rayat shikshan sanstha,karmaveer vidyaprabodhini,madhya vibhag,satara. std-9 th sub- maths...

Rayat Shikshan Sanstha,Karmaveer Vidyaprabodhini,Madhya vibhag,Satara.

Std-9th Sub- Maths (Geometry) Topic- Lines and angles



Our Inspiration

Dr. Karmaveer Bhaurao Patil



Presented By

Mr. Mahamulkar S.M.B.Sc.B.Ed.(A)



Sub.:- MathsStd.:- 9th



Topic

Lines and angles



Sub- Topics

1.1. Basic Geometric Concepts = Slide 7-Slide 171.2. Some Definations = Slide 18- Slide 27



* Basic Geometrical Concepts: In geometry, a point, a line and a plane are undefined terms which

are basic concepts. Lines and planes are sets of points.Each line and each plane contain infinite number of points. Axiom/ Postulate: The simple proper ties which we accept as true

are called axioms or postulates. i.e., the basic facts without proof are axioms or postulates.Theorem: The theorem is a statement which can be proved logically

from the axioms.



1. An infinite number of lines can be drawn through a given point.

Point P is given. Line l, line m, line n, ... pass through the point P.

1.1 Some axioms

nl

m

Py



2. There is one and only one line passing through two distinct points.

A line passing through the points A and B is denoted by line AB.

Meaning of one and only one: One shows existence. At least one means one or more. One and only one means exactly one or uniqueness.

A B



3. When two distinct lines intersect, their intersection is exactly one point.

Line l and line m intersect in the point P .

ml

P



4. There is exactly one plane passing through three non-collinear points. A, B, C are three non-collinear points. Plane E passes through these points.

C B

A

E



5. There is exactly one plane passing through a line and a point not on the line. Plane E is determined by line land the point P which is outside the line.

P

lE



6. There is exactly one plane passing through two distinct intersecting lines. Line 1 and line m intersect in the point P.

Plane E passes through these intersect ing lines.

l

E

m

P



7. When two planes intersect, their intersection is exactly one line. Line l is the intersection of intersecting planes E and F.

lE

F



8.When a line intersects a plane but does not lie in it, then their intersection is a point.

Point P is the point of intersection of the line l and the plane E.



9. A line containing two given points of a plane lies wholly in that plane.

B

E

A



10. If the lines lie in the same plane, then they are called coplanar lines, otherwise they are called non-coplanar lines.

m

l

nm

l

n

EF

Coplanar LinesNon-Coplanar Lines



Some Definitions :-

•Collinear and non-collinear points: If a line passes through three or more distinct points, then the points are called collinear points, otherwise they are called non-collinear points.

In the figure, points A, B, C, D are collinear points.

P, Q, R; P, R, S; Q, R, S; etc. are the sets of non-collinear points.

RP S Q

DCBA



2. Coplanar and non-coplanar points: If four or more non-collinear points lie in the same plane, they are called coplanar points, otherwise they are called non-coplanar points. In the figure A, B, C are coplanar points. PointsT, H, U, A are non-coplanar points.

C B

A

E

T

U

H



3. Concurrent lines: If three or more lines pass through one point, the lines are called concurrent lines. The common point of intersection is called the point of concurrence.

nl

m

pO



In the: figure, l, m, n, r are concurrent lines. Point P is the point of concurrence.

In the above figures, lines l, m, n; lines x, y, z are not con,current lines.

x y



4. Parallel lines: The lines in a plane which are not intersecting are called parallel lines. Parallel lines are always coplanar . In the figure, line lliline m (' II ' symbol for parallel)

l

m

F



5. Parallel planes: Two non-intersecting planes are said to be parallel planes. In the figure, planes ABCD and EFGH are parallel planes.

D C

E F



Every real number can be represented uniquely by a point on a number line. No two distinct points have the same number associated with them. No two distinct numbers are associated with the same point on a number line. Coordinate of a point: The real number associated with a point on the number line is called the coordinate of that point. The distance between two points: Let x and y be the coordinates of the points P and Q respectively. The distance between the points P and Q is the absolute value of the difference between the numbers associated with the point P and Q.

