Contents No. 1
Topi c Name Lines and Angles
Page No. 1
2
Triangles
34
3
Congruence of Triangles
66
4
Circle
104
5
Quadrilaterals
136
6
Co-ordinate Geometry
175
7
Geometric Constructions
202
8
Trigonometry
224
9
Mensuration
248
Question Bank
284
10
Std. IX
Geometry Mr. Biju Babu (B.E Prod.)
Salient Features:
Written as per the new textbook. Exhaustive coverage of entire syllabus. Precise theory for every topic. Covers answers to all textual exercises and problem set. Comprehensive solution to Question Bank. Constructions drawn with accurate measurements. Attractive layout of the content. Self evaluative in nature.
Target
PUBLICATIONS PVT. LTD.
Mumbai, Maharashtra Tel: 022 – 6551 6551 Website : www.targetpublications.in www.targetpublications.org email :
[email protected]
Std. IX
Target Publication s Pvt Ltd.
Second Edition : March 2012
Price :
160/-
Printed at: India Printing Works 42, G.D. Ambekar Marg, Wadala, Mumbai – 400 031
Published by
Target PUBLICATIONS PVT. LTD. Shiv Mandir Sabagriha, Mhatre Nagar, Near LIC Colony, Mithagar Road, Mulund (E), Mumbai - 400 081 Off.Tel: 022 – 6551 6551 email:
[email protected]
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 this 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. It contains answers to textual exercises, problems sets and Question bank. 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 inspire 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
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01
eometry
LINES AND ANGLES
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. The terms or statements whose proofs are not to be asked are called axioms.
OR
Theorem
Important and useful results derived from the axioms are called theorems. The statements we prove from the axioms are called theorems.
OR
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. Some Axioms
1.
m
Infinite number of lines can be drawn through a given point. P n
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.
A
B
P m 4.
E
A
There is exactly one plane passing through three non-collinear points.
C B 5.
A
There is exactly one plane passing through a line and a point, not on the line.
E
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eometry
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There is exactly one plane passing through two distinct intersecting lines.
m
7.
E
When two planes intersect, their intersection is exactly one line.
F
8.
When a line intersects a plane but does not lie in it, then their intersection is a point.
E P
9.
A line containing two given points of a plane lies wholly in that plane.
Q
P
Collinear points, non-collinear points, parallel planes 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.
1.
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.
2.
A
B
C
B
C
A
Coplanar lines: Lines which lie in the same plane are called coplanar lines.
3.
line and line m are coplanar lines.
m
Non-coplanar lines: Lines which do not lie in the same plane are called noncoplanar lines.
4.
line and line m are non-coplanar lines. m
m Concurrent lines: If three or more lines pass through one point, then the lines are called concurrent lines. (Note: The common point of intersection is called point of concurrence)
5.
2
n
o P
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6.
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eometry
Parallel lines: The lines in a plane which are not intersecting are called parallel lines.
m
line || line m. D 7.
Parallel planes: Two non-intersecting planes are said to be parallel planes. Plane ABCD and plane PQRS are parallel planes.
C
A
B S
R Q
P
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. C Solution: i. We can draw three lines through three non collinear points A, B and C. ii. The lines are line AB, line BC and line AC.
B
A 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 points are collinear. ii. Three of these points are collinear. S Solution: R i. We can draw six lines (line PQ, line QR, line SR, line PS, line QS and line PR).
Q
P
ii.
We can draw four lines (line PR, line PS, line QS, line SR).
S
P
3. Observe the given figure and write the sets of all the points which are collinear. Solution: P S Set of collinear points are i. P, S, T, Q F ii. P, F, R, B iii. A, F, S, D R A iv. A, R, E, Q v. B, E, T, D
Lines and Angles
Q
R D T Q E B 3
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eometry
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D
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. ii.
C V
A
B S
R
P
Q
line DC, line PQ and line SR are parallel to line AB. 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 APQB 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).
Co‐ordinates of a point and Distance Co-ordinates of a point
R Q
The real number associated with a point on the number line is called as the co-ordinate of that point. Co-ordinate of point P is −1 and that of point B is 2.
