41
No part of this book may be reproduced or transmitted in any form or by any means, C.D. ROM/Audio Video Cassettes or electronic, mechanical including photocopying; recording or by any information storage and retrieval system without permission in writing from the Publisher.
STD. IX
Geometry
Printed at: Dainik Saamana., Navi Mumbai
Written as per the revised syllabus prescribed by the Maharashtra State Board of Secondary and Higher Secondary Education, Pune.
Fifth Edition: March 2016
10046_10511_JUP
P.O. No. 15197
Salient Features
• Written as per the new textbook. • Exhaustive coverage of entire syllabus. • Topic-wise distribution of all textual questions and practice problems at the beginning of every chapter • Covers answers to all textual exercises and problem set. • Includes additional problems for practice. • Multiple choice questions for effective preparation. • Comprehensive solution to Question Bank. • Constructions drawn with accurate measurements.
Preface Geometry is the mathematics of properties, measurement and relationships of points, lines, angles, surfaces and solids. It is widely used in the fields of science, engineering, computers, architecture etc. It is a vast subject dealing with the study of properties, definitions, theorems, areas, perimeter, angles, triangles, mensuration, co-ordinates, constructions etc. The study of Geometry requires a deep and intrinsic understanding of concepts. Hence, to ease the task we bring to you “Std. IX: Geometry” a complete and thorough guide critically analysed and extensively drafted to boost the students confidence. The question answer format of this book helps the student to understand and grasp each and every concept thoroughly. The book is based on the new text book and covers the entire syllabus. At the beginning of every chapter, topic-wise distribution of all textual questions and practice problems has been provided for simpler understanding of different types of questions. It contains answers to textual exercises, problems sets and Question bank. It also includes additional questions and multiple choice questions for practice. All the diagrams are neat and have proper labelling. The book has a unique feature that all the constructions are as per the scale. Another feature of the book is its layout which is attractive and inspires the students to read. There is always room for improvement and hence, we welcome all suggestions and regret any error that may have occurred in the making of this book.
A book affects eternity; one can never tell where its influence stops.
Best of luck to all the aspirants!
Yours faithfully,
Publisher
PER
No. Topic Name Page No.1 Lines and Angles 1 2 Triangles 39 3 Congruence of Triangles 72 4 Circle 109 5 Quadrilaterals 143 6 Co‐ordinate Geometry 185 7 Geometric Constructions 198 8 Trigonometry 215 9 Mensuration 232
10 Question Bank 256
1
Chapter 01: Lines and Angles
Type of Problems Exercise Q. Nos.
Axioms, Collinear Points,
Non-collinear Points and Parallel
Planes
1.1 Q.1, 2, 3, 4
Practice Problems
(Based on Exercise 1.1) Q.1, 2, 3
Co-ordinates of Points and Distance
1.2 Q.1, 2
Practice Problems
(Based on Exercise 1.2) Q.1
Problem set-1 Q.6
Betweeness, Segment and Ray
1.2 Q. 3, 4, 5, 6, 7, 8, 9, 10, 11
Practice Problems
(Based on Exercise 1.2) Q. 2, 3, 4
Problem set-1 Q.1, 2, 3, 4, 5
Types of Angles, Pairs of Angles
1.3 Q.1, 4, 5
Practice Problems
(Based on Exercise 1.3) Q.3, 4, 5, 6, 7
Practice Problems
(Based on Exercise 1.4) Q.5
Problem set-1 Q.7, 8, 9, 11, 12, 13, 14, 16
For the given measures of angles find
their Complementary/Supplementary
Angles
1.3 Q.2, 3
Practice Problems
(Based on Exercise 1.3) Q.1, 2
Parallel Lines, Corresponding Angles,
Alternate Angles, Vertically Opposite
Angles and Interior Angles
1.4 Q.1, 2, 3, 4
Practice Problems
(Based on Exercise 1.4) Q.1, 2, 3, 4, 6, 7
Problem set-1 Q.10, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25
Lines and Angles01
2
Std. IX : Geometry Introduction Basic Geometry: A point, a line and a plane are undefined terms which are basic concepts in geometry. i. Lines and planes are set of points. ii. Each line and each plane contain infinite number of points. Axioms / Postulates: The simple properties which we accept as true are called Axioms or Postulates. OR The terms or statements whose proofs are not to be asked are called Axioms. Theorem: Important and useful results derived from the axioms are called Theorems. OR The statements that we prove from the axioms are called Theorems. Euclid’s five postulates: 1. A straight line can 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 centre and any radius. 4. All right angles 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.
OR Two distinct intersecting lines cannot be parallel to the same line. 1.1 Some Axioms 1. Infinite number of lines can be drawn through a given point. 2. There is one and only one line passing through two distinct points.
3. When two distinct lines intersect, their intersection is exactly one point. 4. There is exactly one plane passing through three non-collinear points. 5. There is exactly one plane passing through a line and a point, not on the line.
EA
B
C
P
m
n
A B
P
m
A
E
3
Chapter 01: Lines and Angles
6. There is exactly one plane passing through two distinct intersecting lines. 7. When two planes intersect, their intersection is exactly one line. 8. When a line intersects a plane but does not lie in it, then their intersection is a point. 9. A line containing two given points of a plane lies wholly in that plane. 10. If the lines lie in the same plane, then they are called ‘Coplanar lines’, otherwise they are called
‘non-coplanar lines’. 1.2 Collinear Points, Non-collinear points, Parallel planes 1. Collinear Points: If there exists a line containing all the given points, then
those points are called collinear points. Points A, B, C are collinear points. 2. Non-collinear Points: If there does not exist a line containing all the given
points, then those points are called non-collinear points. Points A, B, C are non collinear points. 3. Coplanar Lines: Lines which lie in the same plane are called coplanar lines.
line and line m are coplanar lines.
4. Non-coplanar Lines: Lines which do not lie in the same plane are called
non-coplanar lines.
line and line m are non-coplanar lines.
5. Concurrent Lines: If three or more lines pass through one point, then the
lines are called concurrent lines.
Line , m, n and o are concurrent lines and P is their point of concurrence.
Note: The common point of intersection is called the point of concurrence.
m
m
A
B
C
m n
o
P
P
E
P Q
E
m
P
E
E
F
A B C
4
Std. IX : Geometry
6. Parallel Lines: The lines in a plane which are not intersecting are called parallel lines. line || line m 7. Parallel Planes: Two non-intersecting planes are said to be parallel planes. Plane ABCD and plane PQRS are parallel planes. Exercise 1.1 1. Take any three non-collinear points A, B, C on a paper. How many lines in all can you draw through
different pairs of the points? Name the lines. Solution: We can draw three lines through three non-collinear points A, B and C. The lines are line AB, line BC and line AC. 2. Take four points P, Q, R, S in a plane. Draw lines by joining different pairs of points. How many lines
can you draw in the following cases? i. No three of them are collinear. ii. Three of these points are collinear. Solution: i. We can draw six lines (line PQ, line QR, line SR, line PS, line QS and line PR).
ii. We can draw four lines (line PR, line PS, line QS, line SR). 3. Observe the given figure and write the sets of all the points which are collinear. Solution: Set of collinear points are i. P, S, T, Q ii. P, F, R, B iii. A, F, S, D iv. A, R, E, Q v. B, E, T, D
m
P Q
S R
A B
D C
P Q
RS
A
P
B
Q
D
F
R E
TS
A B
C
P Q R
S
5
Chapter 01: Lines and Angles
4. Observe the given figure and answer the following: i. Name the lines parallel to the line AB. ii. Can you say that line AD and the point R lie in the same plane? Why? iii. Are the points A, S, B, R coplanar? Why? iv. Name the three planes passing through point A. v. Name the points such that the plane containing them does not
contain points P, Q, C and D. Solution:
i. line DC, line PQ and line SR are parallel to line AB.
ii. Yes, line AD and point R lie in the same plane.
[There is exactly one plane passing through a line and a point not on it (axiom)]
iii. Yes, points A, S, B, R are coplanar (since these points are contained in the plane ASRB)
iv. Planes passing through point A are plane APSD, plane ABQP and plane ABCD.
v. Points A, S, R, B, V (reason: Plane containing points P, Q, C and D is plane PQCD and it does not contain points A, S, R, B, V).
1.3 Co-ordinates of Points and Distance Co-ordinates of a Point: The number associated with a point is called the co-ordinate of that point.