Symbolically, d(P,Q)=❘x-y❘



d(P, Q)=x-y, if x >y and d(P, Q)=y-x, if x<y. So, d(P, Q) = IX-YI.

On the number line, N L D 0 A I I I I I

-3 -2 -1 0 1 2 3

(1) the coordinates of the points A and Bare 1 and 3 respectively. ... d(A, B)=11-31=1-21=2. (2) the coordinates of the points D and Tare -1 and 2 respectively. ... d (D, T) = I - 1 - 21 = I - 31 = 3 (3) the coordinates of the points Nand L are - 3 and - 2 respectively. ... d(N, L)=1-3 - (-2)1=1-3+21=1-11=1.

Hence, the distance between any two distinct points is unique non-negative real number. If P and Q are two distinct points, d(P, Q) > O.

Note that d(P, Q) =d(Q, P). If the distance between two points is zero, the points are not distinct. e.g., if d(P, Q) = 0, points P and Q are not distinct. They are one and the same

points.

In fig. (i), d(P, Q)+d(Q, R)=d(P, R). Symbolically: P-Q-R.

In fig. (ii), d(P, R)+d(R, Q)=d(P, Q). Symbolically: P-R-Q.

In fig. (iii), d(Q, P) + d(P, R) = d(Q, R). Symbolically: Q-P-R.

If P, Q, R are any three distinct collinear points and P-Q-R,then the point Q is between the two points P and R.



There exist a relation of betweenness among three distinct points. If the distinct points P, Q and Rare collinear points, there exist three possibilities as shown below:

RQP

R QP

RQ P



1. Take any three non-collinear points A, B,C on a paper. How many lines can you draw through different pairs of points? Name the lines. Ans. Three lines can be drawn. Line l through the points A and B. Line m through the points Band C. Line n through the points A and C.

Exercise 1.1

A C

B

m

n

l



4. Observe the figure. Answer the follow ing questions :

(i) Name the lines parallel to the line AB. (ii) Can you say P Q

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 the point A. (v) Name the points which are such that the plane containing them does not contain points P, Q, C and D. Ans. (i) line DC, line SR and line PQ are parallel to the line AB. (ii) Yes. Line AD and point R lie in the same plane. Point R is not on line AD. There is exactly one plane

passing through a line and a point not on it. (iii) Yes. Points A, S, B, R are coplanar. These points are contained in the plane ASRB. (iv) Plane APSD, plane ABCD and plane APQB pass through the point A. (v) Points A, S, R; A, S, B; A, V, S; B, V, S; etc.



(1) The set of points A and B and all the points between A and B on the line AB is called segment AB and is writ ten as seg AB (or seg BA).

(2) The points A and B are called the end points of seg AB.

(3) A line segment is a subset of a line. (B)

Line segment:-

BA



(B) Length of a line segment: (1) The distance between the end points of a line segment is called the

length of the segment. (2) The length of a segment AB is denoted by I (AB) or merely by AB.

(3) Thus, I (AB) = d (A, B) = AB. (4) The symbol AB is a number .



(C) Congruent segments:

Two line segments are said to be congruent, if they are ofthe same length. If I (AB) = I (CD), then seg AB and seg CD are congruent segments. Symbolically, seg AB ~ seg CD. [Note: If we consider the length of segment AB, we write only AB. If we consider the set ofpoints between A and B, we write seg AB or side AB. ]

11DC

BA



• Properties of congruent segments: (i) Reflexivity: Every segment is congruent to itself.

seg AB ~ seg AB.

(ii) Symmetry: If the first segment is congruent to a second one, then the second segment is congruent to the first one. If seg AB ~ seg CD, then seg CD ~ seg AB.

(iii) Transitivity: If the first segment is congruent to a second one and the second segment is congruent to a third one, then the first segment is congruent to the third one. If seg AB ~ seg CD and seg CD ~ seg PQ, then seg AB ~ seg PQ.



(D) 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), Le., seg AM ~ seg BM. Every line segment has one and only one midpoint.

BA M

(E) Comparison of segments : Segments AB and CD ",--------D are given. If AB < CD, then we say that seg AB is smaller than seg CD. This we write as seg AB < seg CD.

