SESSION 1 basic geometry principal elements

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1 SESSION 1 basic geometry principal elements Point: A point is a location in space. It is represented by a dot. Point are usually named with a upper case letter. For example, we refer to the following as "point A" We can also say a point is the intersection of two lines. Line: A line is a collection of points that extend forever. The following is a line. The two arrows are used to show that it extends forever. We can also say that a line is the intersection between two planes. Line segment: A line segment is part of a line. The following is a segment. A segment has two endpoints. The endpoints in the following segments are A and F. Notice also that the line has no endpoints. Ray: A ray is a collection of points that begin at one point (an endpoint) and extend forever on one direction. The following is a ray. 09_BASIC GEOMETRY Página 2 de 31

2 Angle: Two rays with the same endpoint is an angle. The following is an angle. Plane: A plane is a flat surface like a piece of paper. It extends in all directions. We can use arrows to show that it extends in all directions forever. The following is a plane RELATIONSHIP BETWEEN TWO STRAIGHT LINES Two lines can be: Parallel lines : When two lines never meet in space or on a plane no matter how long we extend them, we say that they are parallel lines The following lines are parallel. Intersecting lines: When lines meet in space or on a plane, we say that they are intersecting lines. The following are intersecting lines. 09_BASIC GEOMETRY Página 3 de 31

3 Perpendicular lines: A line is perpendicular to another if it meets or crosses it at right angles (90 ). SESSION 2 basic geometry principal elements TYPES OF LINE STRAIGHT LINE or CURVED LINE In geometry when we say line it is a straight line. POLYGONAL LINE A broken line composed of a finite number of line segments 09_BASIC GEOMETRY Página 4 de 31

4 SESSION 2 basic geometry ANGLES Types of angles Type of Angle Description Acute Angle an angle that is less than 90 Right Angle Obtuse Angle Straight Angle Reflex Angle an angle that is 90 exactly an angle that is greater than 90 but less than 180 an angle that is 180 exactly an angle that is greater than 180 and less than 360 o Pairs of Angles When parallel lines get crossed by another line (which is called atransversal), you can see that many angles are the same, as in this example: These angles can be made into pairs of angles which have special names. Transversal line 09_BASIC GEOMETRY Página 5 de 31

5 Parallel lines Vertical angles: a=d; c=b; e=h; g=f (Ángulos opuestos por el vértice) Corresponding angles: a=e; b=f; c=g; h=d (ángulos correspondientes) c+e =180 o ; d+f=180 o a+g= 180 o ; b+ h=180 o Complementary Angles Two Angles are Complementary if they add up to 90 degrees (a Right Angle). These two angles (40 and 50 ) are Complementary Angles, because they add up to 90. Notice that together they make a right angle. 09_BASIC GEOMETRY Página 6 de 31

6 But the angles don't have to be together. These two are complementary because = 90 Supplementary Angles Two Angles are Supplementary if they add up to 180 degrees. These two angles (140 and 40 ) are Supplementary Angles, because they add up to 180. Notice that together they make a straight angle. 09_BASIC GEOMETRY Página 7 de 31

7 EXERCISES 1. Plot the following: A (1,1); B( 4,3); C(3,6); D(9,1); E(7,7) 1. A line passing through point A. 2. Two intersecting lines that intersected in point B 3. A line parallel to the first one 4. A perpendicular line from C to the BC 4. Four different lines passing through point C 5. A ray which vertex is in point D 6. A line segment which one of its endpoints is E 7. An acute angle 8. An obtuse angle 9. A right angle 10. A plane angle 12. Two supplementary angles 13. Two complementary angles 14. Two parallels line and a transversal line and the names of pairs of angles 2. The straight lines AB and CD: A are parallel B are not parallel because the two given consecutive interior angles do not add to 180 C D are not parallel because the two given corresponding angles are not equal are not parallel because the two given alternate angles are not equal 09_BASIC GEOMETRY Página 8 de 31

8 3. AB and CD are parallel lines and EH is a transversal. What is the size of angle EFB? 4. AB and CD are parallel lines and EH is a transversal. What is the size of angle DGH? 5. ST and UV are parallel lines. c and e are: A consecutive interior angles B alternate angles C vertical angles D corresponding angles 09_BASIC GEOMETRY Página 9 de 31

9 6. ST and UV are parallel lines g and f are: A consecutive interior angles B alternate angles C vertical angles D corresponding angles 7. ST and UV are parallel lines d and e are: A consecutive interior angles B alternate angles C vertical angles D corresponding angles 8. ST and UV are parallel lines d and h are: A consecutive interior angles B alternate angles C vertical angles D corresponding angles 9. Two angles are supplementary and one of them is 31. What is the size of the other angle? 10. Two angles are complementary and one of them is 31. What is the size of the other angle? 09_BASIC GEOMETRY Página 10 de 31

10 BASIC GEOMETRY SESSION 3 ba asic geometry lugares geométricos: circunferencia, mediatriz y bisectriz Locus is a set of points thatt satisfy a particular condition. PERPENDICULAR BISECTOR_ It s a bisector that is at right angles to the line it halves M is the midpoint (a point that is exactly half way along a line segment) ), so M divides the line segment AB in two equal parts. La mediatriz es el lugar geométrico de los puntos del plano que equidistan a dos puntos fijos y (los dos extremos de un segmento). 1. Plot the perpendicular bisector in every line segment given below: 09_BASIC GEOMETRY Página 11 de 31

