06-Lines and Angle PDF
06-Lines and Angle PDF
06-Lines and Angle PDF
Chapter-6
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INTRODUCTION
In this chapter, you will study the properties of the
angles formed when two lines intersect each other,
and also the properties of the angles formed when
a line intersects two or more parallel lines at
distinct points.
BASIC TERMS AND DEFINATIONS
LINE-SEGMENT
RAY
LINE
COLLINEAR POINTS .
NON-COLLINEAR POINTS
ANGLE
REMARK
Every angle has a measure and unit of measurement
is degree.
One right angle = 90°
1° = 60' (minutes)
1° = 60'' (seconds)
Angle addition axiom
If X is a point in the interior of BAC, then
mBAC = mBAX + mXAC
(i) Right angle
(ii) Acute angel
(iii) Obtuse angle
(iv) Straight angle
(v) Reflex angle
Ex. In figure, ray OC stands on the line AB and
BOC = 125°. Find reflex AOC
(vi) Complementry angles
Ex. In figure, OB OA. AOC and BOC are
complementary angles. Find the value of x and
hence find AOC and BOC.
Ex. Find the measure of the complementary angle of
the following angles :
(i) 22° (ii) 63°
Ex. How many degrees are there in an angle which
equals two-third of its complement ?
Ex. If an angle differs from its complement by 10°,
find the angle.
(vii) Supplementary angles
Ex. In figure, ray AD stands on the line CB,
BAD = (2x + 10)° and CAD = (5x + 30)°, find
the value of x and also write BAD and CAD.
Ex. Find the measure of the supplementary angle of
the following angles :
(i) 45° (ii) 57°
Ex. Two supplementary angles are in ratio 4 : 5, find
the angles.
(viii) Angle Bisectors
POINTS OF ANGELS
(i) Adjacent angles : Two angles are called adjacent
angles, if
(A) They have the same vertex
(B) They have a common arm
(C) Non-common arms are on either side of the
common arm.
(ii) Linear pair of angles
(iii) Vertical opposite angles
Ex. In fig, lines PQ and RS intersect each other at
point O.
If POR : ROQ = 5 : 7, find all the angles.
Ex. In fig. ray OS stands on a line POQ. Ray OR and
ray OT are angle bisectors of POS and SOQ,
respectively. If POS = x, find ROT.
Ex. In Fig. lines AB and CD intersect at O. If AOC +
BOE = 70° and BOD = 40°, find BOE and reflex
COE.
Ex. In Fig. lines XY and MN intersect at O. If POY =
90° and a : b = 2 : 3, find c.
Ex. In Fig., PQR = PRQ, then prove that
PQS = PRT
PARALLEL LINES AND A TRANSVERSAL
Transversal : A line which intersects two or more
given parallel lines at distinct points is called a
transversal of the given lines.
(i) Corresponding angles : Two angles on the same
side of a transversal are known as the
corresponding angles if both lie either above the
two lines or below the two lines, in figure 1 & 5,
4 &8, 2 &6, 3 &7 are the pairs of
corresponding angles. If a transversal intersects
two parallel lines then the corresponding angles are
equal
i.e., 1 = 5, 4 = 8, 2 = 6 and 3 = 7.
(ii) Alternate interior angles : 3 & 5, 2 & 8,
are the pairs of alternate interior angles.
If a transversal intersects two parallel lines then
the each pair of alternate interior angles are
equal
i.e., 3 = 5 and 2 = 8.
(iii) Co-interior angles : The pair of interior angles on
the same side of the transversal are called pairs of
consecutive or co-interior angles. In figure
2 & 5, 3 & 8, are the pairs of co-interior
angles. If a transversal intersects two parallel lines
then each pair of consecutive interior angles are
supplementary
i.e., 2 + 5 = 180° and 3 + 8 = 180°.
LINES PARALLEL TO THE SAME LINE
Theorem 1
If a transversal intersects two lines such that a pair
of corresponding angles is equal, then the two lines
are parallel to each other.
Theorem 2
If a transversal intersects two lines such that a pair
of alternate interior angles is equal, then the two
lines are parallel.
Theorem 3
If a transversal intersects two parallel lines, then
each pair of interior angles on the same side of the
transversal is supplementary.
