Download 7th class geometry

Lesson:
Geometry
General Review
Prepared by:
BASIC CONCEPTS OF
GEOMETRY
POINT
PLANE
LINE
Point
The most basic concept of geometry
is the idea of a point in space.
A point has no length, no width,
and no height, but it does have
location. We will represent a point by
a dot, and we will label points with
letters.
A
Point A
A plane is a flat surface that extends
indefinitely.
Plane
Space extends in all directions
indefinitely.
LINE
A line is straight arragement of
points .
A line has no width ,no thickness
and extends without end in both
direction.
Line AB or AB
A
B
Line Segment AB or AB
A
B
Ray AB or AB
A
B
Two lines in a plane can be either parallel or
intersecting.
Parallel lines never meet
Intersecting lines meet at a point.
The symbol  is used to denote “is parallel to.”
p
q
Parallel lines
p  q
Intersecting
lines
Two lines are perpendicular if they
form right angles when they intersect.
The symbol  is used to denote “is
perpendicular to.”
m
n
Perpendicular lines
nm
ANGLES
We should not confuse difference between angle and angel
An angle is the union of two rays that have a
commen endpoint.
An angle is made up of two rays that share the same
endpoint called a vertex.
A
Vertex
B
x
C
The angle can be named ABC,
CBA, B or x.
Angles are measured by an amount of
rotation
We measure this rotation in units called
degrees .we show it as()
360º
Classifying Angles
Name
Acute
Angle
Right
Angle
Obtuse
Angle
Straight
Angle
Angle Measure
Between 0°
and 90°
Exactly 90°
Between 90°
and 180°
Exactly 180º
Examples
AnGlES(song)
Everywhere you look
There are angles.
Acute, obtuse and right angles.
Everywhere you look
There are angles.
Acute, obtuse and right angles.
Everywhere you look
There are angles.
How many can you find?
Right angles have 90 degrees.
Acute have less than these.
Obtuse angles open wide,
Wider than 90 degrees.
Everywhere you look
There are angles.
Acute, obtuse and right
angles.
Everywhere you look
There are angles.
How many can you find?
Right angles have 90 degrees.
Acute have less than these.
Obtuse angles open wide,
Wider than 90 degrees.
Everywhere you look
There are angles.
Acute, obtuse and right
angles.
Everywhere you look
There are angles.
How many can you find?
When two lines intersect, four angles are
formed. Two of these angles that are opposite
each other are called vertical angles. Vertical
angles have the same measure.
a
d
a = c
c
b
d = b
Two angles that share a common side are
called adjacent angles. Adjacent angles formed
by intersecting lines are supplementary. That
is, they have a sum of 180 °.
a
d
b
c
a and b
b and c
c and d
d and a
Two angles that have a sum of 90° are called
complementary angles.
A
C
B
o
30
o
N
M
60
D
Two angles that have a sum of 180° are called
supplementary angles.
B
A
o
30
C
150o
N
M
D
A line that intersects two or more
lines at different points is called a
transversal.
Parallel Lines Cut by
a Transversal
If two parallel lines are cut by a transversal, then the measures of
corresponding angles are equal(1) and
alternate interior angles are equal(2).
Alternate exterior angles are equal(3).
Same-side interior angles
are suplementary
a b
c d
e f
g h
Corresponding angles are equal.
 a = e
a b
c d
e f
g h
 c= g
 d = h
 b=  f
Same-side interior angles
 c +  e=180 
 d +  f=180
Alternate interior angles are angles on
opposite sides of the transversal between
the two parallel lines.
a b
c d
 c=  f
e f
g h
 e= d
Alternate exterior angles
g= b
 a = h
TRIANGLE: Triangle is a polygon with
three sides. If we connect three noncollinear
poinst we get a triangle
vertices
the plural of vertex is vertices
listening
A TRIANGLE




