geometry 1 2
TRANSCRIPT
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Properties of lines
Intersecting Lines and Angles: If two lines intersect at a point, then opposite
angles are called vertical angles and have the same measure. In the figure,
PRQ and SRT are vertical angles and QRS and PRT are vertical angles.
Also, x + y = 180o, since PRS is a straight line.
Perpendicular Lines: An angle that has a measure of 90o
is a right angle. If two lines intersect at right angels, the lines
are perpendicular. For example:
L1 and L2 above are perpendicular and denoted by L1 L2. A right angle symbol
in an angle of intersection indicates that the lines are perpendicular.
Parallel Lines: If two lines that are in the same plane do not intersect, the two lines
are parallel. In figure below lines L1 and L2 are parallel and denoted by L1L2
Parallel lines cut by a transverse: If two parallel lines L1 and L2 are cut by a third
line called the transverse as shown in the figure below:
1 = 3 (Pair of corresponding angles)
2 = 3 (Pair of alternate angles)
and 3 + 4 = 180 (Sum of interior angles)
L2
L1
L1
L2
R
Q
P
S
Tx y
xy
2
3
41
L2
L1
transverse
11 GEOMETRY
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Polygon
A closed plane figure made up of several line segments that are joined together is
called a polygon. The sides do not cross each other. Exactly two sides meet at
every vertex.
Types of Polygons:
Regular: all angles are equal and all sides are the same length. Regular polygons
are both equiangular and equilateral.
Equiangular: all angles are equal.
Equilateral: all sides are the same length.
Properties of Polygon
POLYGON PARTS
Side one of the line segments that make up
the polygon.
Vertex point where two sides meet. Two or
more of these points are called vertices.Diagonal a line connecting two vertices that
isn't a side.
Interior Angle Angle formed by two adjacent
sides inside the polygon.
Exterior Angle Angle formed by two
adjacent sides outside the polygon.
Property 1Each exterior angle of an n sided regular polygon is
n360
o
degrees
Property 2
Each interior angle of an n sided equiangular polygon is
n
180)2n( odegrees.
Also as each pair of interior angle & exterior angle is linear,
Each interior angle = 180o exterior angle.
Property 3Sum of all the interior angles ofn sided polygon is
( o180)2n
Side
Vertex
ExteriorAngle
Diagonal
InteriorAngle
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Triangles
A triangle is a polygon of three sides.
Triangles are classified in two general ways: by their sides and by their angles.
First, we'll classify by sides:
A triangle with three sides of different lengths is called a scalene triangle. An
isosceles triangle has two equal sides. The third side is called the base. The
angles that are opposite to the equal sides are also equal. An equilateral triangle
has three equal sides. In this type of triangle, the angles are also equal, so it can
also be called an equiangular triangle. Each angle of an equilateral triangle must
measure 60o, since the sum of the interior angles of any triangle must equal to
180o.
Obtuse angled triangle: is a triangle in which one angle is always greater than 90o
e.g.
ABC is an obtuse triangle where C is an obtuse angle
Acute angled triangle: In which all angles are less than 90oe.g.
ABC is a acute triangle
Right Angled Triangle: A triangle whose one angle is 90o
is called a right (angled)
Triangle.
In figure, b is the hypotenuse, and a & c the legs, called base and height
respectively.
A
Ca
B
cb
C
A
110o
B
A
CB
80o
40o 60
o
P
R
Q N
M OD
E
F
scalene
equilateral isosceles
30o, 60o, 90o
triangle: This is a
special case of aright triangle
whose angles are
30o, 60o, 90o.
In this triangle
side opposite to
angle 300 =
Hyp/2.
Do you know?If a, b, c denote the
sides of a triangle,
then
(i) Triangle is acute
angled if c2 < a2 + b2
(ii) Triangle is right
angled if c2 = a2 + b2
(iii) Triangle is obtuse
angled if c2 > a2 + b2.
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Properties of a Triangle
1. Sum of the three angles is 180o.
2. An exterior angle is equal to the sum of the interior opposite angles.
3. The sum of the two sides is always greater than the third side.
4. The difference between any two sides is always less than the third side.
5. The side opposite to the greatest angle is the greatest side and the sideopposite to the smallest angle will be the shortest side.
