Basic Geometry STUDENT

7/29/2019 Basic Geometry STUDENT

1/15

0

Basic GeometryAngle

There are three ways to label an angle:

1. ABC

2. B

3. x

Angle is measured in degrees, the symbol of which is o.

A right-angle = 90o.

Half of a revolution = 180o.

One complete revolution = 360o.

Acute Angle Right Angle Obtuse Angle

Less than 90o 90o Greater than 90o and

smaller than 180o

Straight Angle Reflex Angle

180o Greater than 180o and

smaller than 360o


2/15

1

EEXXEERRCCIISSEE11AA

(a)An acute angle must be less than

(b)An acute angle must be greater than

(c)A right angle equals

(d)An obtuse angle must be less than

(e)An obtuse angle must be greater than

(f) A reflex angle must be greater than

(g)A straight angle equals

(h)The sum of a right angle and an acute angle must be

greater than

EEXXEERRCCIISSEE11BB

Express65 of a straight angle in degrees.

EEXXEERRCCIISSEE11CC

Write down the types of the marked angles in the figure.


3/15

2

EEXXEERRCCIISSEE11DD

In the figure,POQ is a straight line. Find POR.

Triangle

Triangles may be classified according to the angles.

Acute-angled triangle Right-angled triangle Obtuse-angled triangle

All angles are acute. One of the angles is a

right angle.

One of the angles is an

obtuse angle.

Triangles may also be classified according to the length of their sides:

Equilateral triangle Isosceles triangle Scalene triangle

All sides have the same

length.

Two of the sides have

equal length.

No two sides have equal

length.

The sum of the three angles of a triangle is 180

o

. [

]
http://en.wikipedia.org/wiki/File:Triangle.Obtuse.svghttp://en.wikipedia.org/wiki/File:Triangle.Right.svghttp://en.wikipedia.org/wiki/File:Triangle.Acute.svghttp://en.wikipedia.org/wiki/File:Triangle.Scalene.svghttp://en.wikipedia.org/wiki/File:Triangle.Isosceles.svghttp://en.wikipedia.org/wiki/File:Triangle.Equilateral.svghttp://en.wikipedia.org/wiki/File:Triangle.Obtuse.svghttp://en.wikipedia.org/wiki/File:Triangle.Right.svghttp://en.wikipedia.org/wiki/File:Triangle.Acute.svghttp://en.wikipedia.org/wiki/File:Triangle.Scalene.svghttp://en.wikipedia.org/wiki/File:Triangle.Isosceles.svghttp://en.wikipedia.org/wiki/File:Triangle.Equilateral.svghttp://en.wikipedia.org/wiki/File:Triangle.Obtuse.svghttp://en.wikipedia.org/wiki/File:Triangle.Right.svghttp://en.wikipedia.org/wiki/File:Triangle.Acute.svghttp://en.wikipedia.org/wiki/File:Triangle.Scalene.svghttp://en.wikipedia.org/wiki/File:Triangle.Isosceles.svghttp://en.wikipedia.org/wiki/File:Triangle.Equilateral.svghttp://en.wikipedia.org/wiki/File:Triangle.Obtuse.svghttp://en.wikipedia.org/wiki/File:Triangle.Right.svghttp://en.wikipedia.org/wiki/File:Triangle.Acute.svghttp://en.wikipedia.org/wiki/File:Triangle.Scalene.svghttp://en.wikipedia.org/wiki/File:Triangle.Isosceles.svghttp://en.wikipedia.org/wiki/File:Triangle.Equilateral.svghttp://en.wikipedia.org/wiki/File:Triangle.Obtuse.svghttp://en.wikipedia.org/wiki/File:Triangle.Right.svghttp://en.wikipedia.org/wiki/File:Triangle.Acute.svghttp://en.wikipedia.org/wiki/File:Triangle.Scalene.svghttp://en.wikipedia.org/wiki/File:Triangle.Isosceles.svghttp://en.wikipedia.org/wiki/File:Triangle.Equilateral.svghttp://en.wikipedia.org/wiki/File:Triangle.Obtuse.svghttp://en.wikipedia.org/wiki/File:Triangle.Right.svghttp://en.wikipedia.org/wiki/File:Triangle.Acute.svghttp://en.wikipedia.org/wiki/File:Triangle.Scalene.svghttp://en.wikipedia.org/wiki/File:Triangle.Isosceles.svghttp://en.wikipedia.org/wiki/File:Triangle.Equilateral.svghttp://en.wikipedia.org/wiki/File:Triangle.Obtuse.svghttp://en.wikipedia.org/wiki/File:Triangle.Right.svghttp://en.wikipedia.org/wiki/File:Triangle.Acute.svghttp://en.wikipedia.org/wiki/File:Triangle.Scalene.svghttp://en.wikipedia.org/wiki/File:Triangle.Isosceles.svghttp://en.wikipedia.org/wiki/File:Triangle.Equilateral.svg