In adjoining figure, co-ordinate of point P is 1 and that of point B is 2. Distance between Two Points: If x and y are the co-ordinates of point P and Q respectively, then the distance between P and Q is defined as the absolute value of the difference between the number associated with those points. d(P, Q) = x y if x > y
d(P, Q) = y x if x < y
d(P, Q) = | x y |
Thus, distance between any two distinct points is a unique non-negative real number. 1.4 Betweenness If points P, Q and R are three distinct collinear points and if d(P, Q) + d(Q, R) = d(P, R), then the point Q is said to be between points P and R, written as P-Q-R or R-Q-P. 1.5 Segment and Ray Line Segment:
The set consisting of points A and B and all the points between A and B is called segment AB, written as seg AB.
A B
C
P Q
S
D
V
R
A B
P Q R
R Q P O A B C
3 2 1 0 1 2 3
P Q
x y
(x y)
P Q
y x (x > y)
6
Std. IX : Geometry
Note:
i. seg AB and seg BA denote the same line segment.
ii. The points A and B are called the end points of seg AB.
iii. A line segment is a subset of a line. Length of a Line Segment:
The distance between the end points of a line segment is called as the length of the segment.
It is denoted by (AB).
Note: AB = (AB) = d(A, B)
Congruent Segments:
Two line segments are said to be congruent, if they are of the same length.
If (AB) = (CD), then we write seg AB seg CD. Note: i. If we have to consider the length of segment AB, we write only AB or l(AB). ii. If we have to consider the set of points between A and B, we write seg AB or side AB. Properties of Congruent Segments:
i. Reflexivity: seg AB seg AB (Every segment is congruent to itself).
ii. Symmetry: If seg AB seg CD, then seg CD seg AB.
iii. Transitivity: If seg AB seg CD and seg CD seg PQ, then seg AB seg PQ. Midpoint of a Segment:
The point M is said to be the midpoint of seg AB, if A-M-B and d(A, M) = d(M, B).
AM = BM = 1
2AB
Note: Every line segment has one and only one midpoint. Comparison of Segments:
Suppose seg AB and seg CD are given. If AB < CD, then we say that seg AB is smaller than seg CD.
This is denoted by seg AB < seg CD. Ray:
Suppose A and B are two points, then set of all the points of seg AB together with all the points P on the line AB for which B is between A and P is called ray AB. Note: i. Point A is called as the origin of ray AB. ii. The ray is a subset of a line. Opposite Rays:
Two rays which lie on a line having same origin and opposite direction are called opposite rays.
ray OA and ray OB are opposite rays.
BA
DC
A M B
A O B
A B P
A B
C D
7
Chapter 01: Lines and Angles
Exercise 1.2 1. Observe the number line in the figure and answer the following questions: i. Write the co-ordinates of the points C, S, Q, D. ii. Name the points whose co-ordinates are 4, 5, 0, 2. iii. Find d(Q, T), d(E, B), d(O, C), d(O, R). iv. Name the points which are at a distance of 4 from the point O. Solution: i. Co-ordinates of the points C, S, Q, D are 3, 4, 2 and 4 respectively. ii. The points whose co-ordinates are 4, 5, 0, 2 are S, T, O and B respectively. iii. a. d(Q, T) Co-ordinate of point Q is 2 and co-ordinate of point T is 5. 2 < 5 d(Q, T) = 5 2 d(Q, T) = 3 b. d(E, B) Co-ordinate of point E is 5 and co-ordinate of point B is 2. 2 > 5 d(E, B) = 2 (5) = 2 + 5 = 3 d(E, B) = 3 c. d(O, C) Co-ordinate of point O is 0 and co-ordinate of point C is 3. 0 > 3 d(O, C) = 0 (3) = 0 + 3 d(O, C) = 3 d. d(O, R) Co-ordinate of point O is 0 and co-ordinate of point R is 3. 3 > 0 d(O, R) = 3 0 d(O, R) = 3 iv. There are two possibilities: a. The point can be towards the positive side i.e. point S -----[ d(O, S) = 4 0 = 4]
b. The point can be towards the negative side i.e. point D -----[ d(O, D) = 0 (4) = 0 + 4 = 4] 2. The co-ordinates of two points P and Q are x and y respectively. Find d(P, Q) in the following cases: i. x = 7, y = 10 ii. x = 2, y = 11 iii. x = 8, y = 3 iv. x = 5, y = 9 Solution: i. Co-ordinate of point P is x = 7. Co-ordinate of point Q is y = 10. 10 > 7 d(P, Q) = 10 7 = 3 d(P, Q) = 3 ii. Co-ordinate of point P is x = 2. Co-ordinate of point Q is y = 11. 11 > 2 d(P, Q) = 11 (2) = 11 + 2 = 13 d(P, Q) = 13
A OBC DE S TRQP
1 0 23 45 4 5321
8
Std. IX : Geometry
iii. Co-ordinate of point P is x = 8. Co-ordinate of point Q is y = 3. 3 > 8 d(P, Q) = 3 (8) = 3 + 8 = 5 d(P, Q) = 5 iv. Co-ordinate of point P is x = 5. Co-ordinate of point Q is y = 9. 5 > 9 d(P, Q) = 5 (9) = 5 + 9 = 14 d(P, Q) = 14 3. In each of the following, decide whether the relation of betweenness exists among the
points A, B and D. Name the point which lies between the other two. i. d(A, B) = 5, d(B, D) = 8, d(A, D) = 11 ii. d(A, B) = 11, d(B, D) = 6, d(A, D) = 5 iii. d(A, B) = 2, d(B, D) = 15, d(A, D) = 17 Solution: i. d(A, B) + d(B, D) = 5 + 8 = 13 d(A, D) = 11 d(A, B) + d(B, D) d(A, D) The relation of betweenness does not exist among the points A, B and D. ii. d(B, D) + d(A, D) = 6 + 5 = 11 d(A, B) = 11 d(B, D) + d(A, D) = d(A, B) The relation of betweenness exists among the points A, B and D, such that point D lies between A and
B. iii. d(A, B) + d(B, D) = 2 + 15 = 17 d(A, D) = 17 d(A, B) + d(B, D) = d(A, D) The relation of betweenness exists among the points A, B and D, such that point B lies between A and
D. 4. Draw the figures according to the given information and answer the questions:
i. When A-B-C, (AC) = 12, (BC) = 7.5, then (AB) = ?
ii. When R-S-T, (ST) = 3.75, (RS) = 2.15, then (RT) = ?
iii. When X-Y-Z, (XZ) = 5 2 , (XY) = 2 2 , then (YZ) = ?
Solution: i.
(AB) + (BC) = (AC) ----[A-B-C]
(AB) + 7.5 = 12
(AB) = 12 7.5
(AB) = 4.5
B A C 7.5
12
9
Chapter 01: Lines and Angles
ii.
(RS) + (ST) = (RT) ----[R-S-T]
2.15 + 3.75 = (RT)
(RT) = 5.90 iii. (XY) + (YZ) = (XZ) ----[X-Y-Z]
2 2 + (YZ) = 5 2
(YZ) = 5 2 2 2
(YZ) = 3 2 5. In the adjoining figure, (LN) = 5, (MN) = 7, (ML) = 6, (NP) = 11, (MR) = 13, (MQ) = 2, then
find (PL), (NR), (LQ).
Solution:
(PL) +(LN) = (PN) ----[P-L-N]
(PL) + 5 = 11
(PL) = 11 5
(PL) = 6
(MN) + (NR) = (MR) ----[M-N-R]
7 + (NR) = 13
(NR) = 13 7
(NR) = 6
(LM) +(MQ) = (LQ) ----[L-M-Q]
6 + 2 = (LQ)
(LQ) = 8 6. In the adjoining figure, (AC) = 8, (BC) = 5.