6. Centroid:
(a) The point of intersection of the medians of a triangle. (Median is the
line joining the vertex to the mid-point of the opposite side. The
medians will bisect the area of the triangle.)
(b) The centroid divides each median from the vertex in the ratio 2 : 1.
Orthocentre:
The point of intersection of altitudes. (Altitude is a perpendicular drawn from
a vertex of a triangle to the opposite side.)
Circumcentre:
The circumcentre of a triangle is the centre of the circle passing through the
vertices of a triangle. It is also the point of intersection of perpendicular
bisectors of the sides of the triangle.
If a, b, c, are the sides of the triangle, is the area, then abc = 4R
where R is the radius of the circum-circle.
Incentre:
The point of intersection of the internal bisectors of the angles of a triangle.
(a) AC
AB
LC
BL=
(b)a
cb
IL
AI +=
(c) = rs if r is the radius of incircle,
where s = semi-perimeter =2
cba ++and
is the area of the triangle.
R = 5 cm, r =s
=
12
24= 2 cm
F
A
E
DB C
I
A
RS
CD
B
A
I
BL
Ca
bc
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Equilateral TriangleIn an equilateral triangle all the sides are equal and all the angles are equal.
(a) Altitude = a2
3side
2
3=
(b) Area = 22
a4
3
)side(4
3=
(c) Inradius = Altitude3
1
(d) Circumradius = Altitude3
2
Congruent trianglesTwo triangles ABC and DEF are said to the congruent, if they are equal in all
respects (equal in shape and size).
The notation for congruency is or
If A =D, B = E, C = FAB = DE, BC = EF; AC = DF
Then ABC DEF orABC DEF
The tests for congruency
(a) SAS Test: Two sides and the included angle of the first triangle are
respectively equal to the two sides and included angle of thesecond triangle.
(b) SSS Test: Three sides of one are respectively equal to the three sides of
the other triangle.
(c) AAS Test: Two angles and one side of one triangle are respectively equal
to the two angles and one side of the other triangle.
(d) RHS Test: The hypotenuse and one side of a right-angled triangle are
respectively equal to the hypotenuse and one side of another
right-angled triangle.
Similar TrianglesTwo figures are said to be similar, if they have the same shape but not the same
size. If two triangles are similar, the corresponding angles are equal and the
corresponding sides are proportional.
Test for similarity of triangles
(a) AAA Similarity Test: Three angles of one triangle are respectively equal to
the three angles of the other triangle.
B
A
C E
D
F
C
A
Ba
a a
(i) The equilateral
has maximum
area for given the
perimeter,(ii) Of all the
triangles that can
be inscribed in a
given circle, an
equilateral triangle
has maximum area.
Do you know?Congruent triangles
are similar but
similar triangles
need not be
congruent.
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(b) SAS Similarity Test: Two sides of one are proportional to the two sides of
the other and the included angles are equal.
Properties of similar triangles: If two triangles are similar, the following properties
are true:
(a) The ratio of the medians is equal to the ratio of the corresponding sides.(b) The ratio of the altitudes is equal to the ratio of the corresponding sides.
(c) The ratio of the circumradii is equal to the ratio of the corresponding sides.
(d) The ratio of in radii is equal to the ratio of the corresponding sides.
(e) The ratio of the internal bisectors is equal to the ratio of the corresponding
sides.
(f) Ratio of areas is equal to the ratio of squares of corresponding sides.
Pythagoras TheoremThe square of the hypotenuse of a right-angled triangle is equal to the sum of the
squares of the other two sides i.e. in a right angled triangle ABC, right angled at B,
AC2
= AB2
+ BC2
Pythagorean triplets are sets of 3 integers
which can be three sides of a right-angled
triangle.
Examples of Pythagorean triplets are (3, 4, 5),
(5, 12, 13), (7, 24, 25), (9, 40, 41) etc.
Important Theorems
Basic Proportionality Theorem
In a triangle if a line is drawn parallel to one side of a triangle intersecting the other
two sides, then it divides the other two sides proportionally. So if DE is drawn
parallel to BC, it would divide sides AB and AC proportionally i.e.EC
AE
DB
AD=
We can also use the following results:
(i)AE
AC
AD
AB=
(ii) BC
AB
DE
AD=
Mid Point Theorem:
A line joining the mid points of any two sides of a triangle must be parallel to the
third side and equal to half of that (third side).