11 TO BISECT AN ANGLE 1.Con centro en V trazo un arco de circunferencia que corta a los lados del ángulo en 1 y 2 2. Con centro en 1 y radio 12 trazo un arco de circunferencia 3. Con centro en 2 y radio 21 trazo un arco de circunferencia 4. Uno V con el punto que han determinado ambos arcos. 2. Bisect the angles given below: CIRCUMFERENCES AND CIRCLES Look at these shapes and think about the difference: The first and the last are lines and the second and the third are surfaces How do I call them? CIRCUMFERENCE: es el lugar geométrico de los puntos del plano que equidistan de un punto llamado centro. A esa distancia se le llama radio 09_BASIC GEOMETRY Página 12 de 31

12 CIRCLE: Es la región del plano delimitada por una circunferencia y que posee un área definida. PARTS OF A CIRCUMFERENCE 3. Name the parts that are drawn in these circumferences. 4. Plot the following circumferences. 09_BASIC GEOMETRY Página 13 de 31

13 BASIC GEOMETRY SESSION 4 ba asic geometry ángulos circunferencia: lugares geométricos: EXERCISES La medida angular de un arco PQ es la misma que la de su ángulo central correspondiente. El arco PQ tiene una longitud igual a: ANGULO INSCRITO La medida de un ángulo inscrito en una circunferencia es igual a la mitad del arco que abarca, es decir a la mitad del ángulo central correspondiente. Dos ángulos inscritos que abarcan el mismo arco de circunferencia son iguales. Calcula razonadamente el valor de dichos ángulos 09_BASIC GEOMETRY Página 14 de 31

14 Cuánto mide el ángulo inscrito que corresponde a un cuarto de circunferencia? Cuánto mide el ángulo inscrito que corresponde a media circunferencia? Dibuja una semicircunferencia de diámetro AB y varios ángulos inscritos que contengan al diámetro. Defínela como lugar geométrico. Es el lugar geométrico de los puntos del plano desde los cuales se ve dicho segmento bajo un ángulo igual a 90 o ARCO CAPAZ Se llama ARCO CAPAZ del ángulo α para el segmento AB al lugar geométrico de los puntos del plano desde los cuales se ve dicho segmento bajo un ángulo igual a α. Página 189, 22,23 y 24 ( en vez de pentagon, hexagon) Página , 65, 66 y 67 09_BASIC GEOMETRY Página 15 de 31

15 SESSION 5 basic geometry polygons Polygon: When the polygonal line is closed we ve got a polygon. Polygons are 2 dimensional shapes. They are made of straight lines, and the shape is "closed" (all the lines connect up). Which is a polygon? TYPES OF POLYGONES BASED ON WHETHER THEY ARE CONVEX OR CONCAVE The polygons can be.concave or Convex A convex polygon has no angles pointing inwards. More precisely, no internal angles can be more than 180. If there are any internal angles greater than 180 then it is concave. (Think: concave has a "cave" in it) Convex Concave TYPES OF POLYGONES BASED ON WHETHER THEY ARE REGULAR O IRREGULAR Regular or Irregular If all angles are equal and all sides are equal, then it is regular, otherwise it is irregular Regular Irregular ELEMENTOS DE UN POLÍGONO: 09_BASIC GEOMETRY Página 16 de 31

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18 CLASIFICACIÓN DE POLÍGONOS. TYPES OF POLYGONES BASED ON THE NUMBER OF SIDES Name Sides Angles Triangle 3 3 Quadrilateral 4 4 Pentagon 5 5 Hexagon 6 6 Heptagon 7 7 Octagon 8 8 Nonagon 9 9 Decagon _BASIC GEOMETRY Página 19 de 31

19 SUMA DE LOS ÁNGULOS INTERIORES DE UN POLÍGONO REGULAR 09_BASIC GEOMETRY Página 20 de 31

20 CÁLCULO DE LOS ÁNGULOS INTERIORES DE UN POLÍGONO REGULAR 09_BASIC GEOMETRY Página 21 de 31

21 SESSION 6 basic geometry TRIANGLES 09_BASIC GEOMETRY Página 22 de 31

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23 EXERCISES 1. Determinar el lado de un triángulo equilátero cuyo perímetro es igual al de un cuadrado de 12 cm de lado. Serán iguales sus áreas? 2. Calcular el área de un triángulo equilátero inscrito en una circunferencia de radio 6 cm. SUMA DE LOS TRES ÁNGULOS DE UN TRIÁNGULO= 180 O RECTAS NOTABLES DE UN TRIÁNGULO: MEDIANAS, ALTURAS, BISECTRICES Y MEDIATRICES: Medianas: baricentro ; Alturas: ortocentro; Bisectrices: Incentro y Mediatrices: Circuncentro 09_BASIC GEOMETRY Página 24 de 31

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25 SESSION 7 basic geometry TRIANGLES: area y perimeter 09_BASIC GEOMETRY Página 26 de 31

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