4/15

3


In the figure, classify ABC, PQR and XYZaccording to

(a) the sizes of their angles,

(b) the lengths of their sides.


In XYZ, findp.

EEXXEERRCCIISSEE22CC

Find the values of the unknowns in the following figures.


5/15

4

Polygon

A polygonis a closed figure formed by three or more straight lines.

We have special names for polygons with small number of sides:

Number of

sides

3 4 5 6

Name Triangle Quadrilateral Pentagon Hexagon

Number of

sides

7 8 9 10

Name Heptagon Octagon Nonagon Decagon

For polygons with more sides, say 50 sides, we usually call them 50-sided polygons.

A polygon is said to be convexif all its interior angles are smaller than 180o.

Otherwise it is said to be concave.

A convexpolygon A concavepolygonA polygon is said to be equilateralif all the sides have equal length. It is said to be

equiangularif all its interior angles are equal. A polygon which is both equilateral

and equiangular is regular.

Equilateral Equiangular Regular
http://upload.wikimedia.org/wikipedia/commons/8/8f/Simple_polygon.svghttp://en.wikipedia.org/wiki/File:Regular_hexagon.svghttp://en.wikipedia.org/wiki/File:Regular_decagon.svghttp://upload.wikimedia.org/wikipedia/commons/d/dd/Regular_nonagon.svghttp://upload.wikimedia.org/wikipedia/commons/6/66/Regular_octagon.svghttp://upload.wikimedia.org/wikipedia/commons/9/93/Regular_heptagon.svghttp://en.wikipedia.org/wiki/File:Regular_hexagon.svghttp://en.wikipedia.org/wiki/File:Regular_pentagon.svghttp://upload.wikimedia.org/wikipedia/commons/8/8f/Simple_polygon.svghttp://en.wikipedia.org/wiki/File:Regular_hexagon.svghttp://en.wikipedia.org/wiki/File:Regular_decagon.svghttp://upload.wikimedia.org/wikipedia/commons/d/dd/Regular_nonagon.svghttp://upload.wikimedia.org/wikipedia/commons/6/66/Regular_octagon.svghttp://upload.wikimedia.org/wikipedia/commons/9/93/Regular_heptagon.svghttp://en.wikipedia.org/wiki/File:Regular_hexagon.svghttp://en.wikipedia.org/wiki/File:Regular_pentagon.svghttp://upload.wikimedia.org/wikipedia/commons/8/8f/Simple_polygon.svghttp://en.wikipedia.org/wiki/File:Regular_hexagon.svghttp://en.wikipedia.org/wiki/File:Regular_decagon.svghttp://upload.wikimedia.org/wikipedia/commons/d/dd/Regular_nonagon.svghttp://upload.wikimedia.org/wikipedia/commons/6/66/Regular_octagon.svghttp://upload.wikimedia.org/wikipedia/commons/9/93/Regular_heptagon.svghttp://en.wikipedia.org/wiki/File:Regular_hexagon.svghttp://en.wikipedia.org/wiki/File:Regular_pentagon.svghttp://upload.wikimedia.org/wikipedia/commons/8/8f/Simple_polygon.svghttp://en.wikipedia.org/wiki/File:Regular_hexagon.svghttp://en.wikipedia.org/wiki/File:Regular_decagon.svghttp://upload.wikimedia.org/wikipedia/commons/d/dd/Regular_nonagon.svghttp://upload.wikimedia.org/wikipedia/commons/6/66/Regular_octagon.svghttp://upload.wikimedia.org/wikipedia/commons/9/93/Regular_heptagon.svghttp://en.wikipedia.org/wiki/File:Regular_hexagon.svghttp://en.wikipedia.org/wiki/File:Regular_pentagon.svghttp://upload.wikimedia.org/wikipedia/commons/8/8f/Simple_polygon.svghttp://en.wikipedia.org/wiki/File:Regular_hexagon.svghttp://en.wikipedia.org/wiki/File:Regular_decagon.svghttp://upload.wikimedia.org/wikipedia/commons/d/dd/Regular_nonagon.svghttp://upload.wikimedia.org/wikipedia/commons/6/66/Regular_octagon.svghttp://upload.wikimedia.org/wikipedia/commons/9/93/Regular_heptagon.svghttp://en.wikipedia.org/wiki/File:Regular_hexagon.svghttp://en.wikipedia.org/wiki/File:Regular_pentagon.svghttp://upload.wikimedia.org/wikipedia/commons/8/8f/Simple_polygon.svghttp://en.wikipedia.org/wiki/File:Regular_hexagon.svghttp://en.wikipedia.org/wiki/File:Regular_decagon.svghttp://upload.wikimedia.org/wikipedia/commons/d/dd/Regular_nonagon.svghttp://upload.wikimedia.org/wikipedia/commons/6/66/Regular_octagon.svghttp://upload.wikimedia.org/wikipedia/commons/9/93/Regular_heptagon.svghttp://en.wikipedia.org/wiki/File:Regular_hexagon.svghttp://en.wikipedia.org/wiki/File:Regular_pentagon.svghttp://upload.wikimedia.org/wikipedia/commons/8/8f/Simple_polygon.svghttp://en.wikipedia.org/wiki/File:Regular_hexagon.svghttp://en.wikipedia.org/wiki/File:Regular_decagon.svghttp://upload.wikimedia.org/wikipedia/commons/d/dd/Regular_nonagon.svghttp://upload.wikimedia.org/wikipedia/commons/6/66/Regular_octagon.svghttp://upload.wikimedia.org/wikipedia/commons/9/93/Regular_heptagon.svghttp://en.wikipedia.org/wiki/File:Regular_hexagon.svghttp://en.wikipedia.org/wiki/File:Regular_pentagon.svghttp://upload.wikimedia.org/wikipedia/commons/8/8f/Simple_polygon.svghttp://en.wikipedia.org/wiki/File:Regular_hexagon.svghttp://en.wikipedia.org/wiki/File:Regular_decagon.svghttp://upload.wikimedia.org/wikipedia/commons/d/dd/Regular_nonagon.svghttp://upload.wikimedia.org/wikipedia/commons/6/66/Regular_octagon.svghttp://upload.wikimedia.org/wikipedia/commons/9/93/Regular_heptagon.svghttp://en.wikipedia.org/wiki/File:Regular_hexagon.svghttp://en.wikipedia.org/wiki/File:Regular_pentagon.svg


