Basic Mathematics Form One Notes – Geometry
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Points and Lines
The Concept of a Point
Explain the concept of a point
‘A point — is a smallest geometric figure which gives a position of object in a plane
AA Line segment — isa straight line joining two points in a plane
The Concept of a Point to Draw a Line
Extend the concept of a point to draw a line
A line segment — is a straight line joining two points in a plane
AB
oo
A B
A line passing through two points e.g A and B and extends without end (i.e infinitely) in both
directions is denoted by
AB
<>
A B
The Difference Between a Line, a Line Segment and a Ray
Distinguish between a line, a line segment and a ray
A ray – isa line starting from a point, say A and pass through a point, say B and extends without
end in one direction. It is denoted by
a
AB
i
A B
Angles and Lines
Angles
Draw angles
‘An angle ~ is a measure of an amount of turn, For instance, a complete turn has an angle of 360°
360°
Measuring Angles of Different Size Using a Protractor
‘Measure angles of different size using a protractor
There are several types of angles including:- acute, right, complementary, obtuse, supplementary
and reflex angle
2
(a) An acute angle — is an angle less than 90°
A acute
(b) A right angle — is an angle of 90°
(c) Complementary angles —are angles whose sum is 90°
e.g 40° and 50° are complementary angles
40°
50°
a
(d) An obtuse angle — is an angle between 90° and 180°
\ obtuse
(e) A straight angle – is an angle of 180°
an
(f) Supplementary angle —are angles whose sum is 180°
e.g 135° and 45° are complementary angles
PY ‘i 45°
(g) A reflex angle – is an angle between 180° and 360°
a
Example 1
a Two angles are supplementary, One angle is three times the other. What are the angles?
b. Two angles are complementary. One angle is 40° greater than the other. What are the
angles?
Solution
oa
(a) Let one angle be x, the other angle is 3x
Then x Sx=180°
180°
4x = 180°, x= a = 45°
first angle,x = 45°, second angle, 3x =43(
+. The angles are 45° and 135°
(b) Let one angle be x, the other angle is x + 40°
Then x +(x + 40°) = 90°
x+x+40° = 90°
2x + 40° = 90°
2x = 90° — 40°
50°
2x = 50°, x= = 25°
first angle,x = 25°, second angle, x + 40° F :
-. The angles are 25° and 65°
Drawing Angles Using a Protractor
‘Draw angles using a protractor
The angles formed by crossing lines includes vertically opposite angles, alternate angle and
corresponding angles
Vertically opposite angles
The angles on the opposite sides of the crossing lines are equal
6
Consider two line segments AB and CD crossing each other
A D
p–@
C B
ZAPC =ZBPD (vertically opposite)
ZAPD=ZBPC (vertically opposite)
= They are also called X — angles
Alternate angles
Consider a line segment crossing two parallel line segments. This line is called a transversal
Y transversal
The angles within the parallel line segments on the opposite sides of the transversal are equal
66
A E B
c 7 D
ZAEF=ZDFE (alternate)
They are also called Z – angles
Corresponding angles
The angles on the same side of the transversal and on the same side of the parallel lines are
equal. They are called corresponding angles and sometimes called F – angles
H
A E B
Cc F D
G
4FEB=ZGFD_ (corresponding)
There are also three other pairs of corresponding angles in the diagram above. When showing that
two angles are equal you must give reason whether they are vertically opposite, or alternate or
corresponding angles.
Constructions
oa
Construction of a Perpendicular Bisector to a Line Segment
Construct a perpendicular bisector to a line segment
Perpendicular Bisector to a Line Segment is shown below
t Ti 1
A ° B
Contruction of an Angle of 60° Using a Pair of Compasses
Construct an angle of 60° using a pair of compasses
Angle of 60°
/ 60° tk,
Bisection of a Given Angle
Bisect a given angle
Bisect a given angle
Copying a Given Angle by Construction
Copy a given angle by construction
«
Copy a given angle by construction
Parallel Lines
Construct parallel lines
Parallel lines can be shown as below:
° as
:
doe
Different Types of Angles Formed by Parallel Lines and a Transversal
Identify different types of angles formed by parallel lines and a transversal
Different types of angles are shown below
afo
tf
‘fr
sf”
Corresponding Angles:
AnEB=F,C-6,0=H
Alternate interior Angles: Alternate Exterior Angtes:
C=F,D-E A=HB=6
Polygons And Regions
A Polygon and a Region
«
Describe a polygon and a region
Apolygonal regionis defined as a polygon and its interior.