P
O
A
B C
−3 −2 −1
0
1
2
3
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. P
Q
P
Q
y
x
x
y
( x > y) If
( x < y)
≥ y d(P, Q) = x − y x ≤ y d(P, Q) = y − x d(P, Q) = | x − y | x
∴
Thus, distance between any two distinct points is a unique non-negative real number.
Betweenness, segment and ray 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. 4
P
Q
R
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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.
eometry
A
B
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. The 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 seg AB ≅ seg CD.
A
B
C
D
Note:
i.
If we have to consider the length of segment AB, we write (AB) or 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. ii. iii.
Reflexivity: seg AB ≅ seg AB (Every segment is congruent to itself). Symmetry: If seg AB ≅ seg CD, then seg CD ≅ seg AB. 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). Note: Every line segment has one and only one midpoint.
∴
AM = BM =
1 2
M
A
B
AB
Comparison of segments: Suppose seg AB and seg CD are given. If AB < CD, then we say 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 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.
A
B C
A
D
B
P
Note: Point A is called as the origin of ray AB. The ray is a subset of a line. Opposite rays: Two rays which lie on a line having only the origin in common are called opposite rays. ray OA and ray OB are opposite rays.
Lines and Angles
A
O
B
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Exercise 1.2 1.
Observe the number line in the figure and answer the following questions.
E D
C
B
A O
−5 −4 −3 −2 −1
0
P
Q
R S
T
1
2
3
5
4
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 point 0. 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, 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.
∴ c.
∴
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 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 [since d(O, S) = 4 − 0 = 4]. b. The point can be towards the negative side i.e. point D [since d(O, D) = 0 − (−4) = 0 + 4 = 4].
The co-ordinates of two points P and Q are x and y respectively. Find d(P, Q) in the following cases. i. ii. iii. x = 8, = y = 3 iv. x = 5, y = 9 x = 7, y = 10 x = 2, y = 11 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 ∴ 2.
d(P, Q) = 3 6
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ii.
∴
eometry
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
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 betweeness does not exist among the points A, B & D.
ii.
∴ iii.
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 betweeness exists among the points A, B and D such that point D lies between A & B.
∴
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 betweeness exists among the points A, B and D such that point B lies between A & D.
4.
Draw the figure 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.
A
B
7.5 12 (AB) + (BC) = (AC)
∴
(AB)
+ 7.5 = 12
∴
(AB)
= 12 − 7.5
C ----- (A-B-C)
(AB) = 4.5
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eometry
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ii. R
T
S 2.15
(RS)
∴
3.75
+ (ST) = (RT)
----- (R-S-T)
2.15 + 3.75 = (RT) (RT) = 5.90
iii.
X
Y
Z
2 2 5 2 (XY) + (YZ) = (XZ)
∴
2 2 + (YZ) = 5 2
∴
(YZ)
----- (X-Y-Z)
= 5 2 − 2 2
(YZ) = 3 2 5.
In the adjoining figure, (LN) = 5, (MN) = 7, (ML) = 6, (NP) = 11, (MR) = 13, P
(MQ) = 2, then find (PL), (NR), (LQ).
Solution: (PL) +(LN) = (PN)
∴
(PL)
+ 5 = 11
∴
(PL)
= 11 − 5
L
----- (P-L-N) Q
R
M N
(PL) = 6 (MN)
+ (NR) = (MR)
∴
7 + (NR) = 13
∴
(NR)
------ (M-N-R)
= 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:
seg BD ≅ seg CE ≅ seg AC
∴ ∴ 8
(BD)
= (CE) = (AC) = 8
(AB)
+ (BC) = (AC)
(AB)
+5=8
R
P B
C
E
D
S
------ (Given)
------ (A-B-C)
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∴
(AB)
= 8 − 5
∴
(AB)
=3
------ (i)
(BC)
+ (CD) = (BD)
----- (B-C-D)
∴
5 + (CD) = 8
∴
(CD)
= 8 − 5
∴
(CD)
=3
------ (ii)
(CD)
+ (DE) = (CE)
----- (C-D-E)
∴
3 + (DE) = 8
∴
(DE)
= 8 − 3
∴
(DE)
=5
------ (iii)
∴
(BC)
= (DE)
------ [From given and (iii)]
eometry
seg BC seg DE (AB)
= (CD)
------ [From (i) and (ii)]
seg AB seg CD 7.