Has three angles and three sides
The word triangle means “three angles”
symbol of a triangle
has three vertices
Regions of a triangle:
A triangle separrates a plane into three different
regions
These regions are the triangle itself
And the interior and exterior region of the triangle
EXTERIOR REGION
INTERIOR
AUXILIARY ELEMENTS OF A TRIANGLE
MEDIAN
Va
ANGLE BISECTOR
nA
ALTITUDE
ha
Perimeter of a Triangle
Perimeter =The perimeter of a triangle
is the sum of the lenghts of its sides
P( ABC )  a  b  c
Area of a Triangle
The area of a triangle is half of the product of the lenght of
a base and the height of the altitude drawn to that base.
A
m
hd
D
d
hm
ha
aE
A( ADM )  a 2ha
M
d hd
2
m hm
2
Helpful Hint
Perimeter is always
measured in units.
The perimeter of every polygon may be
found by adding all the sides.
Helpful Hints
Area is always measured in square
units.
When finding the area of figures, check
to make sure that all measurements
are the same units before
calculations are made.
TYPES OF TRIANGLE
we can classify triangles accordingto the lengths of their
sides or according to the measures of their angles
listening
A right triangle is a triangle in which one
of the angles is a right angle or measures
90º (degrees).
The hypotenuse of a right triangle is the
side opposite the right angle. The legs of a
right triangle are the other two sides.
hypotenuse
leg
leg
a+b+c=?
Angles on a Triangle
What is the sum of the measures of the
interior angles of a triangle?
a+b+c=?
B
A
C
B
+
A
+
C
The sum of the measures of the interior
angles of a triangle is 180°.
Helpful Hint
The sum of the two acute angles in a
right triangle is 90.
Equilateral triangle’s all angles are 60°
Isosceles triangle’s bases angles are
congruent
Triangle Exterior Angle
Theorem
x y  z
The measure of an exterior angle (z) in a triangle
is equal to the sum of the measures of its two
nonadjacent (y),(x) interior angles.
listening
The sum of the measures of the exterior
angles of a triangle is equal to 360°
Next week on monday
I. exam
RELATIONS BETWEEN
ANGLES AND SIDES
1.LONGER SIDE OPPOSITE
LARGER ANGLE
a  b  c  m(A)  m(B)  m(C )
2.LARGER ANGLE OPPOSITE
LONGER SIDE
m(A)  m(B)  m(C )  a  b  c
If two sides of a triangle are congruent,
the angles opposite these sides are also
congruent.
b  c  m(B)  m(C )
If two angles of a triangle are congruent,
the sides opposite these angles are also
congruent
m(B)  m(C )  b  c
examples
Write the measures of the angles in each triangle
in increasing order.
A
5
4
9
9
6
5
C
B
F
K
M
L
10
S
G
7
7
Write the lenghts of the sides from the smallest to
the biggest of each triangle .
B
L
85
C
55
L
A
60
50
65
70
40
95
20
M
C
C
B
,
Triangle inequality
1- The sum of the lengths of two sides of a triangle is
greater than the length of the third side.
c  a b
b  ac
a bc
The difference between the lengths of
two sides of a triangle is less than the
length of the third side.
ac b
bc abc
bc a
ac bac
ab c
ab cab
example
cm,
cm,.
.
In the given figure,
BC  9
AB  5
cm
find the possible
integer
AC
Values of
ac  b  ac
95  x  95
4
 x  14
AC
5, 6, 7, 8, 9,10,11,12,13
Some properties (((gstrmmd
A
1
If m(A)  90
b
c
B
so a 2  b2  c 2
C
a
A
2
If m(A)  90
c
b
B
so a 2  b 2  c 2
C
a
C
3
If m(A)  90
b
a
so
A
c
B
c 2  b2  a 2
examples
D
A
6
8
y
B
a
C
ıf m(A)  90
Find all the possible
int eger lengths of a ?
solution :
If m(E )  90
Find thebiggest int eger
4
F
E
6
4  6  y  42  62
b 2  c 2  a  (b  c ) 2  y 
52
82  62  a  (8  6) 2  y  2 13 :
10  a  14 so
y7
a  11,12,13

valueof y
7  2 13  8
In a scalene triangle ABC,
,
and
and
In a scalene triangle ABC
A
ha  n A  V a
ha
B
nA
H
K
ha altitude
nA angle bi sec tor
Va
D
hc  nC  Vc
C
Va median
hb  n B  Vb
In an equilateral triangle ABC ha  n A  V a
Unknown words







