54 011-26925013/14 +91-9811134008 +91-9582231489 NTSE, NSO Diploma, XI Entrance Lines and Angles CHAPTER 4 We are Starting from a Point but want to Make it a Circle of Infinite Radius Lines and Angles BASIC GEOMETRICAL CONCEPTS (AXIOMS, THEOREMS AND COROLLARIES) Axioms The basic facts which are taken for granted, without proof, are called axioms. Examples (i) Halves of equal are equal. (ii) The whole is greater than each of its parts. (iii) A line contains infinitely many points. STATEMENTS A sentence which can be judged to be true or false is called a statement. Examples (i) The sum of the angles of a triangle is 180 0 , is a true statement. (ii) The sum of the angles of a quadrilateral is 1800, is a false statement. (iii) x + 10 > 15 is a sentence but not a statement. Theorems A statement that requires a proof, is called a theorem. Establishing the truth of a theorem is known as proving the theorem. Examples (i) The sum of all the angles around a point is 360 0 . (ii) The sum of the angles of a triangle is 180 0 . CAROLLARY : A statement, whose truth can easily be deduced from a theorem, is called its corollary. EUCLID’S FIVE POSTULATES 1. A straight line may be drawn from any point to any other point. 2. A terminated line can be produced indefinitely. 3. A circle can be drawn with any center and any radius. 4. All right angle are equal to one another. 5. If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the angles taken together are less than two right angles. Later on the fifth postulate was modified as under. ‘For every line L and for every point P not lying on L, there exists a unique line M, passing through P and parallel to L’. Clearly, two distinct intersecting lines cannot be parallel to the same line. SOME TERMS RELATED TO GEOMETRY POINT A point is an exact location. A fine dot represents a point. We denote a point by a capital letter – A, B, P, Q, etc. In the given figure, P is a point. Line segment The straight path between two points A and B is called the line segment AB . The points A and B are called the end points of the line segment AB . A line segment has a definite length.
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54
011-26925013/14+91-9811134008+91-9582231489
NTSE, NSO Diploma, XI Entrance
Lines and Angles
CHAPTER
4We are Starting from a Point but want to Make it a Circle of Infinite Radius
Lines and Angles
BASIC GEOMETRICAL CONCEPTS (AXIOMS, THEOREMS AND COROLLARIES)
Axioms The basic facts which are taken for granted, without proof, are called axioms.
Examples (i) Halves of equal are equal.
(ii) The whole is greater than each of its parts.
(iii) A line contains infinitely many points.
STATEMENTS A sentence which can be judged to be true or false is called a statement.
Examples (i) The sum of the angles of a triangle is 1800, is a true statement.
(ii) The sum of the angles of a quadrilateral is 1800, is a false statement.
(iii) x + 10 > 15 is a sentence but not a statement.
Theorems A statement that requires a proof, is called a theorem. Establishing the truth of a
theorem is known as proving the theorem.
Examples (i) The sum of all the angles around a point is 3600.
(ii) The sum of the angles of a triangle is 1800.
CAROLLARY : A statement, whose truth can easily be deduced from a theorem, is called its corollary.
EUCLID’S FIVE POSTULATES
1. A straight line may be drawn from any point to any other point.
2. A terminated line can be produced indefinitely.
3. A circle can be drawn with any center and any radius.
4. All right angle are equal to one another.
5. If a straight line falling on two straight lines makes the interior angles on the same side of it taken
together less than two right angles, then the two straight lines, if produced indefinitely, meet on that
side on which the angles taken together are less than two right angles.
Later on the fifth postulate was modified as under.
‘For every line L and for every point P not lying on L, there exists a unique line M, passing through
P and parallel to L’.
Clearly, two distinct intersecting lines cannot be parallel to the same line.
SOME TERMS RELATED TO GEOMETRY
POINT A point is an exact location.
A fine dot represents a point.
We denote a point by a capital letter – A, B, P, Q, etc.
In the given figure, P is a point.
Line segment
The straight path between two points A and B is called the line segment AB .
The points A and B are called the end points of the line segment AB .
A line segment has a definite length.
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Lines and Angles55
The distance between two points A and B is equal to the length of the line segment AB .
RAY
A line segment AB when extended indefinitely in one direction is the ray
AB .
Ray
AB has one end point A.
A ray has no definite length.
A ray cannot be drawn, it can simply be represented on the plane of a paper.
To draw a ray would mean to represent it.
LINE A line segment AB when extended indefinitely in both the directions is called the line
AB .
A line has no end points. A line has no definite length. A line cannot be drawn, it can simply be represented
on the plane of a paper.
To draw a line would mean to represent it.
Sometimes, we lable lines by small letters l, m, n, etc.
INCIDENCE AXIOMS ON LINES
(i) A line contains infinitely many points.
(ii) Through a given point, infinitely many liens can be drawn.
(iii) One and only one line can be drawn to pass through two
given points A and B.
COLLINEAR POINTS
Three or more than three points are said to be collinear, if there is
a line which contains them all.