The co-ordinates of the points on the number line are as follows. Points Co-ordinates
P 3
Q 5
R 2
S 7
T 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 ∴ (seg PQ) = 8
ii.
∴ ∴ iii.
∴ ∴ iv.
∴ ∴ v.
∴
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 (seg PR) = 5
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 (seg PS) = 4
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 (seg PT) = 12
Co-ordinate of point Q is 5 and co-ordinate of point R is 2. 5>2 d(Q, R) = 5 − 2 = 3 (seg QR) = 3
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td. IX vi.
∴ ∴ vii.
∴
eometry
TARGET Publications
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 (seg QS) = 12
Co-ordinate of point Q is 5 and co-ordinate of point T is 9. 9>5 d(Q, T) = 9 − 5 = 4 (seg 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 ∴
∴ ix.
∴
(seg RS) = 9
Co-ordinate of point R is 2 and co-ordinate of point T is 9. 9>2 d(R, T) = 9 − 2 = 7 (seg 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 (seg 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. 1 AP = AB ----- (By definition) ∴ 2 1 AP = × 7 ∴ 2 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. CD = 2(CQ) ----- (By definition) ∴ CD = 2 × 4.5 (CD) = 9 10. If AB = 7 cm, BP = 4 cm, AP = 5.4 cm, compare the segments. Solution: AB = 7 cm, BP = 4 cm, AP = 5.4 cm 7 > 5.4 > 4 seg AB > seg AP > seg BP. 10
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11.
A
In the figure, length 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. ∴
eometry
5 cm
5 cm B
4 cm
D
5.5 cm C
5.5 cm
4 cm E
Plane separation axiom and angles Plane separation axiom: Given a line in a plane, the points in the plane that do not lie on the line form disjoint sets H 1 and H2. Each set H 1 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. Angle: An angle is the union of two non-collinear rays having the same origin. The origin is called the vertex of the angle. Each ray is known as the arm or side of the angle. An angle is obtained by rotating a ray about its origin.
H1 P
H2
B
A
O Initial arm B
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.
m∠AOB = θ
θ
O
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.
A B
A
O
B
Note:The directed angle BOA is not the same as directed angle AOB.
∠BOA ≠ ∠AOB Positive angle: Anticlockwise directional movement of ray OA about ‘O’ is regarded as positive angle.
O
∠AOB is a positive angle. Negative angle: Clockwise directional movement of ray OA about ‘O’ is regarded as negative angle.
O
Initial arm
Initial arm
A
A
∠AOB is a negative angle. B
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eometry
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System of measuring an angle:
There are two systems of measuring an angle. i. Sexagesimal system (Degree measure) i.
ii.
Circular system (Radian measure)
Sexagesimal system (Degree measure)
The unit of measurement of angle is degree.
⎛ 1 ⎞ th ⎜ 360 ⎟ part of a complete rotation is one degree, denoted by 1 °. ⎝ ⎠ ⎛ 1 ⎞ th ⎜ 60 ⎟ part of a degree is called one minute, denoted by 1'. ⎝ ⎠ ⎛ 1 ⎞ th ⎜ 60 ⎟ part of a minute is called one second, denoted by 1''. ⎝ ⎠ ∴
1° = 60' 1' = 60'' Note: 90° = 89° 59' 60'' 180° = 179° 59' 60'' Minutes and seconds used in angle measurement are different from those used in time measurement.
One complete rotation:
Suppose an intial 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.
O
A
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. Zero angle:
If there is not 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.
B O
A
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.
A B
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 of directed angles co-terminal with a given directed angles. The measures of co-terminal directed angles differ by an integral multiple of
B
O 40°
400°
A
360°. C
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 AB ⊥ line CD 12
A
O
B D
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eometry
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.
C
A
O
C
C B
D ray OC ⊥ ray OB
A
O
B
A
O
D
B D
seg CD ⊥ seg AB
line CD ⊥ ray OB
Note: Point O is called as the foot of the perpendicular drawn from a Point C to line AB. Congruent angles: If the measures of two angles are equal, then the angles are called congruent angles. Since, m∠ABC = m∠PQR ∴ ∠ABC ≅ ∠PQR
A
B
P
30°
30°
C Q
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. Inequality of angles: A If the measure of one of the angles is greater than that of the other, then that angle is said to be greater than the other. 110° since, m∠ABC > m∠PQR ∴ ∠ABC > ∠PQR
P
60°
B
Q
C
Types of angles
R
A
Acute angle: If the measure of an angle is less than 90°, then it is called an acute angle.
