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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
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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
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Express65 of a straight angle in degrees.
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Write down the types of the marked angles in the figure.
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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.svg7/29/2019 Basic Geometry STUDENT
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In the figure, classify ABC, PQR and XYZaccording to
(a) the sizes of their angles,
(b) the lengths of their sides.
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In XYZ, findp.
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Find the values of the unknowns in the following figures.
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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.svg7/29/2019 Basic Geometry STUDENT
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Classify the following polygons.
1. Convex: 2. Concave:
3. Equilateral: 4. Equiangular:
5. Regular:
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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.svg7/29/2019 Basic Geometry STUDENT
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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
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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.
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EEXXAAMMPPLL EE11
Draw the isometric grid of the following figure.
EEXXEERRCCIISSEE55
Draw the oblique and isometric grids of the following figure.
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RevisionExercise
1. Write down the types of the following angles.
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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
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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)
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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.
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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.
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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