# Meeting 1

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### Meeting 1

• 3. Undefined terms : Point, line, and plane It can’t be defined but described Definition Words that can be defined by category and characteristics that are clear, concise, and reversible. Postulates Statements accepted without proof. Theorems Statements that can be proven true. GEOMETRY
• 7.  Garis ℓ dan garis 𝐴𝐵 terletak di bidang 𝑄.  Garis 𝑘 dan garis 𝐴𝐵 terletak di bidang 𝑃.  Garis 𝐴𝐵 merupakan garis yang terletak pada perpotongan bidang 𝑃 dan 𝑄. Garis 𝐴𝐵 disebut garis persekutuan kedua bidang tersebut. UNDEFINED TERMS (CONT)
• 10. NAMING ANGELS 1. Using three letters, the center letter corresponding to the vertex of the angle and the other letters representing points on the sides of the angle. For example, the name of the angle whose vertex is 𝑇 can be angle 𝑅𝑇𝐵 (∡𝑅𝑇𝐵) or angle 𝐵𝑇𝑅 (∡𝐵𝑇𝑅).
• 11. NAMING ANGELS 2. Placing a number at the vertex and in the interior of the angle. The angle may then be referred to by the number. For example, the name of the angle whose vertex is 𝑇 can be ∡1 or ∡𝑅𝑇𝐵 or ∡𝐵𝑇𝑅.
• 12. NAMING ANGELS 3. Using a single letter that corresponds to the vertex, provided that this does not cause any confusion. There is no question which angle on the diagram corresponds to angle A, but which angle on the diagram is angle D? Actually three angles are formed at vertex D: • Angle 𝐴𝐷𝐵 • Angle 𝐶𝐷𝐵 • Angle 𝐴𝐷𝐶
• 13. Line • it's in a straight path. • goes in both directions. • does not end ... so you can't measure it's length. Ray • it's straight. • is part of a line. • has one endpoint. • goes in ONE direction. Line Segment • is straight. • is a part of a line. • has 2 endpoints that show the points that end the line.
• 14. EXAMPLE 1 a. Name the accompanying line. b. Name three different segments. c. Name four different rays. d. Name a pair of opposite rays.
• 15. EXAMPLE 2 Use three letters to name each of the numbered angles in the accompanying diagram.
• 16. Do you think there is another definition in geometry? Apakah dalam geometri hanya istilah-istilah tersebut yang didefinisikan?
• 17. DEFINITIONS The purpose of a definition is to make the meaning of a term clear. A good definition must: • Clearly identify the word (or expression) that is being defined. • State the distinguishing characteristics of the term being defined, using only words that are commonly understood or that have been previously defined. • Be expressed in a grammatically correct sentence.
• 18. DEFINITIONS OF COLLINEAR AND NONCOLLINEAR POINTS Points 𝐴, 𝐵, and 𝐶 are collinear. Points 𝑅, 𝑆, and 𝑇 are not collinear. DEFINITION: • Collinear points are points that lie on the same line. • Noncollinear points are points that do not lie on the same line.
• 19. DEFINITION OF TRIANGLE A triangle is a figure formed by connecting three noncollinear points with three different line segments each of which has two of these points as end points.
• 20. Contoh 3 Susun konsep-konsep berikut dalam urutan pendefinisian: • Segitiga samakaki, segitiga, sudut alas segitiga samakaki Segitiga, segitiga samakaki, sudut alas segitiga samakaki • Sisi miring, segitiga, segitiga siku-siku Segitiga, segitiga siku-siku, sisi miring
• 21. A good definition must be reversible as shown in the following table. The first two definitions are reversible since the reverse of the definition is a true statement. The reverse of the third “definition” is false since the points may be scattered.
• 22. Contoh 4 • Segitiga siku-siku adalah segitiga dengan satu sudutnya siku-siku. (benar) Konvers: Segitiga dengan salah satu sudutnya siku-siku adalah segitiga siku-siku. (benar) DEFINISI • Setiap sudut siku-siku adalah sudut-sudut kongruen (sama besar) (benar) Konvers: Sudut-sudut yang kongruen (sama besar) adalah sudut siku- siku. (salah) BUKAN DEFINISI
• 23. Contoh 5 Susunlah konsep-konsep berikut dalam urutan pendefinisan • Ukuran sudut, sudut, sudut kongruen. • Kaki segitiga samakaki, segitiga samakaki, segitiga • Sudut, segitiga tumpul, sudut tumpul. ---- • Sudut, ukuran sudut, sudut kongruen • Segitiga, segitiga samakaki, kaki segitiga samakaki • Sudut, sudut tumpul, segitiga tumpul
• 24. Contoh 6 Manakah yang merupakan definisi? 1. Segitiga samasisi adalah segitiga di mana ketiga sisinya sama panjang. Definisi 2. Pada segitiga siku-siku, sisi miring adalah sisi di hadapan sudut siku- siku. Definisi
• 25. Postulates Are statements accepted as true without proof. They are accepted on faith alone. They are considered self-evident statements.