Seg BD seg CE seg AC, then determine whether the segments in each of the following pairs are congruent or not.
i. seg BC and seg DE ii. seg AB and seg CD. Solution: i. seg BD seg CE seg AC ----[Given]
(BD) = (CE) = (AC) = 8 ----(i) [ (AC) = 8]
(BC) + (CD) = (BD) ----[B-C-D]
5 + (CD) = 8 ----[From (i) and Given]
S R T 3.75 2.15
X 2 2
5 2
Y Z
A B C D E
Q
P R
S
M N
P
L
RQ
13
11
5 6
2 7
10
Std. IX : Geometry
(CD) = 8 5
(CD) = 3 ----(ii)
(CD) + (DE) = (CE) ----[C-D-E]
3 + (DE) = 8 ----[From (i) and (ii)]
(DE) = 8 3
(DE) = 5 ----(iii)
But (BC) = 5 ----(iv) [Given]
(BC) = (DE) ----[From (iii) and (iv)]
seg BC seg DE ii. (AB) + (BC) = (AC) ----[A-B-C]
(AB) + 5 = 8 ----[From (iv) and Given]
(AB) = 8 5
(AB) = 3 ----(v)
(AB) = (CD) ----[From (ii) and (v)]
seg AB seg CD 7. The co-ordinates of the points on the number line are as follows:
Points P Q R S T Co-ordinates 3 5 2 7 9
Find the lengths of: seg PQ, seg PR, seg PS, seg PT, seg QR, seg QS, seg QT, seg RS, seg RT, seg ST. Solution: i. Co-ordinate of point P is 3 and co-ordinate of point Q is 5. 5 > 3 d(P, Q) = 5 (3) = 5 + 3 = 8
(PQ) = 8 ii. Co-ordinate of point P is 3 and co-ordinate of point R is 2. 2 > 3 d(P, R) = 2 (3) = 2 + 3 = 5
(PR) = 5 iii. Co-ordinate of point P is 3 and co-ordinate of point S is 7. 3 > 7 d(P, S) = 3 (7) = 3 + 7 = 4
(PS) = 4 iv. Co-ordinate of point P is 3 and co-ordinate of point T is 9. 9 > 3 d(P, T) = 9 (3) = 9 + 3 = 12
(PT) = 12
11
Chapter 01: Lines and Angles
v. Co-ordinate of point Q is 5 and co-ordinate of point R is 2. 5 > 2 d(Q, R) = 5 2 = 3
(QR) = 3 vi. Co-ordinate of point Q is 5 and co-ordinate of point S is 7. 5 > 7 d(Q, S) = 5 (7) = 5 + 7 = 12
(QS) = 12 vii. Co-ordinate of point Q is 5 and co-ordinate of point T is 9. 9 > 5 d(Q, T) = 9 5 = 4
(QT) = 4 viii. Co-ordinate of point R is 2 and co-ordinate of point S is 7. 2 > 7 d(R, S) = 2 (7) = 2 + 7 = 9
(RS) = 9 ix. Co-ordinate of point R is 2 and co-ordinate of point T is 9. 9 > 2 d(R, T) = 9 2 = 7
(RT) = 7 x. Co-ordinate of point S is 7 and co-ordinate of point T is 9. 9 > 7 d(S, T) = 9 (7) = 9 + 7 = 16
(ST) = 16 8. If P is the midpoint of seg AB and AB = 7 cm, find AP. Solution: P is the midpoint of seg AB. d(A, P) = d(P, B) ----(i) d(A, P) + d(P, B) = d(A, B) ----[A-P-B]
d(A, P) + d(P, B) = 7 ----[ AB = 7 cm]
d(A, P) + d(A, P) = 7 ----[From (i)] 2[d(A, P)] = 7
d(A, P) = 7
2 = 3.5
(AP) = 3.5 cm 9. If Q is the midpoint of seg CD and d(C, Q) = 4.5, find the length of CD. Solution: Q is the midpoint of seg CD. d(C, Q) = d(Q, D)
d(C, Q) = d(Q, D) = 4.5 ----(i) [ d(C, Q) = 4.5]
A B P
C DQ
12
Std. IX : Geometry
d(C, Q) + d(Q, D) = d(C, D) ----[ C-Q-D] 4.5 + 4.5 = d(C, D) ----[From (i)] d(C, D) = 9
(CD) = 9 10. If AB = 7 cm, BP = 4 cm, AP = 5.4 cm, then compare the segments. Solution: AB = 7 cm, BP = 4 cm, AP = 5.4 cm
4 < 5.4 < 7
BP < AP < AB 11. In the adjoining figure, lengths of the segments are shown. Write the congruent segments. Solution:
(AB) = (AC) = 5 cm
seg AB seg AC
(BC) = (DE) = 5.5 cm
seg BC seg DE
(CD) = (CE) = 4 cm
seg CD seg CE 1.6 Plane Separation Axiom Given a line in a plane, the points in the plane that do not lie on the line form two disjoint sets H1 and H2. Each set H1 and H2 is called a half plane and the line is called the edge of the half plane. If P is any point in any of the half plane, then that half plane is called as the P side of the half plane. 1.7 Angle An angle is the union of two rays having the same origin. The origin is called the vertex of the angle and each of the rays is known as arm or side of the angle.
AOB and BOA are considered as same angles. 1.8 Angles in terms of Rotation An angle is obtained by rotating a ray about its end point. The rotation of a ray may be in clockwise direction or in anticlockwise direction. When the rotation of the ray is in clockwise direction, the angle is regarded as negative. [See adjoining figure.] When the rotation of the ray is in anticlockwise direction, the angle is regarded as positive. [See adjoining figure.]
A
OB
Terminal side
Initial side
A
B C
D
E
5.5 cm
4 cm
P
H1
H2
l
13
Chapter 01: Lines and Angles
Measure of an angle: The amount of rotation of the ray from its initial position to the terminal position is called the measure of the angle.
mAOB = 1.9 Directed Angle The ordered pair of rays (ray OA, ray OB) together with rotation of ray OA to occupy the position of ray OB is called directed angle AOB, denoted by
AOB. In the ordered pair (ray OA, ray OB), the first element ray OA is called initial arm and second element ray OB is called terminal arm. Note:The directed angle BOA is not the same as directed angle AOB. i.e. BOA AOB Positive angle: Anticlockwise direction of ray OA about ‘O’ is regarded as positive angle.
AOB is a positive angle.
Negative angle: Clockwise direction of ray OA about ‘O’ is regarded as negative angle.
AOB is a negative angle.
1.10 One Complete Rotation Suppose an initial arm OA is rotated about its end point O in the anticlockwise direction and takes final position OA again for the first time, then ray OA is said to have formed one complete rotation. Note: The angle traced during one complete rotation in anticlockwise direction is 360°.
The angle traced during two complete rotations in the anticlockwise direction is 2 360 = 720° and so on. 1.11 Zero Angle, Straight Angle, Reflex Angle, Co-terminal Angle Zero Angle: If there is no rotation of initial ray OA, then the directed angle so formed is called a zero angle, i.e. the amount of rotation of ray OA about point O is zero. Straight Angle: If the ray OA and OB of the directed angle AOB are opposite rays, i.e. by rotating the ray OA about O, it occupies the position of the ray OB as shown in the adjoining figure, then the directed angle so formed is called the straight angle.
mAOB = 180
O A
B
AO
B
A O
Initial arm
Initial arm O
A
B
O A
B
O AB
B
AO
Initial side
14
Std. IX : Geometry
Reflex Angle:
If the initial ray OA rotates about O in the anticlockwise direction and takes final position OB before coinciding the ray OA again, such that its
degree measure lie between 180 and 360 then we get the directed angle which is called a reflex angle.
Co-terminal Angle:
The directed angles of different measures having same position of initial ray and terminal ray are called co-terminal angles. There are infinitely many directed angles co-terminal with a given directed angles. The measures of co-terminal directed angles differ by an integral multiple of
360. 1.12 Perpendicularity The two lines are said to be perpendicular to each other when a right angle is formed at the point of intersection of the two lines.
line CD and line AB are perpendicular to each other at the point O and
this is writen as line CD line AB. Perpendicularity of Segments and Rays:
Two rays or two segments or a ray and a segment are said to be perpendicular to each other, if the lines containing them are perpendicular to each other. Note: Point O is called as the foot of the perpendicular drawn from a Point C to line AB. 1.13 Congruent Angles If the measures of two angles are equal, then the angles are called
congruent angles.
Since, mABC = mPQR = 30
ABC PQR Properties of Congruent Angles:
i. Reflexivity: ABC ABC (Every angle is congruent to itself).
ii. Symmetry: If ABC PQR, then PQR ABC.
iii. Transitivity: If ABC PQR and PQR XYZ, then ABC XYZ.
A
C
BO
D
A
B
O
C B
30
A
RQ
30
P
ray OC ray OB
A
C
B O
D
seg CD line AB
A
C
BO
D
line CD ray OB
A
C
B O
D
O400
40
B
A
15
Chapter 01: Lines and Angles
Inequality of Angles:
If the measure of one of the angles is greater than that of the other, then the angle with the greater measure is said to be greater than the angle with smaller measure.
Since, mABC = 110 and mPQR = 60
ABC > PQR 1.14 Types of Angles Acute Angle: If the measure of an angle is less than 90°, then it is called an acute angle. Right Angle: If the measure of an angle is 90°, then it is called a right angle. Obtuse Angle: If the measure of an angle is greater than 90°, then it is called an obtuse angle. Straight Angle: If the measure of an angle is 180°, then it is called a straight angle. 1.15 Pairs of Angles Adjacent Angles:
Two angles are called adjacent angles, if they have a common side and their interiors are disjoint.
