Geometry 1 2

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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

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