11

D

CBA



F) Ray: (i) Let A and B be two given points. Then the 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 the ray AB. (ii) The point A is called the origin of the ray AB. (iii) Ray AB and ray BA are different rays because their origins are different.



(G) Opposite rays: Two rays which lie on a line and having the same origin are called opposite rays. In the figure, ray AB and ray AC are opposite rays. The intersection of ray AB and ray AC is the point A.

BC A

The union of two opposite rays is a line.



1) Write the coordinates of the points C, S, Q,D.

Points C S Q D

Coordinates -3 4 2 -4

Exercise 1.2.

(2) Find d(Q, T), d(E, B), d(O, C), d(O, R).

Ans.- d (Q, T). The coordinate of Q is 2 and the coordinate of T is 5.

.'. d(Q, T) = l 2-5 l = l -3 l= 3 d (E, B). The coordinate of E is - 5 and the coordinate of B is - 2.

... d(E, B)= l -5-(-2) l = l -5+2 l = l -3 l =3. d (0, C). The coordinate of 0 is 0 and the coordinate of C is - 3.

... d(0,C)= l 0-(-3) l = l 0+3 l = l 3 l =3. d(O, R). The coordinate of 0 is 0 and the coordinate of R is 3.

... d(0,R)= l 0-3 l = l -3 l =3.



3. In the figure, l (LN) = 5, I (MN) = 7, .l(ML) = 6, l(NP) = 11, l(MR) = 13, l(MQ) = 2. Find l (PL), l (NR), l (LQ).

Solution: P-L-N ':. d(P, L)+d(L, N) = d(P, N) :. d(P,L)+5=11 :. d(P,L)=1l-5=6 Ans. d(P, L) = l (PI,.) = 6. M-N-R:. d(M, N)+d(N, R)=d(M, R)

... 7+d(N,R)=13 :. d(N,R)=13-7=6 Ans. d(N, R) = l(NR) = 6. L-M-Q .'. D(L, M) +d(M, Q)=d(L, Q)

... 6 + 2 = d (L, Q) :. d (L, Q) = 8

Ans. d (L, Q) = l(LQ) = 8.

P

L

MR

N



4. If Q is the midpoint of seg CD and d (C, Q) = 4.5, imd the length of CD. Solution: Q is the midpoint of seg CD. :. C-Q-D

seg CQ ~ seg DQ. d(C, Q)=d(d, Q)=4.5 d(C, Q)+d(Q, D)=d(C, D)

:. 4.5+4.5=d(C, D) d(C, D)=9 :. I (CD) = 9 Ans. The length of CD is 9.



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 of these sets is called a half plane and the line is called the edge of the half plane.

IfP is a point in any of the half plane, then that half plane is called P side of the half plane.

Plane Separation Axiom:-



Angle :-

(1) 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 the same angles.

(2) Extended concept of the angle: The angles may have positive, negative measures. Such an angle is a directed angle.

B

A

O



Angles in terms of Rotation :- The two hands in a clock make angle with each other. The measure of this

angle depends on the rotation of the hand. An angle is obtained by rotating a ray about its end point. The rotation of a ray may be. in the anticlock wise direction. [Fig. (i)]. In this case the angle is regarded as positive angle.The rotation of a ray may be in the clockwise direction. [Fig. (ii)]. In this case the angle is regarded as negative angle.The amount of rotation of the ray from the initial position to the terminal position is called the measure of the angle.

Terminal si

de

Initial side

Terminal side

Initial side



(1) Definition: The ordered pair of rays (ray OA and ray OB) together with the rotation of ray OA to occupy the position of ray OB is called the directed angle AOB and denoted by AOB. In the ordered pair of ray OA and ray OB, the first element ray OA is called the initial arm and the second element ray OB is called the terminal arm of AOB.

The ordered pair of the directed angle BOA is ray OB and ray OA. Thus the directed angle BOA is not the

same as the directed angle AOB.

Directed Angle :-

Terminal si

de

Initial side AO

B



(2) Positive angle: In the figure, ray OA rotates about 0 in

anticlockwise direction making AOB.