AOB and COB are adjacent angles. Angles in a Linear Pair:
Two adjacent angles are said to form a linear pair of angles, if their non-common sides form a pair of opposite rays.
AOB and AOC form angles in a linear pair. Note:
Sum of the measures of the angles in a linear pair is 180°.
mAOB + mAOC = 180° Vertically Opposite Angles:
Two angles are called vertically opposite angles, if their sides form two pairs of opposite rays.
AOD and BOC, AOC and BOD form pairs of vertically opposite angles. B
A C
D
O
A
B CO
O
A
C
B
A BO
A
O B
A
BO
50°
P
Q R
60°
A
B C
110°
A
BO
110°
16
Std. IX : Geometry
Complementary Angles: If the sum of the measures of the two angles is 90°, then these angles are called complementary angles. mAOB + mPQR = 30 + 60 = 90°. Supplementary Angles: If the sum of the measures of the two angles is 180°, then these angles are called supplementary angles.
mAOB + mPQR = 120 + 60 = 180° Exercise 1.3 1. Answer the following questions and justify. i. Can two acute angles be complement to each other? Solution: Yes, consider two acute angles 30° and 60°. Their sum is 90°. ii. Can two obtuse angles be complement to each other? Solution: No, consider two obtuse angles 100 and 120 (greater than 90°). Their sum is also greater than 90°. iii. Can two right angles be complement to each other? Solution: No, consider two right angles (equal to 90°). Their sum is 180° 90°. iv. Can two acute angles be supplementary? Solution: No, consider two acute angles 40 and 60 (less than 90°). Their sum is less than 180. v. Can two obtuse angles be supplementary? Solution: No, consider two obtuse angles 120 and 100 (greater than 90°). Their sum is greater than 180°. vi. Can two right angles be supplementary? Solution: Yes, consider two right angles (equal to 90°). Their sum is 180°. vii. Can two adjacent angles be supplementary? Solution: Yes, if the sum of the measures of the two adjacent angles is 180°. viii. Can two adjacent angles be complementary? Solution: Yes, if the sum of the measures of the two adjacent angles is 90°. ix. Can two obtuse angles be adjacent angles? Solution:
Yes, AOB and BOC are obtuse and adjacent angles.
P
Q R
60°
A
120°
B O
A
B C
100°
120°
O
A
O
30
B
R
P
Q60°
17
Chapter 01: Lines and Angles
x. Can an acute angle be adjacent to an obtuse angle? Solution: Yes, AOB is obtuse and AOC is acute. Together they form adjacent angles. xi. Can two acute angles form a linear pair? Solution: No, the sum of the measures of the two acute angles is not equal to 180° (will be less than 180°). xii. Can two obtuse angles form a linear pair? Solution: No, the sum of the measures of the two obtuse angles is not equal to 180° (will be greater than 180°). xiii. Can two right angles form a linear pair? Solution: Yes, the sum of the measures of the two right angles is equal to 180°. 2. Measures of some angles are given below. Find the measures of their supplementary angles.
i. 60 ii. 138 iii. th
3
5
of a right angle iv. (180 r)
v. (90 + r) vi. 87 vii. 124 viii. 108 Solution: i. Measure of the supplementary angle = 180 60 = 120. Measure of the supplementary angle is 120. ii. Measure of the supplementary angle = 180 138 = 42 Measure of the supplementary angle is 42°.
iii. th
3
5
of a right angle = 3
5 90 = 3 18 = 54
Measure of the supplementary angle = 180 54 = 126 Measure of the supplementary angle is 126. iv. Measure of the supplementary angle = 180 (180 r) = 180 180 + r = r Measure of the supplementary angle is r. v. Measure of the supplementary angle = 180 (90 + r) = 180 90 r = (90 r) Measure of the supplementary angle is (90 r). vi. Measure of the supplementary angle = 180 87 = 93 Measure of the supplementary angle is 93. vii. Measure of the supplementary angle = 180 124 = 56 Measure of the supplementary angle is 56. viii. Measure of the supplementary angle = 180 108 = 72 Measure of the supplementary angle is 72
A
C
O B
20°
110°
18
Std. IX : Geometry
3. Measures of some angles are given below. Find the measures of their complementary angles.
i. 58 ii. 16 iii. rd
2
3
of a right angle iv. (a + b)
v. (90 r) vi. 78 vii. 68 viii. 56 Solution: i. Measure of the complementary angle = 90 58 = 32 Measure of the complementary angle is 32. ii. Measure of the complementary angle = 90 16 = 74 Measure of the complementary angle is 74.
iii. rd
2
3
of a right angle = 2
3 90 = 2 30 = 60
Measure of the complementary angle = 90 60 = 30
Measure of the complementary angle is 30. iv. Measure of the complementary angle = 90 (a + b) = (90 a b) Measure of the complementary angle is (90 a b). v. Measure of the complementary angle = 90 (90 r) = 90 90 + r = r Measure of the complementary angle is r vi. Measure of the complementary angle = 90 78 = 12 Measure of the complementary angle is 12. vii. Measure of the complementary angle = 90 68 = 22 Measure of the complementary angle is 22°. viii. Measure of the complementary angle = 90 56 = 34 Measure of the complementary angle is 34. 4. In the adjoining figure, find the measures of AOC and BOC. Solution: AOC + BOC = 180° ----(Angles in a linear pair)
(3x + 5) + (2x 25) = 180°
3x + 2x + 5 25 = 180 5x 20 = 180°
5x = 180 + 20 5x = 200
x = 200
5 = 40°
Now, AOC = (3x + 5) = (3 40 + 5) = (120 + 5) = 125°
BOC = (2x 25) = (2 40 25) = (80 25) = 55°
AOC = 125° and BOC = 55°
A O B
C
(3x + 5)° (2x 25)°
19
Chapter 01: Lines and Angles
5. In the adjoining figure, COD = 90°, BOE = 72° and AOB is a straight angle. Find the measures of the following angles. AOC, BOC, AOE Solution: AOC = BOE ----[Vertically opposite angles] But, BOE = 72 ----[Given] C = 72
AOC + BOC = 180 ----[Angles in a linear pair] 72 + BOC = 180 BOC = 180 72 BOC = 108 ----(i)
AOE = BOC ----[Vertically opposite angles] AOE = 108 ----[From (i)] AOC = 72, BOC = 108 and AOE = 108. 1.16 Geometric Sentences and their Proofs If-then form of Statement: Any statement stated in the ‘if-then’ form is said to be a conditional statement. The part of ‘if-then’ form of statement, which follows, ‘if’ is called antecedent and that which follows ‘then’ is called consequent. Example: General Statement: Two intersecting lines are contained in one plane. Conditional Statement: If two lines intersect each other then they are contained in one plane. Broadly speaking: ‘If p then q.’ If -------p---------- then -----------q ---------- antecendent Consequent Data (Given) to be proved sufficient for consequent necessary for antecedent Converse of a Statement: A statement obtained by interchanging the antecedent and consequent is called the converse of the original statement. Example: General statement: If two lines intersect, then they are in a plane. Converse: If two lines are in a plane, then they intersect each other. Note: If a property is true, then its converse may or may not be true. (For the above example its converse is not true.) Proof: The process of establishing the conclusion by logical reasoning on the basis of axioms and previously proved theorems is called proof. There are two types of proofs. i. Direct proof: If from an antecedent, we reach upto the consequent using axioms or previously proved
theorems, then it is called a direct proof. ii. Indirect proof: In this method, we suppose that the consequent is false and proceed logically and arrive at
a step which contradicts what is given (antecedent) or some well known fact, then we agree that the consequent is true.
Corollary: A property that can be readily derived from a theorem is called a corollary.
OA B
E
C D
72
20
Std. IX : Geometry
1.17 Parallel Lines Parallel Lines: Non intersecting coplanar lines are called parallel lines.
line || line m || line n Transversal: A line intersecting two or more coplanar lines in distinct points is called as a transversal.
Line P is a transversal intersecting line m, line n and line at points A, B, C
respectively. Intercept: A segment cut off on a transversal by two distinct lines is called as an intercept.
AB is an intercept on transversal n.
