE 4 Make and test a conjecture Numbers such as 3, 4, and 5 are called consecutive integers. Make and test a conjecture about the sum of any three consecutive integers. SOLUTION STEP 1 Find a pattern using a few groups of small numbers. 3 + 4 + 5 = 12 7 + 8 + 9 = 12 10 + 11+ 12 = 33 16 + 17 + 18 = 51

  11. = 101 3 = 0 3 EXAMPLE 4 Make and test a conjecture STEP 1 Test your conjecture using other numbers. For example, test that it works with the groups –1, 0, 1 and 100, 101, 102. 100 + 101 + 102 = 303 –1 + 0 + 1 = 0

  12. 3. Make and test a conjecture about the sign of the product of any three negative integers. ANSWER Conjecture: The result of the product of three negative numbers is a negative number. Test: Test conjecture using the negative integers –2, –5 and –4 –2 –5 –4 = –40 for Examples 3 and 4 GUIDED PRACTICE

  13. Patterns and Inductive Reasoning • Counterexample • An example that shows a conjecture is false. All Math teachers are male. Mrs. Beery, Ms. Wildermuth, Mrs. Hodge, Mrs. Cherry, Mrs. Frimer, Mrs. Dolezal are all counterexamples.

  14. EXAMPLE 5 Find a counterexample A student makes the following conjecture about the sum of two numbers. Find a counterexample to disprove the student’s conjecture. Conjecture: The sum of two numbers is always greater than the larger number. SOLUTION To find a counterexample, you need to find a sum that is less than the larger number.

  15. –5 > –2 ANSWER Because a counterexample exists, the conjecture is false. EXAMPLE 5 Find a counterexample –2 +–3 =–5

  16. 5. Find a counterexample to show that the following conjecture is false. = 12 14 14 12 ( )2 > ANSWER Because a counterexample exist, the conjecture is false for Examples 5 and 6 GUIDED PRACTICE Conjecture: The value of x2 is always greater than the value of x.

  17. Unit 1-Basics of Geometry 1.2: Points, Lines and Planes

  18. Points, Lines, and Planes • Definition • Uses known words to describe a new word. • Undefined terms • Words that lack a formal definition. • In Geometry it is important to have a general agreement about these words. • The building blocks of Geometry are undefined terms.

  19. Points, Lines, and Planes • The 3 Building Blocks of Geometry: • Point • Line • Plane • These are called the “building blocks of geometry” because these terms lay the foundation for Geometry.

  20. Points, Lines, and Planes Point • The most basic building block of Geometry • Has no size • A location in space • Represented with a dot • Named with a Capital Letter

  21. Points, Lines, and Planes Example: point P P

  22. Points, Lines, and Planes Line • Set of infinitely many points • One dimensional, has no thickness • Goes on forever in both directions • Named using any two points on the line with the line symbol over them, or a lowercase script letter

  23. l Points, Lines, and Planes Example: line AB, AB, BA or l B A **2 points determine a line

  24. Points, Lines, and Planes Plane • Has length and width, but no thickness • A flat surface that extends infinitely in 2-dimensions (length and width) • Represented with a four-sided figure like a tilted piece of paper, drawn in perspective • Named with a script capital letter or 3 points in the plane

  25. Points, Lines, and Planes Example: Plane P or plane ABC A C B P **3 noncollinear points determine a plane

  26. C B A Points, Lines, and Planes • Collinear • Points that lie on the same line Points A, B, and C are Collinear

  27. E F D Points, Lines, and Planes • Coplanar • Points that lie on the same plane Points D, E, and F are Coplanar

  28. Points, Lines, and Planes • Line Segment • Two points (called the endpoints) and all the points between them that are collinear with those two points Named line segment AB, AB, or BA line AB segment AB  A B A B

  29. Points, Lines, and Planes • Ray • Part of a line that starts at a point and extends infinitely in one direction. • Initial Point • Starting point for a ray. • Ray CD, or CD, is part of CD that contains point C and all points on line CD that are on the same side as of C as D • “It begins at C and goes through D and on forever”

  30. E D A B F C Segments and Their Measures • Between • When three points are collinear, you can say that one point is between the other two. Point B is between A and C Point E is NOT between D and F

  31. A C B Points, Lines, and Planes • Opposite Rays • If C is between A and B, then CA and CB are opposite rays. • Together they make a line.

  32. Points, Lines, and Planes C Y D C Y D C Y D Line CD Ray DC Ray CD CD and CY represent the same ray.  Notice CD is not the same as DC. ray CD is not opposite to ray DC

  33. Points, Lines, and Planes • The intersection of two lines is a point. • The intersection of two planes is a line.

  34. Unit 1-Basics of Geometry 1.3: Segments and Their Measures

  35. Segments and Their Measures • Postulates • Rules that are accepted without proof. • Also called axioms

  36. Segments and Their Measures • Ruler Postulate • The points on a line can be matched one to one with the real numbers. • The real number that corresponds to a point is called the coordinate of the point. • The distance between points A and B, written as AB, is the absolute value of the difference between the coordinates of A and B. • AB is also called the length of AB.

  37. Segments and Their Measures • Segment length can be given in several different ways. The following all mean the same thing. • A to B equals 2 inches • AB = 2 in. • mAB = 2 inches

  38. Segments and Their Measures • Example 1 • Measure the length of the segment to the nearest millimeter. D E

  39. E D A B F C Segments and Their Measures • Between • When three points are collinear, you can say that one point is between the other two. Point B is between A and C Point E is NOT between D and F

  40. AC A B C AB BC Segments and Their Measures • 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.

  41. Segments and Their Measures • Example 2 • Two friends leave their homes and walk in a straight line toward the others home. When they meet, one has walked 425 yards and the other has walked 267 yards. How far apart are their homes?

  42. Segments and Their Measures • The Distance Formula • A formula for computing the distance between two points in a coordinate plane. • If A(x1,y1) and B(x2,y2) are points in a coordinate plane, then the distance between A and B is

  43. Segments and Their Measures • Example 3 • Find the lengths of the segments. Tell whether any of the segments have the same length.

  44. Segments and Their Measures • Congruent • Two segments are congruent if and only if they have the same measure. • The symbol for congruence is . • We use = between equal numbers and  between congruent figures.

  45. Segments and Their Measures Markings on figures are used to show congruence. Use identical markings for each pair of congruent parts. A 2.5 B AB = DC = 2.5 AB  DC D 2.5 C AD  BC

  46. B(x2, y2) c |y2 – y1| a A(x1, y1) C(x2, y1) b |x2 – x1| Segments and Their Measures • Distance Formula and Pythagorean Theorem (AB)2 = (x2 – x1)2 + (y2 – y1)2 c2 = a2 + b2

  47. Segments and Their Measures • Example 4 • On the map, the city blocks are 410 feet apart east-west and 370 feet apart north south. • Find the walking distance between C and D. • What would the distance be if a diagonal street existed between the two points?

  48. Unit 1-Basics of Geometry 1.4: Angles and Their Measures

  49. Angles and Their Measures • Angle • Formed by two rays that share a common endpoint. • Sides • The rays that make the angle. • Vertex • The initial point of the rays.