6/15

5


Classify the following polygons.

1. Convex: 2. Concave:

3. Equilateral: 4. Equiangular:

5. Regular:


Name the following polygons:

1.2. 3.
http://en.wikipedia.org/wiki/File:Regular_pentagon.svghttp://upload.wikimedia.org/wikipedia/commons/9/93/Regular_heptagon.svghttp://upload.wikimedia.org/wikipedia/commons/d/dd/Regular_nonagon.svghttp://en.wikipedia.org/wiki/File:Regular_pentagon.svghttp://upload.wikimedia.org/wikipedia/commons/9/93/Regular_heptagon.svghttp://upload.wikimedia.org/wikipedia/commons/d/dd/Regular_nonagon.svghttp://en.wikipedia.org/wiki/File:Regular_pentagon.svghttp://upload.wikimedia.org/wikipedia/commons/9/93/Regular_heptagon.svghttp://upload.wikimedia.org/wikipedia/commons/d/dd/Regular_nonagon.svghttp://en.wikipedia.org/wiki/File:Regular_pentagon.svghttp://upload.wikimedia.org/wikipedia/commons/9/93/Regular_heptagon.svghttp://upload.wikimedia.org/wikipedia/commons/d/dd/Regular_nonagon.svghttp://en.wikipedia.org/wiki/File:Regular_pentagon.svghttp://upload.wikimedia.org/wikipedia/commons/9/93/Regular_heptagon.svghttp://upload.wikimedia.org/wikipedia/commons/d/dd/Regular_nonagon.svghttp://en.wikipedia.org/wiki/File:Regular_pentagon.svghttp://upload.wikimedia.org/wikipedia/commons/9/93/Regular_heptagon.svghttp://upload.wikimedia.org/wikipedia/commons/d/dd/Regular_nonagon.svghttp://en.wikipedia.org/wiki/File:Regular_pentagon.svghttp://upload.wikimedia.org/wikipedia/commons/9/93/Regular_heptagon.svghttp://upload.wikimedia.org/wikipedia/commons/d/dd/Regular_nonagon.svghttp://en.wikipedia.org/wiki/File:Regular_pentagon.svghttp://upload.wikimedia.org/wikipedia/commons/9/93/Regular_heptagon.svghttp://upload.wikimedia.org/wikipedia/commons/d/dd/Regular_nonagon.svghttp://en.wikipedia.org/wiki/File:Regular_pentagon.svghttp://upload.wikimedia.org/wikipedia/commons/9/93/Regular_heptagon.svghttp://upload.wikimedia.org/wikipedia/commons/d/dd/Regular_nonagon.svghttp://en.wikipedia.org/wiki/File:Regular_pentagon.svghttp://upload.wikimedia.org/wikipedia/commons/9/93/Regular_heptagon.svghttp://upload.wikimedia.org/wikipedia/commons/d/dd/Regular_nonagon.svg