AOB is a positive angle.

3) Negative angle: In the figure, ray OA rotates about 0 in clockwise direction making AOB. AOB is a negative angle.

A



One Complete Rotation :-

Suppose the initial ray OA is rotated about its end point 0 in the anticlockwise direction and takes the final position OA again for the first time, then the ray OA is said to have formed one complete rotation. The angle traced during one complete rotation in anticlockwise direction is 360°. The angle traced during two complete rotations in anticlockwise direction is 720° (2 x 360°) and so on.

BO



Systems of Measuring an Angle :-There are two systems of measuring an angle (i)Sexagesimal system (Degree measure) (ii)(ii) Circular system (Radian measure) We will study only sexagesimal system. System of measurement of angles: (Degree measure) In this system, the unit of measurement of angles is degree which is th part of one 360 comple.te rotation. One degree is denoted by 1°.

th part of a degree is called one minute and is denoted by 1'. th part of one minute is called one second and is denoted by 1".

1 ° = 60'; l' = 60" .



Zero Angle, Straight Angle :-

(1) Zero angle: If there is no rotation of initial ray OA, then the directed angle so

formed is called a zero angle.

B

O

A

(2) Straight angle: If for the directed angle AOB, the ray OA rotates in the anti clock wise direction, so that the initial arm i.e., ray OA and the terminal arm, i.e., ray OB are opposite rays, then the angle so formed is called the straight angle.

,

AO



(3) Reflex angle: If the initial ray OA rotates about 0 in the anticlockwise direction and takes the final position OB before coinciding the ray OA again, such that its degree measure lies between 180° and 360°, then the directed angle AOB is called a reflex angle.

A

O

B

4) Coterminal angles: The directed angles of different measures having the same position of initial rays and the terminal rays are called coterminal angles. The measures of coterminal directed angles differ by an integral multiple of 360°.



Exercise 1.3.

•Answer the following questions and justify: (i) Can two acute angles be complement to each other? Ans. Yes. If the sum of their measures is 90°. Acute angles of measure 30° and 60° are complement to each other. (ii) Can two obtuse angle be complement to each other? Ans. No. The sum of the measures of complementary angles is 90'j, while the measure of an obtuse angle is greater than 90°. So, the measure of two obtuse angles will be greater than 180°. (iii) Can two right angles be complement to each other? Ans. No. The sum of the measures of complementary angles is 90°, while the



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 ... AOC = 72° ... (Given: BOE = 72°) ... (1

AOC + BOC = 180° ... (Angles forming a linear pair ..72° + BOC = 180° [From (1)

... BOC = 180° _72° = 108° ... (2 AOE= BOC ... (Vertically opposite angles ... AOE = 108° ... [From (2)



Perpendicularity :-

(a) Perpendicular lines: 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. The statement, line AB is perpendicu lar to line CD, is written as

line AB ⊥. line CD.

A

B

DC O



(b) Perpendicularity of rays and segments: 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. In each of the above figures, the angle at the point 0 is a right angle. The point o is called the foot of the perpendicular drawn from the point A on the line CD.



If the measures of two angles are equal, the angles are called congruent angles. If ABC and PQR are congruent, we write it as

ABC PQR

Congruent Angles :-

• Properties of Congruent Angles:

(i) Reflexivity: Every angle is congruent· to itself, e.g., ABC ~ ABC. (ii) Symmetry: If the first angle is congruent to a second angle, then the second

angle is congruent to the first one. e.g., if ABC ~ PQR, then PQR ~ ABC. (iii) Transitivity: If the first angle is congruent to a second angle and the second

one is congruent to a third angle, then the fIrst angle is congruent to the third angle,

e.g., if ABC ~ PQR and PQR ~ LMN, then ABC ~ LMN .



* Inequality of Angles: If the measure of one of the angles is greater than

that of the other one, then the angle with greater measure is said to be greater than the angle with smaller measure.

If m∠ABC = 47° and m∠ PQR = 74°, then

∠ PQR > ∠ ABC.



