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GEOMETRY
TYPES OF ANGLES
1. ACUTE ANGLE
An angle which is less than 90º
2. RIGHT ANGLE
An angle is 90º
3. OBTUSE-ANGLED
An angle between 90º and 180º
4. REFLEX ANGLE
An angle greater than 180º
5. COMPLEMENTARY ANGLES
a
These are angles whose sum is 90º
b
e.g. a + b = 90º
6. SUPPLEMENTARY ANGLES
These are angles whose sum is 180º
e.g. a + b = 180º
1
b
a
PROPERTIES OF ANGLES & STRAIGHT LINES
Consider a straight line:
180º
The total angle on this line is 180º.
If another straight line is drawn and meets the other line at some point “p”, then the
two angles adjacent to each other “a” and “b” will sum to 180º. They are also known
as SUPPLEMENTARY ANGLES.
b
a
p
INTERSECTION OF TWO STRAIGHT LINES
When two straight lines intersect each other, then the angles opposite each other at the
intersection are equal in value.
∠a = ∠c and ∠b = ∠d
c
b
d
a
PARALLEL LINES CUT BY A TRANSVERSAL
The arrows on the horizontal lines show they are in parallel, therefore the
corresponding angles are equal:
∠a = ∠w , ∠b = ∠x , ∠c = ∠y , ∠d = ∠z
a
d
Now as ∠a = ∠c and ∠b = ∠d
then ∠c = ∠w and ∠d = ∠x (i.e. alternate angles)
Also as ∠a = ∠w and ∠b = ∠d ,
then ∠w + ∠d = 180º and ∠c + ∠x = 180º
These are known as supplementary angles
2
w
z
y
x
b
c
PROOF OF PARALLEL LINES
If two straight lines are cut by a transversal, then the two lines are parallel if any of
the following conditions apply:
1. Two corresponding angles are equal. ∠a = ∠w
2. Two alternate angles are equal. ∠d = ∠x
3. Two interior angles are supplementary. ∠c + ∠x = 180º
Ex 1: Calculate the value of angle ά if line BF bisects ∠ABC .
D
θ = 80º
γ
A
B
β
φ = 38º
ά
F
C
X
1.
2.
3.
4.
5.
6.
E
Z
Y
Lines DX, BZ & EY are parallel to each other as they are all at 90º to line XY.
∠DAB = ∠ABZ or θ = γ = 80º (i.e. alternate angles)
β = φ = 38º (i.e. alternate angles)
∠ABC = γ + β = 80º+38º=118º
As line FB bisects angle ∠ABC , then angle ∠FBC = 59º
β + α = 59º, hence α = 59º − β = 59º − 38º = 21º
3
Ex 2: In the diagram below, prove that ∠x = ∠a + ∠b .
b (alternate angle)
a
x
180º- (a + b)
b
a (alternate angle)
Sum of all the angles around the cross = 360º
∴ 360º = 2 x + 2 {180º − ( a + b )}
∴ 2 x = 360º − 2 {180º − ( a + b )}
∴ x = 180º − {180º − ( a + b )}
= 180º − 180º + ( a + b ) = a + b
∴∠x = ∠a + ∠b when the two lines are in parallel.
a
a
x
x
b
b
∠x = ∠a + ∠b
∠x = ∠a + ∠b
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TRIANGLES
TYPES OF TRIANGLES
1. ACUTE-ANGLED
All angles are less than 90º
2. RIGHT-ANGLED
One angle is 90º
The side opposite this angle is known as the hypotenuse
3. OBTUSE-ANGLED
Has one angle greater than 90º
4. EQUILATERAL
60º
All its sides and angles are equal, hence each angle = 60º
60º
60º
5. ISOSCELES
a
b
Two sides and two angles are equal.
N.B. The equal angles are opposite the equal sides
N.B. The sum of all the angles in any triangles = 180º
5
B
A
CONGRUENT TRIANGLES
Consider the following two triangles:
A
D
B
C
E
F
Side AB = Side DE
Side BC = Side EF
Side AC = Side DF
Also angles:
Λ
Λ
Λ
A=D
Λ
Λ
B=E
Λ
C=F
As these triangles are equal in every respect, then they are said to be congruent. Also
if 2 triangles are congruent, then their areas are equal.
i.e.
ΔABC = ΔDEF
Although two triangles may be congruent, it is not necessary to prove that all the sides
and angles are equal. The following may be used to prove congruency:
1. One side and 2 angles are equal (similar location)
A
B
D
E
C
6
F
2. Two sides and an angle between them are equal.
D
A
B
E
C
F
3. Three sides of one triangle equal three sides of the other.
D
A
B
E
C
F
4. The hypotenuse and one side of a right-angled triangle are equal.
A
B
D
E
C
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F
N.B. If 3 angles on one triangle are equal to the angles on the other triangle, this does
not prove congruency as a minimum of one side is required. By moving the
hypotenuse “DF” to the left, it can be seen that the angles are still the same but
the triangles are not congruent.
D
A
B
E
C
F
Also two sides and a non-included angle do not prove congruency. In triangles OBA
and OCA, angle θ and side OA are common. Also sides OB and OC are equal,
however the triangles are not congruent. See in diagram below:
C
B
θ
O
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A
PYTHAGORAS THEOREM
( AC )
2
A
2
= ( AB ) + ( BC )
2
∴ AC =
2
( AB ) + ( BC )
2
B
First of all, this theorem is only applicable to right-angled triangles. It can be seen that
if you take the length of each side adjacent to the right angle, square them and then
add them together, the resulting sum will equal the longest side (hypotenuse) squared.
Example
A ladder is placed against a vertical wall that makes a right angle
with the horizontal ground. The base of the ladder is 1m away from the
wall and the top is at a height of 4m. Calculate the length of the
ladder.
T
4m
1m
TB =
B
2
2
( 4 ) + (1)
TB = 16 + 1 = 17 = 4.12m
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C
SIMILAR TRIANGLES
Triangles that have the same angles as each other are considered to be similar even
though the lengths of the sides are different. Consider the example below. Both
triangles are right-angled. The value of θ in the smaller triangle is 90o - 60o = 30o. In
the larger triangle, the value of φ is 90o - 30o = 60o, so the triangles are similar, hence
the sides of “a” and “b” in the larger triangle can be calculated.
30o
θo
b
69.282cm
10cm
φo
60o
5cm
a
Firstly, the unknown side of the smaller triangle needs to be calculated using
Pythagoras Theorem:
102 − 52 = 75 = 8.66cm
The ratio of the larger triangle to the smaller one is:
69.282
=8
8.66
As the ratio of the larger triangle to the smaller one, then side “a” = 8 x 5 = 40cms and
side “b” = 8 x 10 = 80cm.
N.B. You can also use Pythagoras Theorem to calculate the length of a side once you
have the lengths of the other two sides.
e.g.
b = 402 + 69.2822 = 6400 = 80cm
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