A
B C
D E
A
B C
Do you know?If you multiply the
Pythagorean triplets
by constant, then
resultant will also be
Pythagorean triplets
e.g. (6, 8, 10), (18,
24, 30) etc.
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Angle Bisector Theorem:
The internal bisector of an angle of a triangle divides the opposite side internally in
the ratio of the sides containing the angle. i.e. In an ABC in which AD is the
bisector ofA, thenDC
BD
AC
BA= .
Intercept theorem:
Intercepts made by two transversals (cutting lines) on three or more parallel lines
are proportional. In the figure, lines l and m are transversals to three parallel lines
AB, CD, EF. Then, the intercepts (portions of lengths between two parallel lines)
made, AC, BD & CE, DF, are resp. proportional.
DF
CE
BD
AC=
Quadrilaterals
A polygon with 4 sides, is a quadrilateral
1. In a quadrilateral, sum of the four angles is equal to 360.
2. The area of the quadrilateral = one diagonal x sum of the perpendicular
to it from vertices.
Cyclic Quadrilateral:
If a quadrilateral is inscribed in a circle, it is said to be cyclic quadrilateral.
1. In a cyclic quadrilateral, opposite angles are supplementary.
2. In a cyclic quadrilateral, if any one side is extended, the exterior angle so
formed is equal to the interior opposite angle.
Summary of the properties of different types of quadrilaterals.
Name Properties Remarks
Parallelogram Opposite sides are
equals and parallel.
Opposite angles are
equal.
Diagonals bisect each
other.
Sum of any two adjacent
angles is 180. Each
diagonal divides the
parallelogram into two
triangles of equal area.
Rhombus All sides are equal,
opposite sides are
parallel.
Opposite angles are
equal.
Diagonals bisect each
other at right-angled. But
they are not equal.
Area =2
1 d1 d2
(side)2
=2
21 d2
1d
2
1
+
A
C D
E F
l
m
B
d1
d2
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Trapezium
Only one pair of
opposite sides is
parallel.
Area =2
1 sum of the
parallel side h
The median is equal to half
of the sum of the parallel
sides.Rectangle Opposite sides are
equal.
Each angle is 90o.
Diagonals are equal and
bisect each other (not at
right angles).
Area = l b
Perimeter = 2 ( l + b)
Diagonal = 22 b+l
Square
All sides are equal.
All angles are 90o
Diagonals are equal and
bisect at right angles
When it is inscribed in a
circle, the diagonal is equal
to the diameter of the circle
when circle is inscribed in a
square, the side of the
square = diameter of the
circle.
CirclesIf O is a fixed point in a given plane, the set of points in the plane which are at equal
distances from O is a circle.
Properties of the circle
1. Angles inscribed in the same arc of a circle are equal.AP1B = AP2B
2. If two chords of a circle are equal, their corresponding arcs have equal
measure.
3. A diameter perpendicular to a chord bisects the chord.
4. Equal chords of a circle are equidistant from the centre.
5. When two circles touch, their centres and their point of contact are collinear.
6. If the two circles touch externally, the distance between their centres is equal
to sum of their radii.
7. If the two circles touch internally, the distance between the centres is equal
to difference of their radii.
A B
P1 P2
h
l
b
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8. Angle at the centre made by an arc is equal to twice the angle
made by the arc at any point on the remaining part of the circumference.
Let O be the centre of the circle.
BOC = 2 P, when BAC = P
9. The angle inscribed in a semicircle is 90
o
.
10. Angles in the alternate segments are equal.
In the given figure, PAT is tangent to the circle and makes
angles PAC & BAT resp. with the chords AB & AC.
Then, BAT =ACB & ABC = PAC
11. If two chords AB and CD intersect externally at P, then PA PB = PC PD
12. If PAB is a secant and PT is a tangent, then PT2
= PA PB
13. If chords AB and CD intersect internally, then PA PB = PC PD
14. The length of the direct common tangent (PQ)
= 2212 )rr()centrestheirbetweencetandisThe(
The length of the transverse common tangent (RS)
= 2212 )rr()centrestheirbetweencetandisThe( +
C
A
B
D
P
AB
P
T
A
BC
DP
A
B C2P
P
O
QP
O O
r1 r2
P A T
C B
O
OR
S
r2r