AC is an intercept on transversal . Angles formed by the transversal Alternate Angles: ABC and BCD Interior Angles: PBC and QCB Corresponding Angles:
Properties for Parallel Lines:
1. For two lines (intersected by a transversal) to be parallel,
i. at least a pair of alternate angles should be congruent (alternate angles test).
ii. at least a pair of corresponding angles should be congruent (corresponding angles test).
iii. at least a pair of interior angles should be supplementary (interior angles test). 2. If a transversal intersects two parallel lines, then
i. all pairs of alternate angles are congruent.
ii. all pairs of corresponding angles are congruent.
iii. all pairs of interior angles are supplementary.
m
n
A B
C D
m
n
p
A
B
C
P
m
n
A B
DC
A
C
B P
QD
ABC and BPQ
Q
C
A
B
P
A
B
P Q
C
A
P
B C
Q
21
Chapter 01: Lines and Angles Theorems and Corollaries Theorem 1.1 If two lines intersect each other, then the vertically opposite angles are congruent. Given: line AB and line CD intersect each other at point O. To prove: AOC BOD and AOD BOC Proof: line AB and line CD intersect each other at point O. mAOC + mBOC = 180° ----(i) [Angles in a linear pair] mBOC + mBOD = 180° ----(ii) [Angles in a linear pair] mAOC + mBOC = mBOC + mBOD ----[From (i) and (ii)] mAOC = mBOD AOC BOD
Similarly, it can be proved that AOD BOC. Theorem 1.2 (Alternate angles test) If a pair of alternate angles formed by a transversal of two coplanar lines is congruent, then the lines are parallel. Given: Transversal n intersects coplanar lines, line and line m in
points P and Q respectively. APQ PQD To prove: line || line m
Proof: (Indirect proof) Suppose line is not parallel to line m.
It means that the coplanar lines AB and CD are not parallel. Then, they intersect in say point R. Now, points P, Q, R are non-collinear, so we get PQR. APQ + RPQ = 180° ----(i) [Angles in a linear pair] RPQ + PQR + PRQ = 180° ----(ii) [Sum of measures of angles of a triangle]
APQ + RPQ = RPQ + PQR + PRQ ----[From (i) and (ii)] APQ = PQR + PRQ APQ > PQR
i.e., APQ > PQD ----[Q-D-R] This contradicts the given, APQ PQD. So, our assumption is wrong.
line || line m. Theorem 1.3 (Corresponding angles test) If a pair of corresponding angles formed by a transversal of two coplanar lines is congruent, then the lines are parallel. Given: Transversal n intersects coplanar lines, line and line m at points P and Q respectively.
EPB PQD To prove: line || line m
Proof: EPB PQD ----(i) [Given] EPB APQ ----(ii) [Vertically opposite angles] PQD APQ ----[From (i) and (ii)] line || line m ----[Alternate angles test]
A P B
C D
E
Q
F
m
n
A
B
D
C
O
A B
C DQ
F
P
n
E
m
F
AP
E
B
n
RD
C Q
22
Std. IX : Geometry
Theorem 1.4 (Interior angles test) If a pair of interior angles formed by a transversal of two coplanar lines is supplementary, then the lines are parallel.
Given: Transversal n intersects coplanar lines, line and line m respectively.
APQ + PQC = 180°
To prove: line || line m
Proof:
APQ + PQC = 180° ----(i) [Given]
PQC + PQD = 180° ----(ii) [Angles in a linear pair]
APQ + PQC = PQC + PQD ---- [From (i) and (ii)]
APQ = PQD
APQ PQD
line || line m ----[Alternate angles test for parallel lines] Theorem 1.5 (Converse of alternate angles test) If the two lines are parallel, then the alternate angles formed by a transversal are congruent.
Given: Transversal n intersects line and line m at points P and Q respectively.
line || line m
To prove: APQ PQD Proof: (Indirect proof)
Suppose APQ is not congruent to PQD.
Then draw a line r passing through point P such that SPQ PQD.
line r || line m ----[Alternate angles test for parallel lines]
But, line || line m ----[Given]
This means there are two parallel lines, parallel to line m, passing through the same point P which is outside the line m.
This contradicts the uniqueness of parallel lines.
Our assumption is wrong.
APQ PQD. Theorem 1.6 (Converse of corresponding angles test) If the two lines are parallel, then the corresponding angles formed by a transversal are congruent.
Given: Transversal n intersects line and line m at points P and Q respectively.
line || line m
To prove: EPB PQD Proof:
line || line m ----[Given]
PQD APQ ----(i) [Alternate angles]
EPB APQ ----(ii) [Vertically opposite angles]
EPB PQD ----[From (i) and (ii)]
A P B
C Q D
E
m
n
F
A P B
C D
E
Q
F
m
n
A B
C D
P
Q
ES
F
T
r
n
m
23
Chapter 01: Lines and Angles
Theorem 1.7 (Converse of interior angles test): If two lines are parallel, then the interior angles formed by a transversal are supplementary.
Given: Transversal n intersects line and line m in points P and Q respectively.
line || line m
To prove: BPQ + PQD = 180° Proof:
line || line m ----[Given]
PQD APQ ---- (i) [Alternate angles]
BPQ + APQ = 180° ----(ii) [Angles in a linear pair]
BPQ + PQD = 180° ----[From (i) and (ii)] Corollary 1: If two lines are parallel to the same line, then they are parallel to each other.
Given: line , line m, line n are coplanar lines.
line || line m, line || line n
To prove: line m || line n
Construction: Draw a transversal q intersecting lines , m and n in points P, Q
and R respectively. Proof:
line || line m and line q is a transversal.
GPB PQD ----(i) [Corresponding angles]
line || line n and line q is a transversal.
GPB QRF ----(ii) [Corresponding angles]
PQD QRF ----[From (i) and (ii)]
line m || line n ----[Corresponding angles test for parallel lines] Corollary 2: If a line coplanar with two parallel lines is perpendicular to one of them, it is also perpendicular to the other.
Given: line || line m and line n intersects them at P and Q respectively.
line n line
To prove: line n line m Proof:
line || line m and line n is a transversal.
EPB PQD ----(i) [Corresponding angles]
line n line ----[Given]
EPB = 90° ----(ii)
PQD = 90° ----[From (i) and (ii)]
line n line m
n
A P B
C Q D
E
m
F
A B
C D
E
P
Q
m
n
F
A B
C D
E F
G
P
Q
R H
q
m
n
24
Std. IX : Geometry
Corollary 3: If a line is perpendicular to two coplanar lines, then those two lines are parallel to each other. Given: line n line line n line m To prove: line || line m
Proof: line n line ----[Given]
EPB = 90° ----(i) line n line m ----[Given] PQD = 90° ----(ii) EPB PQD ----[From (i) and (ii)] line || line m ----[Corresponding angles test for parallel lines] Properties of Corresponding, Alternate and Interior Angles When a line intersects two coplanar lines, i. If one pair of corresponding angles is congruent, then the remaining pairs of corresponding angles are
congruent. ii. If one pair of alternate angle is congruent, then the other pair of alternate angles is also congruent. iii. If one pair of interior angle is supplementary, then the other pair of interior angles is also supplementary. iv. When two lines are parallel and a transversal intersects then a. A pair of corresponding angles is congruent. b. A pair of alternate angles is congruent. c. A pair of interior angles is supplementary. Exercise 1.4 1. i. If a transversal intersects two parallel lines, then state the relation between alternate angles. ii. If each of the two lines are parallel to the third line, then what is the relation between them? iii. If a transversal intersects two parallel lines and the corresponding sides of two angles are
parallel, then what is the relation between these angles? iv. If AB is a line and P is a point outside it, then how many lines can be drawn through P and
parallel to AB? v. If a transversal intersects two parallel lines such that the ratio between the interior angles on
one of its side is 2 : 7, then find the measure of the greater angle. Solution: i. Both the pairs of alternate angles are congruent. ii. The three lines are parallel to each other. iii. The two angles are congruent. iv. One and only one line can be drawn.
v. Let the measures of the interior angles be 2x and 7x. Since, interior angles are supplementary,
2x + 7x = 180 9x = 180
x = 180
9
= 20
The measure of the greater angle is 7x = 7 20 = 140°.
Measure of the greater angle is 140°.
n
A B
C D
E
P
Q
m
F
25
Chapter 01: Lines and Angles
2. In the adjoining figure,
line || line m and line p is the transversal.
If r = 20°, then find a : b. Solution: r = a ----[Vertically opposite angles]
a = 20° ----[ r = 20°]
Now, a + b = 180 ----[Interior angles] 20 + b = 180 b = 180 20 = 160
a
b =
20
160
= 2
16=
1
8
a : b = 1 : 8 3. In the given figure,
If line || line m || line n, then find A.
Solution:
line || line n and seg OC is a transversal.