Different Types of Triangles
A triangle ~ is a polygon with three sides. The sides connect the points called vertices
A right — angled triangle — has one angle equal to 90°
KX c
2 ABC = 90°
An osctes triangle has two equal sides and two equal angles
ne
A
ZL».
ZABC =ZBCA
(AB = AC
‘An equilateral triangle —has three equal sides and all angles equal
A
B C
ZABC = ZBCA=2ZCAB
AB = BC =CA
NOTE:<A triangle with all sides different and all angles different is called scalene triangle
.
A triangle with vertices 4, B and C is denoted as
A ABC
A triangle has two kinds of angles
a. Interior angles
b. Exterior angles
Interior angle — is an angle inside the triangle. The sum of interior angles of a triangle is
Example, consider the triangle below
A
7 VAN é
ZABC + 2 BCA + 2 CAB = 180°
Exterior angle – is an angle outside the triangle Consider the triangle below
n
Cc
E
B
A
2 CBE —is an exterior angle of a triangle
From the triangle above
2 CBE + 2 CBA = 180°
Z CBE = 180°—ZCBA] (angles onastraight line)
Example 2
Find the angles x and y in the diagrams below
@) ) Oo Cas
Solution
(a) x + 41° + 73° = 180°
x + 114° = 180°
x = 180° — 114°
x = 66°
(b) y +52° + 90° = 180°
y + 142° = 180°
y = 180° — 142°
y = 38°
B
Different Quadrilaterals
Construct different quadrilaterals
‘A quadrilateral — is a polygon with four sides.Examples of quadrilaterals are a square, a
rectangle, a rhombus, a parallelogram, a kite and a trapezium
A square ~ has equal sides and all angles are 90°
A rectangle — has two pairs of opposite sides equal and all angles are 90°
A rhombus — has all sides equal, Opposite angles are also equal
A parallelogram — has two pairs of opposite sides equal Opposite angles are also equal
”
A kite —has two pairs of adjacent sides equal One pair of opposite angles are also equal
A trapezium — has one pair of opposite sides pair
Any quadrilateral is made up of two triangles. Consider the below quadrilateral
B
‘Sum of angles of quadrilateral = 2 «180° = 360°
Example 3
Find the angles x and y in the diagrams below
(a) x (b)
150° D
Q) 80°
60°
, C\
Solution
76
(a) x + 150° + 80° + 60° = 360°
x + 290° = 360°
x = 360° — 290°
x= 70°
(b) y + + 70° + 90° = 360°
2y + 160° = 360°
2y = 360° — 160°
2y = 200°
_ 200° _ 100°
ar rte
y = 100°
Circles
A Circle
Draw a circle
To make a circle: Draw a curve that is “radius” away from a central point.
Roos
3S
ye
ow
&
Centere
And so:All points are the same distance from the center.
‘You can draw it yourself:Put a pin in a board, put a loop of string around it, and insert a pencil
into the loop. Keep the string stretched and draw the circle!
7
Different Parts of a Circle
Describe different parts of a circle
The radius of the circle is a straight line drawn from the center to the boundary line or the
circumference. The plural of the word radius is radii
{ Radius
{ or |
The diameter is the line crossing the circle and passing through the center. It is the twice of the
length of the radius,
/ \
| Diameter |
\ o
The circumference of a circle is the boundary line or the perimeter of the circle
B
{
( . |
6
Cremderence
‘An are is a part of the circumference between two points or a continuous piece of a circle. The
shorter arc between and is called the minor are. The longer arc between and is called the major
Minor Are
2 Ne
{ . |
\ 6
Major Are
The chord is a straight line joining two points on the circumference points of a circle. The
diameter isa special kind of the chord passing through the center.
Chord
{ . |
\ 6
~
Asemi-cireleis an are which is half of thecircumference.
Semi Circle
——
/ ~\
\ 0
A tangent is a straight line which touches the circle. It does not cut the circumference. The point
at which it touches, is called the point of contact.
\
+ |
a |
Tangent Line
80