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LINES AND ANGLES
If there are two different rays with common initial point, then
the rays form an angle. In other words, an angle is made by
two non-collinear rays with common initial point.
The angle is denoted by ‘ ∠ ’.
1 Minute (1’ ) = 60 seconds (60”)
Types Of Angle
Angles are categorized in various categories, depending upon
their characteristics.
1. Acute Angle
An angle whose measure is greater than 0° and less than 90°
is called acute angle.
C
C
B
A
A
Ray AC and ray AB are two rays with common initial point A.
The angle so formed is denoted by ∠ BAC or ∠ CAB or ∠ A.
Arms Of Angle
Rays forming the angle are called arms of angle, so AB and AC
are the arms of the angle.
B
2. Right Angle
An angle whose measure is 90° is called a right angle.
C
Vertex Of Angle
The common initial point is called the vertex of angle. So, A is
the vertex of angle.
Interior Of Angle
The area included in two arms of angle is called interior of
angle.
Exterior Of Angle
The area outside the angle is called exterior of the angle.
A
3. Obtuse Angle
An angle whose measure is more than 90° and less than 180°
is called obtuse angle.
C
Angular Region
The arms of angle and the interior of angle combined together
are called angular region.
Note
1. Arms of angle are not included in the interior of angle.
2. Arms of angle are not included in the exterior of angle.
Measure Of Angle
We use different units to measure different types of quantities,
like Kg to measure weight, meter to measure length. Similarly,
we use the unit ‘Degree’ to measure an angle and it is denoted
by ‘°’.
A
A
B
C
A
B
5. Reflex Angle
An angle whose measure is more than 180° and less than 360°
is called a reflex angle.
A
If ∠ BAC has measure of 70°, then we write it as ∠ BAC = 70°
or M( ∠ BAC) = 70°.
Note
1° = 60 Minutes (60’ )
B
4. Straight Angle
An angle whose measure is 180° is called a straight angle.
C
700
B
C
B
6. Vertically Opposite Angles
If two lines intersect each other, then they make four angles.
The angles opposite to each other are called vertically opposite
angles. Two intersecting lines form Four angles. That means,
two pairs of vertically opposite angles are made by two
intersecting lines.
9. Supplementary Angles
If the sum of two angles is 180°, then the angles are called
supplementary angles.
F
Vertically opposite angles have a common vertex and no arm
common, i.e. their arms form two pair of opposite rays.
C
1100
700
D
B
E
C
A
C
O
B
D
1100 700
D
AB and CD are two lines intersecting at point O. Four angles
∠ AOC, ∠ COB, ∠ BOD and ∠ DOA are made.
(i) ∠ AOC and ∠ BOD are vertically opposite to each other.
(ii) ∠ AOD and ∠ COB are vertically opposite to each other.
7. Adjacent Angles
Two angles are adjacent if:(i) They have a common vertex.
(ii) They have a common arm.
(iii) There non-common arms are on the either sides of
common arm. In other words, their areas are not
overlapping.
A
Sum of the linear pair of angles is 180°.
C
1100 700
A
B
∠ BAC and ∠ CAD are two adjacent angles. The non-common
arms AD and AB are opposite rays, .i.e. non-common arms
form a line.
A
The sum of ∠ BAC and ∠ CAD is 180°.
B
∠ BAC and ∠ CAD are two angles with a common vertex A
and common arm AC. Non-common arm AB of ∠ BAC and AD
of ∠ CAD are on the either sides of common arm AC. That
means, areas of ∠ BAC and ∠ CAD are not overlapping.
8. Complementary Angles
If the sum of two angles is 90°, then the angles are called
complementary angles.
(i)
B
10. Linear Pair Of Angles
If there are two adjacent angles such that their non-common
arms are opposite rays, then the angles form a linear pair. In
other words, if non-common arms of two adjacent angles
make a line, then the angles make a linear pair.
D
C
D
C
Note
Sum of both supplementary angles and linear pair of angles is
180°. The difference between them is that supplementary
angle may be adjacent or non-adjacent, i.e. they need not be
adjacent, but linear pair of angles are always adjacent.
Transversal
If there are two lines and a third line intersects them at two
distinct points, then the third line is called a transversal. So, a
line intersecting two or more lines at distinct pints is called
transversal.
r
F
0
60
300
A
B
A
D
E
m
1 2
4 3
n
5 6
8 7
(ii)
D
C
Lines m and n are intersected by line r at two distinct points.
So, r is a transversal made on lines m and n.
300
0
60
A
B
A transversal intersecting two lines made eight angles, four
angles on each line.
Note
1. It is not necessary that transversal is always parallel lines.
Transversal may be on parallel lines or on non-parallel
lines as well.
2. Transversal must intersect two lines at two distinct points.
If a line intersects two lines at same point, then it is not a
transversal.
2.
Each pair of alternative interior angles are equal.
Or
If alternative interior angles are equal, then the
lines are parallel.
3.
Each pair of consecutive interior angles are equal.