POC OCD ----[Alternate angles]
OCD = 110° ----[POC = 110]
OCA + ACD = OCD ----[Angle addition property] 30° + ACD = 110° ACD = 110°° ACD = 80° line m || line n and seg AC is a transversal. ACD + CAB = 180° ----[Interior angles] 80° + CAB = 180° CAB = 180 80 CAB = 100° A = 100° 4. In the adjoining figure, line || line m and line PQ is the transversal.
If PEB = 70°, find the measures of each of the remaining angles. Solution: AEF PEB ----[vertically opposite angles]
AEF = 70° ----[PEB = 70]
PEB + PEA = 180° ----[Angles in a linear pair] 70° + PEA = 180° PEA = 180° 70° PEA = 110° BEF PEA ----[Vertically opposite angles] BEF = 110° DFQ BEF ----[Corresponding angles] DFQ = 110° CFE DFQ ----[Vertically opposite angles] CFE = 110° CFQ AEF ----[Corresponding angles] CFQ = 70° EFD CFQ ----[Vertically opposite angles] EFD = 70°
E
Q
F
P
m
A B
C D
D
110°
P O
A B
C
m
n
30
a
b
r
p
m
26
Std. IX : Geometry
Problem Set - 1 1. Points P and Q are on line PS, but not on seg RS. Point Q is a point on ray RS but not on ray SR. Point P is a point of ray SR, but not of ray RS. Draw the figure. Solution: Point Q is on ray RS but not on ray SR. R-S-Q Point P is on ray SR but not on ray RS. P-R-S Points P and Q are not on seg RS. P-R-S-Q 2. In the adjoining figure, A, B, C, D and E are the points of line . i. Name the opposite rays with point A as origin. ii. Name the ray opposite to ray BE. iii. Find the intersection of ray CE and ray EC. Solution: i. Ray AB and ray AC are the opposite rays with point A as their origin. ii. Ray BA is the opposite ray of ray BE. iii. Intersection of ray CE and ray EC is seg CE. 3. Given that a point X is between the points Y and Z and the point Y is in between the points R and X.
Draw the figure and write the betweenness of the points R, X, Y and Z on the line. Solution: Point X is between the points Y and Z. YXZ Point Y is between the points R and X. RYX The betweeness on the line is RYXZ. 4. In a queue for a bus, 25 persons are standing. How many of them do not stand in between any two of
them? How many of those 25 are standing between any two of them? If the number of persons in the queue is M, write down the answers for the above questions.
Solution: 1. If there are 25 persons in a queue. i. 2 persons are not standing in between any two of them [Reason: since the first and the last person are
not between any two persons] ii. The remaining 23 persons are standing in between any two of them. 2. If there are M persons in a queue. i. 2 persons are not standing in between any two of them. ii. The remaining (M 2) persons are standing in between any two of them. 5. The co-ordinate of a point R on a number line is 8. The point S is on the same line which is to the left
of R and at a distance of 7 units from R. Find the co-ordinate of S. If P is the midpoint of seg RS, find the co-ordinate of point P.
Solution: Let the co-ordinate of point S be x. The co-ordinate of point R is 8, Since, S is to the left of point R, 8 > x d(S, R) = 8 x
R Y X Z
S P R3.5
x y 87
D C A B E
P R S Q
27
Chapter 01: Lines and Angles
7 = 8 x x = 8 7 = 1 Co-ordinate of point S is 1. P is the midpoint of seg SR.
PR = 1
2SR =
1
2× 7
PR = 3.5 units Let the co-ordinate of point P be y. The co-ordinate of point R is 8. Since, P lies to the left of point R, 8 > y d(P, R) = 8 y 3.5 = 8 y y = 8 3.5 = 4.5 Co-ordinate of point P is 4.5. Given A-B-C, the co-ordinate of point A is the least. If d(A, B) = 7.5, d(B, C) = 13.8 and the
co-ordinate of point B is 8.5, then find the co-ordinates of points A and C. Solution: Let the co-ordinate of point A be x. The co-ordinate of point B is 8.5. Since, A is to the left of point B 8.5 > x d(A, B) = 8.5 – x 7.5 = 8.5 x x = 8.5 7.5 = 1 Co-ordinate of point A is 1. Let the co-ordinate of point C be y. The co-ordinate of point B is 8.5. Since, B lies to the left of point C y > 8.5 d(B, C) = y 8.5 13.8 = y 8.5 13.8 + 8.5 = y y = 22.3 Co-ordinate of point C is 22.3. 7. In the given figure, BAC = 75°, ABC = 35°. Find the
measures of BAZ, CBX and ACY. Solution: BAZ + BAC = 180° ----[Angles in a linear pair] BAZ + 75 = 180° BAZ = 180 75 BAZ = 105° CBX + ABC = 180° ----[Angles in a linear pair] CBX + 35 = 180° CBX = 180 35 CBX = 145° In ABC, BAC + ABC + ACB = 180° ----[Sum of the measures of the angles of a triangle is 180]
75
35 C Y
AZ
B
X
A B C
x 8.5 y 7.5 13.8
28
Std. IX : Geometry
75 + 35 + ACB = 180° 110° + ACB = 180° ACB = 180° 110° ACB = 70° ACY + ACB = 180° ----[Angles in a linear pair] ACY + 70 = 180° ACY = 180 70 ACY = 110° 8. In the given figure, XLN = 100°, YMN = 165°, ZNM = 95°.
Find the measures of MLN, LMN and LNM. Solution: MLN + XLN = 180° ----[Angles in a linear pair] MLN + 100° = 180° MLN = 180° 100° MLN = 80° LMN + YMN = 180° ----[Angles in a linear pair] LMN + 165° = 180° LMN = 180° 165° LMN = 15° LNM + ZNM = 180° ----[Angles in a linear pair] LNM + 95° = 180° LNM = 180° 95° LNM = 85° 9. In the given figure, if ABC = 135°, ABX = 90°, XCD = 55°, BCD = 100°, then determine
whether XBC and XCB are congruent to each other. Solution: XBC + ABX = ABC ----[Angle addition property] XBC + 90° = 135° XBC = 135° 90° XBC = 45° ----(i) XCB + XCD = BCD ----[Angle addition property] XCB + 55° = 100° XCB = 100 55 XCB = 45° ----(ii) XBC = XCB ----[From (i) and (ii)] XBC XCB 10. In the adjoining figure, lines RS, VP and TQ intersect in point O, VOS = 100°, ROT = 122°. Find
ROQ, QOP and POS. Solution:
VOR + VOS = 180° ----[Angles in a linear pair] VOR + 100° = 180° VOR = 180° 100° VOR = 80° ----(i) POS VOR ----[Vertically opposite angles] POS = 80° ROT + SOT = 180° ----[Angles in a linear pair]
A
B C
D
X
16595
L
M N
X
Y Z
100
Q
O
P
S
TV
R
29
Chapter 01: Lines and Angles
122° + SOT = 180° SOT = 180° 122° SOT = 58° ROQ SOT ----[Vertically opposite angles] ROQ = 58° VOR + VOT = ROT ----[Angle addition property] 80° + VOT = 122° ----[From (i) and Given] VOT = 122° – 80° VOT = 42° QOP VOT ----[Vertically opposite angles] QOP = 42° 11. In the given figure, if AOC BOD, AOB = 45°,
BOD = 60° and AOB COD, then find BOC. Solution: AOC BOD ----[Given]
C = 60° ----[ BOD = 60°]
BOC + AOB = AOC ----[Angle addition property] BOC + 45° = 60° BOC = 60° 45° BOC = 15° 12. If XPY = 135°, XPZ = 175° and the point Y is in the interior of ZPX, then find YPZ. Solution: YPZ + XPY = XPZ ----[Angle addition property] YPZ + 135° = 175° YPZ = 175° 135° YPZ = 40° 13. If POR = 120° and P-O-L, the points S and T be on the R-side of line PL, such that ROS SOL
and ROT TOP. Draw the figure and find TOS. Solution: POR + LOR = 180° ----[Angles in a linear pair] 120° + LOR = 180° LOR = 180° 120° LOR ° ----(i) ROS SOL ----[Given] ray OS bisects LOR.
ROS = 1
2 LOR
ROS = 1
2 60° ----[From (i)]
ROS = 30° ROT TOP ----[Given] ray OT bisects POR.
ROT = 1
2 POR
P
O
L
T
R
S
120°
°°
X P
175°
135°
Y
Z
A B
C
D O60
45
30
Std. IX : Geometry
ROT = 1
2 120°
ROT = 60° TOS = ROT + ROS ----[Angle addition property] TOS = 60° + 30° TOS = 90° 14. Ray PR and ray PQ are perpendicular to each other. The point B is in the interior of QPR and the
point A is in the exterior of RPQ such that ray PB and ray PA are perpendicular to each other. Draw the figure and write down the pairs of (i) Complementary angles (ii) Supplementary angles (iii) Congruent angles.