(i) Acute angle (MeasU!e of the angle is less than 90°) (ii) Right angle (Measure of the angle is 90°)

(iii) Obtuse angle (Measure of the angle is greater than 90° but less than 180°)

(iv) Straight angle (Measure of the angle is 180°)

Types of Angle :-



Pairs of Angle :-Q T

(i) Adjacent angles: Two angles are called adjacent angles if they have (a) common vertex (b) a common side and their interiors are disjoint.

In the adjoining figure, ∠PQR and ∠RQT are adjacent angles.

TO



(ii) Linear pair of angles: ∠ MPN and ∠RPN are adjacent angles. Their non common sides PM and PR form a pair of opposite rays. Hence, these angles are linear pair of angles. The angles in a linear pair are adjacent angles, but adjacent angles may not form a linear pair. Linear pair axiom: The sum of the measures of the angles in a linear pair is 180°.

RM P

N



(iii) Supplementary angles: If the sum of the measures of two angles is 180°, then these angles are called supplementary angles. Each of them is said to be the supplement of the other. (iv) Complementary angles : If the sum of the measures of two angles is 90°, then these angles are called complementary angles. Each of them is said to be the complement of the other.

MM



(v) Vertically opposite angles : Two angles are called vertically opposite angles, if their sides form two pairs of opposite rays. In the above figure, the pair of angles, ∠PMS, ∠ RMQ and ∠ PMR, ∠ SMQ are vertically opposite angles.

R

P

Q

S

M



Geometric sentences and their profiles:-

Conditional sentences : Any sentence stated in 'when-then' 0: 'if-then' form is said to be a

conditional sentence. The part of the sentence which follow 'when' or 'if' is called an antecedent and the which follows' then' is called a consequent. statement obtained by interchangin, antecedent and consequent is called th converse of the original statement. Proof: In geometry, proof of theorems are dividel into two types : (1)Direct proof and (2)Indirect proof. (1) Direct proof: If from antecedent w reach up to consequent using axioms

0previously proved theorems, then it is a direct proof.



(2) Indirect proof: In this method, we suppose that the consequent is false and proceed logically and arrive at a step which contradicts either what is given (antecedent) or some well known fact. Conclusion regarding the truth of the consequent is reached indirectly. Corollary : A property that can be easily derived from a

theorem is called corollary.



• Theorem 1.1: If two lines intersect each other, then vertically opposite angles are congruent. Given: Line AB and line CD inter sect each other at the point O. To prove: (i) AOC BOD (ii) BOC AOD. Proof: AOC + AOD = 1800 ... (Angles in a linear pair) ... (1) BOD + AOD = 1800 ... (Angles in a linear pair) ... (2) From (1) and (2), AOC + AOD = BOD + AOD AOC = BOD i. e., AOC BOD Similarly, we can prove that BOC AOD.



(a)Parallel lines: Non-intersecting coplanar lines are parallel lines.

(b) Transversal and intercept: Transversal : A line intersecting two other coplanar lines in two distinct

points is called a transversal of the two lines. In the figure, line a and line b are the transversal of lines c and d. Line c and

line d are the transversals of lines a and b.

Parallel Line:-

Intercept : A segment cut off by a transversal on two distinct lines is called an intercept. In the figure, seg PQ is the intercept

made by the transversal a on the lines c and d. Seg PR is the intercept made by the

line d on the lines a and b.



(c) Angles formed by the transversal: In the figure, line n is the transversal of lines land m. a, b, c and d are formed at the point A. e, f, g and h are formed at the point B. Thus, eight angles are formed. Of them, one side or ray of each angle is contained in the transversal. Taking one angle from each of these two groups- (1) Four pairs of corresponding angles are formed: (i) a, e (ii) b, f (iii)

c, g (iv) d, h. (2) Two pairs of alternate angles are formed: (i) c, e (ii) \ b, h. (3) Two pairs of interior angles are formed: (i) c, h (ii) b, e.

aa



• Tests for Parallel Lines Alternate angles test : Theorem 1.2 : 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 two coplanar lines l and m in points P and Q respectively and ∠ APQ ≅ ∠ PQD.