O
50° B
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.
R
A
O
B
A 110° O
Straight angle: If the measure of an angle is 180°, then it is called a straight angle.
A
O
B
are disjoint.
Lines and Angles
B
A
Adjacent angles: Two angles are called adjacent angles, if they have a common side and their interiors
∠AOB and ∠COB are adjacent angles.
B
O
C
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eometry
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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.
A
∠AOB and ∠AOC form angles in a linear pair. B
Note:
O
C
Sum of the measures of the angles in a linear pair is 180°. m∠AOB + m∠AOC = 180° Vertically opposite angles: Two angles are called vertically opposite angles, if their sides form two pairs of opposite rays.
A
C O
∠AOD and ∠BOC, ∠AOC and ∠BOD form pairs of vertically opposite angles.
B
D
Complementary angles: A If the sum of the measures of the two angles is 90°, then these angles are called 30° complementary angles.
P B
m∠AOB + m∠PQR = 90°. O Supplementary angles: If the sum of the measures of the two angles is 180°, then these angles are called supplementary angles.
60°
Q
P
A 120°
m∠AOB + m∠PQR = 180° B
O
R
60° Q
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 (greater than 90°) each. Their sum is also greater than 90° (not ≠ 90°). iii. Can two right angles be complement to each other? Solution: No, consider two right angles (equal to 90°) each. Their sum is 180° ≠ 90°. iv. Can two acute angles be supplementary? Solution: No, consider two acute angles (less than 90°) each. Their sum is less than 180 ° (≠180°). v. Can two obtuse angles be supplementary? Solution: No, consider two obtuse angles (greater than 90°) each. Their sum is greater than 180° ( ≠180°). vi. Can two right angles be supplementary? Solution: Yes, consider two right angles (equal to 90°) each. Their sum is 180°.
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Lines and Angles
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td. IX -
TARGET Publications
∴ ∴ ∴
∠VOR + ∠VOT = ∠ROT 80° + ∠VOT = 122° ∠VOT = 122° – 80° ∠VOT = 42° ∠QOP = ∠VOT
eometry
------ (Angle addition property)
------ (Vertically opposite angles)
QOP = 42° 11.
In the given figure, if BOD = 60° and AOB Solution: ∠AOC ≅ ∠BOD ∴ ∠ΑΟC = 60° ∠BOC + ∠AOB = ∠AOC ∴ ∠BOC + 45° = 60° ∴ ∠BOC = 60° − 45°
AOC BOD, COD, then find
AOB = 45°, BOC.
B
A
C
------ (Given) ------ (Since ∠BOD = 60°)
D O
BOC = 15° 12. If XPY = 135°, XPZ = 175° and the point Y is in the interior of Solution: ------ (Angle addition property) ∠YPZ + ∠XPY = ∠XPZ ∴ ∠YPZ + 135° = 175° ∴ ∠YPZ = 175° − 135° YPZ = 40° 13.
If
ZPX, then find
YPZ.
Y 175° 135°
Z
P
X
POR = 120° and P-O-L, the points S and T be on the R-side of line PL, such that
and ROT TOP. Draw the figure and find TOS. Solution: P ------ (Angles in a linear pair) ∠POR + ∠LOR = 180° 120° + ∠LOR = 180° ∴ ∴ ∠LOR = 180° − 120° ∴ ∠LOR = 60° ------ (Given) ∠ROS ≅ ∠SOL ray OS bisects ∠LOR. ∴ 1 ∴ ∠ROS = ∠LOR 2 1 ∴ ∠ROS = × 60° 2 ∴ ∠ROS = 30° ------ (Given) ∠ROT ≅ ∠TOP Ray OT bisects ∠POR. 1 ∴ ∠ROT = × ∠POR 2 1 ∴ ∠ROT = × 120° 2 ∴ ∠ROT = 60° ------ (Angle addition property) ∠TOS = ∠ROT + ∠ROS ∴ ∠TOS = 60° + 30°
ROS
SOL
T
120° ° O °×
R
×
S L
TOS = 90°
Lines and Angles
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