SESSION 1 basic geometry principal elements

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SESSION

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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.

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BASIC GEOMETRY

09_BASIC GEOMETRY Página 3 de 31

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.

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Perpendicular lines: A line is perpendicular to another if it meets or crosses it at right angles (90°).

SESSION

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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

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BASIC GEOMETRY

SESSION

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basic geometry| ANGLES

Types

of

angles

Type

of

Angle

Description

Acute Angle an angle that is less than 90°

Right Angle an angle that is 90° exactly

Obtuse Angle

an angle that is greater than 90° but less than 180°

Straight Angle an angle that is 180° exactly

Reflex Angle

an angle that is greater than 180° and less than 360o

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.

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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

http://www.youtube.com/watch?v=8eD3wODClh8

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.

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BASIC GEOMETRY

But the angles don't have to be together.

These two are complementary

because 27° + 63° = 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.

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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 are not parallel because the two given corresponding angles are not equal

D are not parallel because the two given alternate angles are not equal

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BASIC GEOMETRY

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

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6. ST and UV are parallel lines g and f are:

B alternate angles

C vertical angles

D corresponding angles

7. ST and UV are parallel lines d and e are:

B alternate angles

C vertical angles

8. ST and UV are parallel lines d and h are:

B alternate angles

C vertical 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?

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BASIC 09_BA

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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

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BASIC GEOMETRY

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.

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SESS

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BASIC GEOMETRY

Æ¿Cuánto mide el ángulo inscrito que corresponde a un cuarto de circunferencia?

Æ¿Cuánto mide el ángulo inscrito que corresponde a media circunferencia?

ÆDibuja 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 90o

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

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Página

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65,

66 y

67

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SESSION

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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

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BASIC GEOMETRY

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BASIC GEOMETRY

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

10 10

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SUMA DE LOS ÁNGULOS INTERIORES DE UN POLÍGONO REGULAR

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BASIC GEOMETRY

CÁLCULO DE LOS ÁNGULOS INTERIORES DE UN POLÍGONO REGULAR

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SESSION

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basic geometry| TRIANGLES

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BASIC GEOMETRY

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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:

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BASIC GEOMETRY

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SESSION

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basic geometry| TRIANGLES: area y perimeter

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BASIC GEOMETRY