• 26. Some Important Terms • Exists-there is at least one “chairs exist in this room” • Unique-there is no more than one “In this room, the computer is unique, the chairs are not” • One and only one-exactly one; shows existence and uniqueness “In this room, there is one and only one fire extinguisher”
• 27. INITIAL POSTULATES In building a geometric system, not everything can be proved since there must be some basic assumptions, called postulates (or axioms), that are needed as a beginning. Postulate 1.1 implies that through two points exactly one line may be drawn while.
• 28. INITIAL POSTULATES Postulate 1.2 asserts that a plane is defined when a third point not on this line is given.
• 29. #1 Ruler Postulate • A] The points on a line can be paired with the real numbers in such a way that any two points can have coordinates 0 and 1. We know this as the number line. 0- 4 -2 642 Whole numbers and fractions are not enough to fill up the points on a line. The spaces that are missing are filled by the irrational numbers. 43 2 , 3, 7, 11, ,etc
• 30. #1 Ruler Postulate • B] Once a coordinate system has been chosen in this way, the distance between any two points equals the absolute value of the difference of their coordinates. This is the more important part. a b a bDistance =
• 31. # 2 Segment Addition Postulate B is between A and C so AB + BC = AC A B C Note that B must be on AC.
• 32. #3 Protractor Postulate • On AB in a given plane, chose any point O between A and B. Consider OA and OB and all the rays that can be drawn from O on one side of AB. These rays can be paired with the real numbers from 0 to 180 in such a way that: • OA is paired with 0. and OB is paired with 180. • If OP is paired with x and OQ with y, then m POQ x y  
• 33. Relax! You don’t have to memorize this. Restated: 1] All angles are measured between 00 and 1800. 2] They can be measured with a protractor. 3] The measurement is the absolute values of the numbers read on the protractor. 4] The values of 0 and 180 on the protractor were arbitrarily selected.
• 34. Protractor Postulate Cont. 0180 Q P B O A x y m POQ x y  
• 35. #4 Angle Addition Postulate • If point B is in the interior of , thenAOC m AOB m BOC m AOC     O A B C 1 2 1 2m m m AOC    
• 36. #4 Angle Addition Postulate • is a straight angle and B is any point not on AC , so AOC O A B C 0 180m AOB m BOC    These angles are called “linear pairs.” 12 0 1 2 180m m   
• 37. Postulate #5 •A line contains at least 2 points; • a plane contains at least 3 non- collinear points; • Space contains at least 4 non- coplanar points.
• 38. Postulate #5 •A line is determined by 2 points. • A plane is determined by 3 non- collinear points. • Space is determined by 4 non- coplanar points.
• 39. Postulate # 6 •Through any two points there is exactly one line. Restated: 2 points determine a unique line.
• 40. Postulate # 7 •Through any three points there is at least one plane. •And through any three non-collinear points there is exactly one plane.
• 41. Three collinear points can lie on multiple planes. M While three non-collinear points can lie on exactly one plane.
• 42. Three collinear points can lie in multiple planes – horizontal and vertical.
• 43. Three collinear points can lie in multiple planes – Slanted top left to bottom right and bottom left to top right.
• 44. With 3 non-collinear points, there is only one plane – the plane of the triangle. B A C
• 45. Postulate # 8 • Two points of a line are in a plane and the line containing those points in that plane.
• 46. Notice that the segment starts out as vertical with only 1 pt. in the granite plane. As the top endpoint moves to the plane… The points in between move toward the plane. When the two endpoints lie in the plane the whole segment also lies in the plane.
• 47. Postulate # 9 • The two planes intersect and their intersection is a line. H G F E D CB A Remember, intersection means points in common or in both sets.
• 48. Postulate # 9 •The two planes intersect and their intersection is a line. H G F E D CB A Remember, intersection means points in common or in both sets.
• 49. Final Thoughts • Postulates are accepted as true on faith alone. They are not proved. • Postulates need not be memorized. • Those obvious simple self-evident statements are postulates. • It is only important to recognize postulates and apply them occasionally.