Solution: i. Complementary angles: APR + RPB = APB = 90° and RPB + BPQ = RPQ = 90° a. APR and RPB are complementary angles. b. RPB and BPQ are complementary angles. ii. Supplementary angles: APB + RPQ = 90 + 90 = 180° APB and RPQ are supplementary angles. iii. Congruent angles: a. APB RPQ ----[Each is 90°] b. APB = RPQ ----[Given] APR + RPB = RPB + BPQ ----[Angle addition property] APR = BPQ APR BPQ 15. In the given figure, coplanar lines , m and n intersect each other in the same
point. p, q, r, s, t and u denote the measures of the angles as shown. If p = 97°, r = 29°, then find the values of q, s, t and u. Solution: s = p ----[Vertically opposite angles] s = 97° u = r ----[Vertically opposite angles] u = 29° u + p + q = 180° ----[Angles in a linear pair] 29° + 97° + q = 180° 126° + q = 180° q = 180° 126° q = 54° t = q ----[Vertically opposite angles] t = 54° 16. In the given figure, ray OC bisects AOB.
Prove that: COD = 1
2 [BOD AOD]
Proof: BOC AOC . ----(i) [Ray OC bisects AOB] COD + BOC = BOD ----[Angle addition property]
O
B
A
D
C
R
P
A
B
Q
u p
q
rs
m
n
t
31
Chapter 01: Lines and Angles
COD = BOD BOC ---- (ii) COD + AOD = AOC ----[Angle addition property] COD = AOC AOD ---- (iii) COD + COD = BOD BOC + AOC AOD ----(iv) [Adding (ii) and (iii)] 2COD = BOD BOC + BOC AOD ----[From (i) and (iv)] 2COD = BOD AOD
COD = 1
2 [BOD AOD]
17. In the given figure, AB || CD. Find the values of x, y and z. Solution: seg AB || seg CD and seg AD is a transversal. CDA = BAD ----[Alternate angles] z = 75° seg AB || seg CD and seg BC is a transversal. ABC = DCB ----[Alternate angles] x = 35° In DCO, D + C + O = 180° ----[Sum of the measures of angles of a triangle is 180°] z + 35 + y = 180°
75° + 35° + y = 180° ----[ z = 75]
110° + y = 180° y = 180° 110° y = 70° 18. In the given figure, measure of some angles are given. Which two lines
are parallel and which two are not. Justify. Solution: 105° + x = 180° ----[Angles in a linear pair] x = 180° 105° = 75 Transversal k intersects line and line m.
Corresponding angles so formed are congruent. ----[Each is 75°]
line || line m ----[Corresponding angle test
for parallel lines] Transversal k intersects line m and line n.
Corresponding angles so formed are not congruent. ----[
Line m is not parallel to line n. Transversal k intersects line and line n.
Corresponding angles so formed are not congruent. ----[75 65]
Line is not parallel to line n. 19. In the given figure , AB || CD. Find the values of x and y. Solution: line AB || line CD and ray QP is a transversal
PGH = PEF ----[Corresponding angles]
x = 85°
DHF = QHC ----[Vertically opposite angles]
DHF = 115°
B
A
O
D
C
y
x
75°
z
35°
75°
105°
65° n
m
x
k
A R
C
Q
D B P
85° x
H
y F
115°
E G
32
Std. IX : Geometry
line AB || line CD and ray QR is a transversal. BFH + DHF = 180° ----[Interior angles] y + 115° = 180° y = 180° 115° y = 65° 20. In the given figure, a b, x y. Prove that line || line n.
Proof:
a and b are the corresponding angles formed by the line and line m.
a b ----[Given]
line || line m ---- (i) [Corresponding angles test for parallel lines]
x and y are alternate angles formed by the line m and line n. x y ----[Given) line n || line m ----(ii) [Alternate angles test for parallel lines] From (i) and (ii),
line || line n ----[If two lines are parallel to the same line, then they are parallel to each other] 21. In the figure, if x = 70° and y = 71°, state with reason whether the
line m || line n or not. Justify. Solution: x = 70° and y = 71° 70 71 x is not congruent to y. The corresponding angles are not congruent. Line m is not parallel to line n. 22. In the given figure, measures of two angles are given. If line ED || seg AB
and E-C-D, then find the values of x, y and z. Solution: line ED || seg AB and seg AC is a transversal. ACE BAC ----[Alternate angles] x = 65° line ED || seg AB and seg BC is a transversal. BCD ABC ----[Alternate angles] z = 35° In ABC, A + B + C = 180° ----[Sum of the measures of the angles of a triangle is 180°] 65° + 35° + y = 180° 100° + y = 180° y = 180 100 y = 80° 23. If AB || CD, then find PCD and CPD from the adjoining figure. Solution: CPD + CPA = 180° ----[Angles in a linear pair] CPD + 100° = 180° CPD = 180° 100° CPD = 80° ----(i) Ray AB || ray CD and seg AD is a transversal. ADC = BAD ----[Alternate angles]
x
y
m
n
m
a b
xy
n
A B
DC
56° P 100°
E D
BA
65° 35°
x zy
C
33
Chapter 01: Lines and Angles
ADC = 56° ----[BAD = 56]
PDC = 56° ----(ii) [A-P-D] In PCD, CPD + PDC + PCD = 180° ----[Sum of the measures of the angles of a triangle is 180°] 80° + 56° + PCD = 180° ----[From (i) and (ii)] 136° + PCD = 180° PCD = 180° 136° PCD = 44° 24. If a transversal intersects two parallel lines, then show that the bisectors of any pair of alternate
angles are also parallel.
Given: Transversal n intersects line and line m at points P and Q respectively. line || line m.
Ray PM and ray QN bisects APQ and PQD respectively. To prove: ray PM || ray QN Proof:
MPQ = 1
2APQ ----(i) [ray PM bisects APQ]
NQP = 1
2PQD ----(ii) [ray QN bisects PQD]
line || line m and line n is a transversal.
APQ = PQD ----[Alternate angles)
1
2 APQ =
1
2PQD ----(iii) [Multiply both sides by
1
2]
MPQ = NQP ----[From (i), (ii), (iii)] These are alternate angles on ray PM and ray QN, when line n is the transversal. ray PM || ray QN ----[Alternate angles test for parallel lines] 25. In the adjoining figure, AB || CD. Find the value of x. Solution: Construction: Extend ray AB to point S to intersect seg CE in point R. AB || CD ----[Given] line AS || ray CD ----[A-B-S] line AS || ray CD and seg CR is a transversal. SRE = DCR ----[Corresponding angles] SRE = 100° BRE + SRE = 180° ----[Angles in a linear pair] BRE + 100° = 180° BRE = 180 100 BRE = 80° ----(i) RBE + ABE = 180° ----[Angles in a linear pair] RBE + 120° = 180° RBE = 180 120 RBE = 60° ----(ii) In BER, BRE + RBE + BER = 180° ----[Sum of the measures of the angles of a triangle is 180°] 80° + 60° + x = 180° ----[From (i) and (ii)] 140° + x = 180° x = 180 140 = 40 x = 40
A B
C D
E
F
Q
n
M
P
m
°
°
N
A B
C D
EP Q
R S
x120°
100°
34
Std. IX : Geometry
One-Mark Questions 1. Draw figure related to following axiom:
Line AB and line AC have one and only one common point.
Solution:
2. The co-ordinates of the points P and Q are
2 and 0 respectively. Find d(P, Q). Solution: 0 > 2 d(P, Q) = 0 (2) = 0 + 2 d(P, Q) = 2 3. When PQR, (PQ) = 3 2 , (PR) = 5 2 ,
find (QR).
Solution: d(P, R) = d(P, Q) + d(Q, R) ----[P-Q-R] 5 2 = 3 2 + d(Q, R) d(Q, R) = 5 2 3 2 d(Q, R) = 2 2
(QR) = 2 2 4. Measure of an angle is (60 r). Find the
measure of its complement. Solution: Complement angle of (60 r) = 90 (60 r) = 90 60 + r = 30 + r 5. Observe the figure and find the opposite
ray of ray QR. Solution: Ray QP 6. Point B has co-ordinate 3 on the number
line. Find the co-ordinates of the points which are at a distance 6 from B.
Solution: Let A and C be the points which are at a
distance 6 from B as shown in the figure below.
The co-ordinate of A is 3 6 = 9 and the co-ordinate of C is 3 + 6 = 3.