7/15

6

SolidA figure in three dimensional space is often called a solid. A solid often has a

number ofvertices, edgesand faces.

A solid is called a polyhedron is every face is a polygon.

For all polyhedron

VE+ F= 2

Below are some common solids.

Cube Cuboid Triangular Prism Rectangular Pyramid

Cylinder Cone Sphere


8/15

7

If we cut a solid along a direction which is parallel to its base, we obtain the

cross-section.

EEXXEERRCCIISSEE44

The following figure shows two polyhedra, which all the edge lengths are identical.

Draw the cross sections.

There are various ways to represent solids. One way is to use the grids.

Isometric grids

An isometric grid is formed by three sets of lines: vertical lines and two other sets of

lines which form 60o with the vertical lines.

Oblique grids

An oblique grid is formed by three sets of lines: vertical lines, horizontal lines and

lines which are 45o to the horizontal lines.


9/15

8

EEXXAAMMPPLL EE11

Draw the isometric grid of the following figure.

EEXXEERRCCIISSEE55

Draw the oblique and isometric grids of the following figure.


10/15

9

RevisionExercise

1. Write down the types of the following angles.


11/15

10

2.

(a)Which angle is an acute angle?

A. AOE

B. BOFC. DOF

D. COF

(f) FOB =

A. 105o

B. 110oC. 115o

D. 120o

(b)Which angle is a right angle?

A. AOE

B. BOE

C. COE

D. DOG

(g) COA =

A. 10o

B. 15o

C. 20o

D. 25o

(c)Which angle is an obtuse angle?

A. AOE

B. AOD

C. AOC

D. AOB

(h) DOF =

A. 75o

B. 70o

C. 65o

D. 60o

(d)EOG =

A. 55o

B. 50o

C. 45

o

D. 40o

(i) EOC =

A. 70o

B. 75o

C. 80

o

D. 85o

(e) AOG =

A. 155o

B. 150o

C. 145o

D. 140o

(j) BOD =

A. 45o

B. 50o

C. 55o

D. 60o


12/15

11

3. Write down the number of vertices, edges and faces for the following polyhedra.

(a)

(b)

4. Draw the cross-section obtained when the following solids are cut at the dotted

lines along the given planes.

(a)

(b)


13/15

12

6. Draw the uniform cross-sections of the following figures.

(a)

(b)

7. Draw the 2-D representation of the following figure on an isometric grid paper.

8. Draw the 2-D representation of the following figure on an oblique grid paper.


14/15

13

9. Without using a protractor, find the marked angle in each of the following clocks.

(a)

(b)

10. Draw the isometric grids and oblique grids of the following figures.


15/15

14

degree Cube

acute angle Cuboid

right angle triangular prism

obtuse angle rectangular pyramid

straight angle cylinder

reflex angle cone

scalene triangle sphere

isosceles triangle isometric grid

equilateral triangle oblique grid

polygon

quadrilateral

pentagon

hexagon

heptagon

octagon

nonagon

decagon

convex polygon

concave polygon

edge

vertice

face

GLOSSARY

Documents

Basic Geometry STUDENT