Or
If consecutive interior angles are equal, then the
lines are parallel.
The angles formed by the transversal on the lines are classified
as follows: -
Note
1. These properties apply only in case of parallel lines.
2. Vertically opposite angles will be equal in any case, i.e. if
the lines are parallel or it the lines are not parallel.
1. Vertically Opposite Angles
Transversal also makes vertically opposite angles on the two
lines.
(i) ∠ 1, ∠ 3
(ii) ∠ 2, ∠ 4
(iii) ∠ 5, ∠ 7
(iv) ∠ 6, ∠ 8
2. Corresponding Angles
If two angles are on the same side of transversal and both are
either above the lines or both are below the lines, then they
are called corresponding angles.
(i) ∠ 1, ∠ 5
(ii) ∠ 2, ∠ 6
(iii) ∠ 4, ∠ 8
(iv) ∠ 3, ∠ 7
3. Alternative Interior Angles
Two angles on the internal sides of lines, on the opposite side
of transversal and on different lines are called Alternative
Interior Angles.
(i) ∠ 4, ∠ 6
(ii) ∠ 3, ∠ 5
4.
Consecutive Interior Angles / Interior Angles On
Same Side Of Transversal / Co-Interior Angles On
Same Side Of Transversal
Two angles on the internal sides of lines, on the same side of
transversal and on different lines are called consecutive interior
angles. Consecutive Interior angles are also called.
(i) ∠ 4, ∠ 5
(ii) ∠ 3, ∠ 6
Transversal On Parallel Lines
When the transversal is on parallel lines, then there are special
relations between the angles of transversal.
r
m
n
1 2
4 3
5 6
8 7
Triangle
A plane figure formed by three closed lines is called triangle.
In other words, a triangle is area bounded by three lines.
B
C
Triangle ABC is made of three closed lines AB, AC and BC.
Triangle ABC can be represented as Δ ABC.
Sides Of Triangle
Lines making the triangle are called sides of triangle.
AB, AC and BC are sides of triangle.
Angles Of Triangle
The angles made by three sides of triangles are called angles
of triangle.
∠ ABC, ∠ ACB and ∠ BAC are angles of triangle. Also, ∠ ADC,
∠ ACB and ∠ BAC can simply be referred as ∠ B, ∠ C and ∠ A
respectively.
Vertices Of Triangle
Points of contacts of the sides of triangle are called vertices of
triangles.
A, B and C are vertices of triangle.
Exterior Angles Of Triangle
If we produce a side of triangle, then the produced side forms
an angle with its adjoining side outside the triangle. The
angles so formed in the exterior of triangle is called exterior
angle of triangle.
Angles which are on the opposite interior sides of exterior
angle are called opposite interior angles or remote interior
angles.
If transversal intersects two parallel lines, in such cases: 1.
Each pair of corresponding angles are equal.
Or
If corresponding angles are equal, then the lines
are parallel.
B
In Δ ABC,
C
D
∠ ACD is exterior angle when side BC is produced to D. ∠ A,
∠ B are opposite interior angles of ∠ ACD.
Types Of Triangle
Triangles are given various names depending upon their
characteristics. The triangles are classified on the basis of their
sides and angles.
B
1. On Basis Of Sides
(i) Scalene Triangle
A triangle whose all sides are different in length is called
scalene triangle.
B
B
C
1.
If a ray stands on a line, then the sum of the two
adjacent angles so formed is 180° or the angles
form a linear pair.
2.
If sum of two adjacent angles is 180°, then noncommon arms of angles are opposite rays.
Or
Two adjacent angles form a linear pair if and only if
they are supplementary.
3.
If two lines intersect, then the vertically opposite
angles are equal.
4.
If a transversal intersects two parallel lines, then
each pair of corresponding angles are equal.
Or
In a transversal intersects two lines making a pair
of equal corresponding angles, then the lines are
parallel.
5.
If a transversal intersects two parallel lines, then
each pair of alternative interior angles are equal.
Or
If a transversal intersects two lines making a pair
of equal alternative angles, then the lines are
parallel.
6.
If a transversal intersects two parallel lines, then
each pair of consecutive interior angles are
supplementary.
Or
If a transversal intersects two lines making a pair
of supplementary consecutive interior angles, then
the lines are parallel.
7.
The sum of the three angles of triangle is 180°.
8.
If a side of a triangle is produced, then exterior
angle so formed is equal to the sum of the two
opposite interior angles.
2. On Basis Of Angles
(i) Acute Angled Triangle
A triangle whose all angles are acute is called Acute Angled
Triangle.
B
C
(ii) Obtuse Angled Triangle
A triangle whose one angle is obtuse is called Obtuse Triangle.
C
Axioms And Properties
C
(iii) Equilateral Triangle
A triangle whose all sides are equal in length is called
Equilateral Triangle.
B
B
C
(ii) Isosceles Triangle
A triangle whose two sides are equal in length is called
isosceles triangle.
C
(iii) Right Angled Triangle
A triangle whose one angle is right angle or 90° is called Right
Angled Triangle.