Download grade 10 mathematics session 7: euclidean geometry

GRADE 10 MATHEMATICS
SESSION 7: EUCLIDEAN
GEOMETRY
Euclidean
Geometry
Key Concepts
 Classifying Angles
 Parallel lines and transversal lines
 Classifying Triangles
 Properties of Triangles
 Relationship between angles
 Congruency
 Similarity
 Pythagoras
 Mid-point Theorem
 Properties of Quadrilaterals
Terminology
Acute angle:
Greater than 00 but less than 900
Right angle:
Angle equal to 900
Obtuse angle:
Angle greater than 900
Straight angle:
Angle equal to 1800
Angle greater than 1800 but less than 3600
Reflex angle:
Revolution:
Sum of the angles around a point, equal to 3600
Adjacent angles:
Angles that share a vertex and a common side.
Vertically opposite angles: Angles opposite each other when two lines intersect.
They share a vertex and are equal.
Supplementary angles: Two angles that add up to 1800.
Complementary angles: Two angles that add up to 900.
Parallel lines
Lines that are always the same distance apart
A transversal line A line that intersects two or more parallel lines.
Interior angles
Angles that lie in between the parallel lines.
Exterior angles
Angles that lie outside the parallel lines.
Corresponding angles
Angles on the same side of the lines and the same side
of the transversal.
Co-interior angles
Angles that lie in between the lines and on the same side
of the transversal.
Alternate interior angles Interior angles that lie inside the line and on opposite
sides of the transversal.
X-planation
Properties of the angles formed by a transversal line intersecting two parallel lines
If the lines are parallel 


the corresponding angles will be equal
the co-interior angles are supplementary
the alternate interior angles will be equal.
If the corresponding angles will be equal or the co-interior angles are supplementary or the
alternate interior angles will be equal
 the lines are parallel
Classifying Triangles
There are four kinds of triangles:
Scalene Triangle
No sides are equal in length
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||
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Isosceles Triangle
Two sides are equal
Base angles are equal
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Equilateral Triangle
All three sides are equal
All three interior angles are equal
Right-angled triangle
One interior angle is 90
A
60
||
||
60
||
60
B
C
Relationship between angles
Sum of the angles of a triangle
Exterior angle of a triangle
c
a
b
b
a
a  b  c  180
c
c a  b
Congruency of triangles
Rule 1
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Two triangles are congruent if three
sides of one triangle are equal in
length to the three sides of the other
triangle. (SSS)
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Rule 2
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Two triangles are congruent if two
sides and the included angle are
equal to two sides and the included
angle of the other triangle. (SAS)
Rule 3
Two triangles are congruent if two
angles and one side are equal to
two angles and one side of the other
triangle. (SAA)
|
|


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Two right-angled triangles are congruent
if the hypotenuse and a side of the one
triangle is equal to the hypotenuse and a
side of the other triangle. (RHS)
|
Rule 4
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Similarity
Rule 1 (AAA)
If all three pairs of corresponding angles of two
triangles are equal, then the triangles are similar.
Rule 2 (SSS)
If all three pairs of corresponding sides of two
triangles are in proportion, then the triangles
are similar.
The Theorem of Pythagoras
A
2
AC

AB2  BC2
or
2
AB

AC2  BC2
or
2
BC

AC2  AB2
B
C
Mid-Point Theorem
The line joining the mid-points of two sides of a triangle is parallel to the third side and equal
to half the length of the third side.
Properties of quadrilaterals
Trapezium
A
 Two sides are parallel.
>
=
=
B
D
>
C
=
E
D
=
A
C
|||
|||
E
|||
|||
= =
B
A
Opposite sides are parallel.
All sides equal in length.
Diagonals bisect each other at right angles.
Diagonals bisect the opposite angles.
C
D
|||
=




=
>
 Opposite sides parallel and equal in length.
 Diagonals are equal in length and bisect
each other.
 Interior angles are right angles.
Rhombus
|||
|||
B
Rectangle
D
>
>
>
 Opposite sides parallel and equal.
 Opposite angles equal.
 Diagonals bisect each other.
A
=
Parallelogram
E
|||
|||
=
|||
=
45
45
A
|||
E
C
=
B
C
|||
Adjacent pairs of sides are equal in length
The longer diagonal bisects the opposite angles.
The longer diagonal bisects the other diagonal.
The diagonals intersect at right angles.
B
45
=




E
45
D
|||
Kite
45
45
=
Opposite sides parallel.
All sides equal in length.
Diagonals are equal in length.
Diagonals bisect each other at right angles.
Interior angles are right angles.
Diagonals bisect interior angles
(each bisected angle equals 45 )
45
45
|||






A
=
Square
C
=
B
D
X-ample Questions
Question 1
Calculate the size of the angles marked with small letters:
(a)
(b)
49
70
x y
x
(c)
(d)
70
100
x
x
Question 2:
Calculate the size of the angles marked with small letters:
(a)
(b)
80
30
y
x
40
x
Question 3:
Prove that ABC  ADC
..
A
1 2
|
|
B
1 2
C
D
Question 4:
Consider the diagram below. Is ΔABC ||| ΔDEF? Give reasons for your answer.
Question 5:
In ΔMNP, M = 900, S is the mid-point of MN and T is the mid-point of NR.
(a) Prove U is the mid-point of NP.
(b) If ST = 4 cm and the area of ΔSNT is 6 cm2, calculate the area of ΔMNR.
(c) Prove that the area of ΔMNR will always be four times the area of ΔSNT, let ST = x
units and SN = y units.
Question 6
(a)
Using the information provided
on the diagram, prove that
AD||BC.
(b) What type of quadrilateral is
ABCD? Give a reason.
2x
x
Question 7
In the diagram, PQRS is a parallelogram.
P̂1 45 and PR bisects R̂
Prove that PQRS is a square.
Question 8
Prove that ABCD is a
Question 9
trapezium.
ABCD is a parallelogram with diagonal AC.
Given that AF = HC, show that:
ΔAFD Ξ ΔCHB
45
120
X-ercise
1. Calculate the value of a and b
2. Find the value of x
10
x
24