Ray :A ray is a straight line which extends infinitly in one
direction from a fixed point.
Amount , a collection or mass especially of something which
can not be counted
Flat:level and smooth,with no curved
Angle:The space between two lines or surface at the point
at which they touch each other .
Angel:A spiritual creature in religions.
Share:a part of something that has been divided
between several people.
Polygon:A flat shape with three or more straight sides.
Unknown words
Corresponding(shesabamisi)=The points, lines, and angles which
match perfectly
when two congruent figures are placed one on top of the other
The difference of a winner and a loser
CONGRUENCE (n)(kongruenteloba)
CONGRUENT (adj) (kongruentuli)means
equal in all respects(things).
Objects which have same size and same shape
are called congruent objects.
The figures have the same shape but they have different size.
So, the figures are not congruent
The figures have the same size and same shape.
So they are all congruent figures

Congruent Triangles
if the corresponding angles and
corresponding sides are congruent,
∆KLM
 ∆ XYZ
K corresponds to X
L corresponds to Y
M corresponds to Z
then these triangles are called
congruent triangles.
.
ABC  DEF
A   D
B   E
C  F
AB  DE
BC  EF
AC  DF
corresponding angles are congruent to each other
all corresponding sides are congruent to each other.
Example
:
MNP  STX
state the congruent parts without drawing the triangles.
sides MN and ST
which sides are congruent sides NP and TX
sides PM and XS
Angles M and S
which angles are congruent Angles N and T
Angles P and X
We can write the congruence ABC  DEF in six different ways
ABC  DEF
ACB  DFE
BAC  EDF
BCA  EFD
CAB  FDE
CBA  FED
B.WORKING WITH CONGRUENT TRIANGLES
1.The Side-Angle-Side (SAS) Congruence Postulate
example
homework
2.Angle- Side- Angle Congruence Postulate
3.Side-Side-Side Congruence Postulate
example
Theorems
If a line parallel to one side of a triangle bisects another side of the
triangle ,ıt also bisects the third side.
A
A
step1
N
step2
N
K
d
K
d
B
C
T
AN=NB
B
AK=KC
C
A
step3
N
d
K
NAK=
B
AN=NB
AK=KC
T
C
TKC
Angle N and angle T corresponding
angles and they are congruent
ASA
A
step4
N
d
K
NAK=
TKC
B
T
C
AN=NB
AK=KC
According to ASA side NK =TC
TC =BC/2 and side NK=BT
And side KT =NB and KT=AN
Because AN=NB
Triangle Midsegment Theorem
The line segment which joins the midpoint of two sides of a triangle is
called a midsegment of the triangle
It is parallel to the third side and its lenght is equal to half the lenght of
the third side
A
D
E
C
DC is parallel to EM
M
DC=EM/2
example1
example2
A
In the figure points D,M, C are
the midpoint of AE ,EM,AK
respectively
AE=10 cm
AK=12 cm
D
E
Find the perimeter of the triangle
DMC
C
M
K
EK=14 cm
18 cm
20 cm
22 cm
16 cm
homework


Page 163 (check yourself15) .2and 3
Page 191 . 17and 18
ISOSCELES;EQUILATERAL AND
RIGHT TRIANGLES
Properties of Isosceles and Equilateral Triangles
1* *(REMEMBER)
Properties
A
1.In any isosceles triangle
D
Va
nA

M

h
a
E
example
A
AB  AC
find the measure of x
O
20
9 cm
Find the P ( ABC )
X
B
5 cm
K
C
2.ABC is an isosceles triangle with
AB  AC and PE AB and PD AC
then PE  PD  AB  AC
A
E
D
B
P
C
example
AB  AC
find the P ( ABC )
A
E
D
12 cm
B
10 cm
P
BC=20 cm
C
solution
3.ABC is an isosceles triangle with
AB  AC
then PD  PH  hc  hb
A
D
B
H
P
C
1.In any equilateral triangle
AB  AC  BC and PE AB and PD AC
then PE  PD  AB  AC  BC
A
A
E
D
E
D
example
b
a
B
B
P
C
P
C
AB  AC  BC
AB  BC  AC  60 cm
Find a  b
2.In any equilateral triangle
If PF BC and PD AC and EP AB
then PE  PD  PF  AB  AC  BC
example
A
A
D
D
P
9 cm
F
P
F
11 cm
8 cm
B
E
C
B
E
ABC is an equilateral triangle
Find the P( ABC )
C
3.In any equilateral triangle
A
D
E
C
B
F
PE  PD  PF  AH where AH  BC
homework

Page 171 (check yourself17) .1 .2.3.4.6
Now time to listen a friend of
Pythagoras
Properties of Right Triangles
TIME TO LAUGH -1
General exam questions