In the given figure A,B,C are collinear points, while P,Q,R are non-collinear.
INTERSECTING LINES
Two lines having a common point are called intersecting lines.
In the given figure, the lines AB and C intersect at a point O.
CONCURRENT LINES Three or more lines intersecting at the same point are said to be
concurrent.
In the given figure, lines l, m, n pass through the same point P and
therefore, they are concurrent.
PLANE
A plane is a surface such that every point of the line joining any
two points on it, lies on it.
Examples The surface of a smooth wall; the surface of the top of the table; the surface of a smooth
blackboard; the surface of a sheet of paper etc., are close examples of a plane. These
surfaces are limited in extent but the geometrical plane extends endlessly in all
directions.
Parallel Lines Two lines l and m in a plane are said to be parallel, if they have no point in common and we write, l || m.
The distance between two parallel lines always remains the same.
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NTSE, NSO Diploma, XI Entrance
Lines and Angles56
Questions
1. (i) How many lines can be drawn to pass through a given point?
(ii) How many lines can be drawn to pass through two given points?
(iii) In how many points can the two lines at the most intersect?
(iv) If A, B, C are three collinear points, name all the line segments determined by them.
2. Which of the following statements are true?
(i) A line segment has no definite length.
(ii) A ray has no end point.
(iii) A line has a definite length.
(iv) A line
AB is the same as line
BA .
(v) A ray
AB is the same as ray
BA .
(vi) Two distinct points always determine a unique line.
(vii) Three lines are concurrent if they have a common point.
(viii) Two distinct lines cannot have more than one point in common.
(ix) Two intersecting liens cannot be both parallel to the same line.
(x) Open half-line OA is the same thing as ray
OA .
(xi) Two lines may intersect in two points.
(xii) Two lines l and m are parallel only when they have no point in common.
ANGLES AND THEIR PROPERTIES
ANGLE
Two rays OA and OB having a common end point O form angle
AOB, written as AOB .
OA and OB are called the arms of the angle and O is called its
vertex.
INTERIOR OF AN ANGLE
The interior of AOB is the set of all points in its plane, which lie
on the same side of OA as B and also on the same side of OB as
A, e.g., P is a point in the interior of AOB . Any point on any
arm or vertex is said to lie on the angle, e.g., Q is a point on
AOB .
EXTERIOR OF AN ANGLE
The exterior of an angle AOB is the set of all those points in its
plane, which do not lie on the angle or in its interior. In the given
figure, R is a point in the exterior of AOB .
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Lines and Angles57
MEASURE OF AN ANGLE
The amount of turning from OA to OB is called the measure of
AOB , written as m AOB . An angle is measured in degrees
denoted by.
AN ANGLE OF 3600
If a ray OA starting from its original OA, rotates about O, in the
anticlockwise direction and after making a complete revolution it
comes back to its original position, we say that it has rotated
through 360 degrees, written as 3600.
This complete rotation is divided into 360 equal parts. Each part measures 10.
10 = 60 minutes, written as 60’.
1’ = 60 seconds, written as 60’’.
We use a protractor to measure an angle.
KINDS OF ANGLE
(i) RIGHT ANGLE An angle whose measure is 900 is called a right angle.
(ii) ACUTE ANGLE An angle whose measure is more than 00 but less than 90
0 is called an acute angle.
(iii) OBTUSE ANGLE An angle whose measure is more than 900 but less than 180
0 is called an obtuse
angle.
(iv) STRAIGHT ANGLE An angle whose measure is 1800 is called a straight angle.
(v) REFLEX ANGLE An angle whose measure is more than 1800 but less than 360
0 is called a reflex
angle.
(vi) COMPLETE ANGLE An angle whose measure is 3600 is called a complete angle.
EQUAL ANGLES
Two angles are said to be equal, if they have the same measure.
Bisector of an angle A ray OC is called the bisector of AOB , if
m AOC = m BOC .
In this case, AOC = BOC = 2
1AOB
COMPLEMENTARY ANGLES
Two angles are said to be complementary, if the sum of their measures is 900.
Two complementary angles are called the complement of each other.
Example Angles measuring 550 and 35
0 are complementary angles.
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NTSE, NSO Diploma, XI Entrance
Lines and Angles58
SUPPLEMENTARY ANGLES
Two angles are said to be supplementary, if the sum of their measures is 1800.
Example Angles measuring 620 and 118
0 are supplementary angles.
Example 1 Find the measure of an angle which is 240 more than its complement.
Solution Let the measure of the required angle be x0.
Then measure of its complement = (90 – x)0.
x – (90 – x) = 24 2x = 114 x = 57.
Hence, the measure of the required angle is 570.
Example 2 Find the measure of an angle which is 320 less than its supplement.
Solution Let the measure of the required angle be x0.
Then measure of its complement = (180 – x)0.
x – (180 – x) = 32 2x = 148 x = 74.
Hence, the measure of the required angle is 740.
Example 3 Find the measure of an angle, if six times its complement is 120 less than twice its
supplement.
Solution Let the measure of the required angle be x0.