Theorem 4
If a transversal intersects two lines such that a pair
of interior angles on the same side of the
transversal is supplementary, then the two lines are
parallel.
Theorem 5
Lines which are parallel to the same line are
parallel to each other. Let us draw a line t
transversal for the lines, l, m and n. It is given that
line m || line l and line || n line l. Therefore, 1 = 2
and 1 = 3 (Corresponding angles axiom)
So, 2 = 3 (Why ?)
But 2 and 3 are corresponding angles and they
are equal.
Therefore, you can say that
Line m || Line n
(Converse of corresponding angles axiom)
Ex. In figure ||m, n||p and 1 = 85º find 2.
Ex. As shown in the figure AB || CD. If point P is
between these two lines such that mABP = 50°
and mCDP = 70° then find BPD
Ex. In fig. AB || CD and CD || EF. Also EA AB. If
BEF = 55º, find the values of x, y and z.
Ex. In Fig. find the values of x and y and then show
that AB || CD.
Ex. In Fig. if PQ || ST, PQR = 110° and RST = 130°,
find QRS.
Ex. In Fig. if AB || CD, APQ = 50° and PRD = 127°,
find x and y.
Ex. In figure, if m||n and 1 and 2 are in the ratio
3 : 2, determine all the angles from 1 to 8.
WORKSHEET - 1
Ex. 1
In figure, OP and OQ bisects BOC and AOC
respectively. Prove that POQ = 90°.
Ex. 2
In figure, lines AB, CD and EF intersect at O. Find
the measures of AOC, DOE and BOF
Ex. 3
In the given figure AB || CD. Find FXE.
Ex. 4
If a transversal intersects two lines such that the
bisectors of a pair of corresponding angles are
parallel, then prove that the two lines are parallel.
Ex. 5
In fig, if PQ || RS, MXQ = 135° and MYR = 40°,
find XMY.
Ex. 6
In figure two straight lines PQ and RS intersect
each other at O. If POT = 75°, find the values of
a, b and c.
TRIANGLE
A plane figure bounded by three lines in a plane is
called a triangle. Every triangle have three sides
and three angles. If ABC is any triangle then AB,
BC & CA are three sides and A, B and C are
three angles.
Types of Triangles
(A) On the basis of sides we have three types of
triangle.
1. Scalene triangle : A triangle in which no two sides
are equal is called a scalene triangle
2. Isosceles triangle : A triangle having two sides
equal is called an isosceles triangle.
3. Equilateral triangle : A triangle in which all sides
are equal is called an equilateral triangle.
(B) On the basis of angles we have three types :
1. Right angle : A triangle in which any one angle is
right angle is called right angle.
2. Acute triangle :A triangle in which all angles are
acute is called an acute triangle.
3. Obtuse triangle : A triangle in which any one angle
is obtuse is called an obtuse triangle.
SOME IMPORTANT THEOREMS
Theorem 1
The sum of interior angles of a triangle is 180°.
To prove :
A + B + C = 180° or 1 + 2 + 3 = 180°.
Construction
Through A, draw a line parallel to BC.
Ex. In a ABC, B = 105°, C = 50°. Find A.
Ex. If the angles of a triangle are in the ratio 2 : 3 : 4,
determine all the angles of triangle.
Theorem 2
If the bisectors of angle ABC and ACB of a
triangle ABC meet at a point O, then
BOC = 90° + A
Given : AABC such that the bisectors of ABC
and ACB meet at a point O.
BOC = 90° + A
Theorem 3 (Exterior Angle of a Triangle)
If the side of the triangle is produced, the exterior
angle so formed is equal to the sum of two interior
opposite angles.
Given : A triangle ABC. D is a point on BC produced
forming exterior angle 4
To Prove : 4 = 1 + 2
i.e., ACD = CAB + CBA
Ex. An exterior angle of a triangle is 115° and one of
the opposite angles is 35°. Find the other two
angles.
Ex. In fig. if QT PR, TQR = 40° and SPR = 30°,
find x and y.
Corollary
An exterior angle of a triangle is greater than
either of the interior opposite angles.
Theorem 4
The sides AB and AC of a ABC are produced to P
and Q respectively. If the bisectors of PBC and