• 50. Theorem 1.1  If two lines intersect, then they intersect in exactly one(one and only one) point. The point exists(there is at least one point) and is unique(no more than one point exists). A .
• 51. Theorem 1.1 If 2 lines intersect, then they intersect in exactly one point. This is very obvious. To be more than one the line would have to curve. But in geometry, all lines are straight.
• 52. Theorem 1.2 (Know the meaning not the number)  Through a line and a point not in the line, there is exactly one(one and only one) plane. The plane exists(there is at least one plane) and is unique(no more than one plane exists). A This is not so obvious.
• 53. Theorem 1.2 Through a line and a point not on the line there is exactly 1 plane that contains them. If you take any two points on the line plus the point off the line, then… The 3 non-collinear points mean there exists a exactly plane that contain them. If two points of a line are in the plane, then line is in the plane as well. A B C
• 54. Theorem 1.3 (Know the meaning not the number)  If two lines intersect, then exactly one (one and only one) plane contains the lines. The plane exists(there is at least one plane) and is unique(no more than one plane exists).
• 55. Theorem 1.3 If two lines intersect, there is exactly 1 plane that contains them. This is not so obvious.
• 56. Theorem 1.3 If two lines intersect, there is exactly 1 plane that contains them. If you add an additional point from each line, the 3 points are noncollinear. Through any three noncollinear points there is exactly one plane that contains them.
• 57. Quick Quiz  Two points must be ___________ Collinear  Three points may be __________ Collinear  Three points must be __________ Coplanar  Four points may be __________ Coplanar
• 58. Quick Quiz  Three noncollinear points determine a ___ Plane  A line and a point not on a line determine a __________ Plane  A line and a plane can 1)__________ 2)_________ or 3)____________ Be Parallel, Intersect in exactly one point, or the plane can contain the line  Four noncoplanar points determine __________ Space
• 59.  For Kepler, a devout Christian, mathematics was itself a religious undertaking. He wrote in Harmonice Mundi (1619):-  Geometry existed before the creation; is co-eternal with the mind of God; is God himself ... Where there is matter there is geometry. ... geometry provided God with a model for the Creation and was implanted into man, together with God's own likeness - and not merely conveyed to his mind through the eyes. ... It is absolutely necessary that the work of such a Creator be of the greatest beauty...
• 60. LATIHAN 1. Perhatikan gambar berikut. a. Ada 6 segmen yang berbeda, sebutkan! b. Ada 12 sinar yang berbeda, sebutkan! c. Sebutkan nama garis di atas menurut 6 cara! 2. Sebutkan semua sudut yang tersebar pada gambar di bawah ini!
• 61. LATIHAN 3. Gunakan gambar berikut untuk mengisi pertanyaan di bawah. a. 𝑇𝑅 ≅ ⋯ b. … ≅ 𝑌𝐴 c. … titik tengah 𝐴𝐵 4. Sebutkan nama-nama segitiga yang terdapat pada gambar berikut (ada 16 segitiga). Kemudian kelompokan segitiga-segitiga tersebut berdasarkan jenis sudutnya (∆lancip, ∆ siku- siku, ∆ tumpul). S Y
• 62. LATIHAN 5. Susunlah urutan istilah berikut menurut pendefinisiannya. a. Segitiga samakaki, segitiga, titik puncak segitiga samakaki b. Sudut-sudut kongruen, garis bagi sudut, sudut 6. Diketahui bidang I dan bidang II keduanya memuat titik 𝐴, 𝐵 dan 𝐶. Buktikan bahwa 𝐴, 𝐵 dan 𝐶 kolinear.

### Hinweis der Redaktion

1. Jelas mengidentifikasi kata (atau ekspresi) yang didefinisikan. Negara karakteristik yang membedakan dari istilah yang didefinisikan, hanya menggunakan kata-kata yang umum dipahami atau yang sudah ditetapkan sebelumnya. Dinyatakan dalam kalimat tata bahasa yang benar.
2. Untuk Kepler, seorang yang taat Kristen, matematika itu sendiri usaha agama. Dia menulis di Harmonice Mundi (1619): - Geometri ada sebelum penciptaan; adalah co-kekal dengan pikiran Allah; adalah Allah sendiri ... mana ada masalah ada geometri. ... Tuhan Dilengkapi dengan model geometri untuk Penciptaan dan ditanamkan ke manusia, bersama-sama dengan rupa Allah sendiri - dan tidak Hanya disampaikan kepada pikirannya melalui mata. ... Hal ini mutlak diperlukan bahwa karya Sang Pencipta seperti menjadi keindahan terbesar ...
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