7. Measure of an angle is (60 r). Find the measure of its supplementary angle.
Solution: Supplementary angle of (60 r) = 180 (60 r) = 180 60 + r = 120 + r 8. If d(A, B) = 6, d(B, D) = 5, d(A, D) = 11,
then state the betweenness among the points A, D and B.
Solution: d(A, B) + d(B, D) = 6 + 5 = 11 = d(A, D) d(A, D) = d(A, B) + d(B, D) ABD. 9. Draw a figure showing two adjacent angles
as well as complementary angles. Solution:
10. Ray AD is the bisector of BAC. Name the
two angles which are congruent. Solution: BAD and DAC. 11. Two complementary angles are of equal
measures. State their measures. Solution: Measure of each angle is 45. 12. If initial arm of the directed angle is rotated
20 in anticlockwise direction, then state the measure of this angle.
Solution: Measure of angle is 20. 13. Draw the figure related to statement given
below: Two angles are adjacent but they are neither complementary nor supplementary.
Solution:
P Q R S
P
QR
S
30 60
B A 3
C
A
B
C
30 25
D
A
B
C
35
Chapter 01: Lines and Angles
14. Name the alternate angle of i. PTU ii. RUT Solution: i. The alternate angle of PTU is TUS. ii. The alternate angle of RUT is QTU. Additional Problems for Practice Based on Exercise 1.1 1. Draw the figure related to the following axiom : i. When two planes intersect, their
intersection is exactly one line. ii. When a line intersects a plane but does
not lie in it, then their intersection is a point.
iii. Infinite number of lines can be drawn through a given point.
2. Take any five points in a plane such that any 3
points are non-collinear. How many lines can be drawn through these points taking two at a time?
3. Draw the figure related to the following
statement : i. A plane F is determined by two parallel
lines l and m. ii. Line AB and a plane E have one and
only one common point R. iii. Lines p and q are in a plane E and they
have one and only one common point R. Based on Exercise 1.2 1. The co-ordinates of some points on a line are
given in the following table :
Points P Q R S T Co-ordinates 8 6 2 5 1
Find : d(P, Q), d(Q, R), d(Q, S), d(Q, T) 2. In each of the following, decide whether the
relation of betweenness exists among the points. Name the point which lies between the other two. i. d(P, Q) = 10, d(Q, R) = 3, d(P, R) = 7 ii. d(D, E) = 5, d(E, F) = 8, d(D, F) = 6
iii. d(A, B) = 16, d(C, A) = 9, d(B, C) = 7 iv. d(X, Y) = 15, d(Y, Z) = 7, d(X, Z) = 8
3. When X–Y–Z, (XY) = 17, (YZ) = 8 then
find (XZ). 4. If B is the midpoint of seg PQ and
(PQ) = 7 cm, find (PB). Based on Exercise 1.3 1. Measures of some angles are given below.
Find the measures of their supplementary angles.
i. 45 ii. (90 – x) iii. r iv. 75 2. Measures of some angles are given below.
Find the measures of their complementary angles.
i. 35 ii. (a – b)
iii. th2
5 of a right angle iv. 53
3. Draw the figures satisfying the following
conditions. i. Two angles are adjacent as well as
supplementary. ii. Two angles are adjacent but not
supplementary. iii. Two angles are supplementary of each
other but not adjacent. 4. The measure of an angle exceeds four times
the measure of its supplement by 30. Find its measure.
5. The measure of an angle is less by 5 than four times the measure of its complement. Find its measure.
6. In the given figure, find the measures of BOC and BOA.
7. Ray QT is bisector of PQR and
mPQR = 130. Find mPQT.
C O A
B
(13x + 30) (5x + 6)
T
W
U
V
m
n
P Q
S R
P
Q
T
R
36
Std. IX : Geometry
E
AB
F
DC
Based on Exercise 1.4 1. If two parallel lines are intersected by a
transversal, then prove that the bisectors of any two corresponding angles are parallel.
2. In the figure, EAB DBA. Ray AF and
ray BC are bisectors of EAB and ABD respectively. Prove that ray AF ray BC
3. In the figure, AB CD. Find the value of x.
4. In the figure, m ABC = 65, m DCE = 35,
mCEF = 145, m BCE = 30. Prove that ray AB ray EF.
5. In the figure, side QP and side QR of PQR
are produced to points S and T respectively. If SPR = 135, PQT = 110. Find PRQ.
6. In the figure, line AB line CD. Find the
values of x, y and z. 7. In the figure, line AB line ED and
line EF line DC. Prove that ABC and
DEF are supplementary angles.
Multiple Choice Questions 1. How many lines can be drawn passing through
two distinct points? (A) Two (B) Only two (C) Only one (D) Infinite 2. There is exactly one plane passing through (A) two distinct points (B) a line and a point on that line (C) three collinear points (D) two distinct intersecting lines 3. The intersection of two planes is exactly ____. (A) a point (B) a line (C) a plane (D) a line and a point not on the line 4. In the given figure which of the following set
of points are collinear? (A) C, K, L, A (B) B, I, J, K (C) G, L, K, C (D) D, E, G, F
A B
65
30
35
145
E F
C D
S
P
T Q R
110
135
B P
C Q R
y
D
110
A
65z
x
E
A
F
B D CA B
O
D
135
145
x
C
A
GF L E
H K
CJ
D
B I
37
Chapter 01: Lines and Angles
5. If the points P, T and A are collinear and d(P, A) = 13, d(T, A) = 5 and d(T, P) = 8, then which of the following can be true?
(A) A P T (B) T A P (C) A T P (D) All of the above 6. The angle traced during one complete rotation
in anticlockwise direction is _______. (A) 90 (B) 180 (C) 270 (D) 360 7. The directed angle POR, as shown in the
figure, is also called as (A) zero angle (B) straight angle (C) reflex angle (D) co-terminal angle 8. If there is no rotation of initial ray OA, then
the directed angle so formed is called as (A) one complete rotation (B) zero angle (C) straight angle (D) reflex angle 9. The directed angle TOR, as shown in the
figure, is regarded as (A) negative angle (B) positive angle (C) reflex angle (D) co-terminal angle 10. If the directed angles with measures 50, 410,
770 have same initial arm and same terminal arm, then such angles are called
(A) reflex angles (B) straight angles (C) zero angles (D) co-terminal angles 11. Which of the following is not a property of
congruent angles? (A) Reflexivity (B) Symmetry (C) Transitivity (D) Inequality 12. The pair of angles (90 x) and (90 + x) are (A) supplementary (B) complementary (C) congruent (D) vertically opposite angles 13. The complementary angle of 34 is (A) 146 (B) 56 (C) 90 (D) 180
14. In the given figure, if L O M, then the measure of LON is
(A) 50 (B) 110 (C) 70 (D) 180 15. If two lines intersect each other then (A) the pair of alternate angles are congruent (B) the pair of vertically opposite angles are
congruent (C) the pair of corresponding angles are
congruent (D) the pair of interior angles are
supplementary 16. If a transversal intersects two parallel lines,
then which of the following is not true? (A) a pair of corresponding angles is
congruent (B) a pair of interior angles is
complementary (C) a pair of alternate angles is congruent (D) a pair of interior angles is
supplementary Answers to additional Problems for Practice Based on Exercise 1.1
1. i.
ii. iii.
EFl
•
E
P
l
O
l
m n
r
O P
R
O
T
RTerminal arm
O M
N
L
(2x 30)(2x + 10)
38
Std. IX : Geometry
2. Ten 3. i.
ii. iii. Based on Exercise 1.2 1. d(P, Q) = 14, d(Q, R) = 4, d(Q, S) = 11,
d(Q, T) = 5 2. i. Yes, P–R–Q ii. No iii. Yes, A–C–B iv. Yes, X–Z–Y 3. 25 units 4. 3.5 cm Based on Exercise 1.3 1. i. 135 ii. (90 + x) iii. (180 – r) iv. 105 2. i. 55 ii. (90 – a + b) iii. 54 iv. 37 3. i.
ii.
iii.
4. 150 5. 71 6. BOC = 134 and BOA = 46 7. 65 Based on Exercise 1.4 3. 80 5. 65 6. x = 65, y = 115, z = 45
Answers to Multiple Choice Questions 1. (C) 2. (D) 3. (B) 4. (A) 5. (C) 6. (D) 7. (C) 8. (B) 9. (A) 10. (D) 11. (D) 12. (A) 13. (B) 14. (B) 15. (B) 16. (B)
D
C B A
60120
•
• •
•
P
QR
T
3040
C
A
B
•
•
P
Q R
•
•
F
l m
•
E
R
A•
•B
E
q
p
R •