To prove : Line l // line m. Proof(Indirect proof) Suppose line l is not parallel to line m. i.e., coplanar lines AB and CD are not parallel Then, they must intersect. Suppose they intersect in point R. Now, points P, Q and Rare non collinear points.

.'. we get ∆ PQR. ∠ APQ + ∠ R PQ= 180° ... (Angles in a linear pair) ... (1) ∠ RPQ + ∠ PQR + ∠ PRQ = 180° ... (The sum of the measures of the angles of a triangle) ... (2) From (1) and (2), ∠ APQ+ ∠ RPQ = ∠ RPQ + ∠ PQR + ∠ PRQ :. ∠ APQ= ∠ PQR+ ∠ PRQ :. ∠ APQ > ∠ PQR i.e., ∠ APQ > ∠ PQD

.'. it contradicts the given, because ∠ APQ= ∠ PQD. So, our assumption that line l is not parallel to line m is wrong. .'. line l // line m.

PA

Q

B

F

C

E

Dm

l

RC

A

DE

B

QP



• Corrsponding angles test: Theorem 1.3 : If a pair of corresponding angles

formed by a transversal of two coplanar lines is congruent, then the lines are parallel.

Given: Line n is the transversal of two coplanar lines I and m and EPB ~ PQD.

To prove: Line I line m. Proof: EPB ~ PQD ... (Given) ... (1) EPB ~ APQ ... (Vertically

opposite angles) ... (2) From (1) and (2), APQ ~ PQD

But these are alternate angles. .'. line I line m ... (Alternate angles test for

parallel lines)

A B

C

Q

P

D

F

E



Interior angles test : Theorem 1.4 : If a pair of interior angles formed by a transversal of two coplanar lines is supplementary, then the lines are parallel. Given: Line n is the transversal of two coplanar lines I and m and APQ + PQC = 180°. To prove : Line I line m. Proof: APQ + PQC = 1800 ... (Given) ... (1) PQC+ PQD = 1800 ••• (Angles in a linear pair) ... (2)

A B

C

Q

P

D

F

EFrom (1) and (2), APQ + PQC = PQC + PQD. :. APQ = PQD But these are alternate angles . . '. line I line· m ... (Alternate angles test

for parallel lines )



• Converse of alternate angles test: Theorem 1.5: If two lines are parallel, then alternate angles formed by a transversal are congruent. Given: Line AB line CD. Transversal EF intersects them in the points P and Q respectively.

To prove: APQ ~ PQD. Proof: (Indirect proof) Suppose APQ is not congruent to PQD. Then, draw a line ML passmg through the point P such that MPQ ~ PQD. Line ML line CD ... (Alternate angles test for parallel lines ) But line AB line CD .,. (Given) This means that there are two parallel lines~ parallel to line CD and passing through a point P

outside the line CD. This is contradictory to the uniqueness of parallel lines our assumption is wrong. APQ ~ PQD.

A B

C

Q

P

D

F

E



Converse of corresponding angles test: Theorem 1.6: If two lines are parallel, then the corresponding angles formed by a transversal are congruent.

Given: Line l line m and line EF is 1 transversal. To prove: EPB ~ PQD.

Proof: Line AB line CD ... (Given):. APQ ~ PQD ... (Alternate angles) ." APQ ~ EPB .. . (Vertically opposite angles) ... ' From (1) and (2), EPB ~ PQD .

A B

C

Q

P

D

F

E



• Converse of interior angles test: Theorem 1.7: If two lines are paralll then the interior angles formed by transversal are supplementary. Given: Line AB II line CD and line EF is tl transversal. To prove: BPQ + PQD = 180°. Proof: Line AB line CD ... (Give n) APQ ~ PQD ... (Alternate angles) ... (1) BPQ + APQ = 180°... (Angles in a linear pair) ... (2) From (1) and (2), BPQ + PQD = 180°.

A B

C

Q

P

D

F

E



Corollary 1 : If two lines are parallel t the same line, then they are parallel to each other. Given: Three lines l, m, n are coplanar. Line l line m and line l line n. To prove: Line m line n.

Construction : Draw a transversal q, intersecting the lines I, m and n in the points P, Q and R respectively.

Proof: Line l line m ... (Given) Line q is a transversal ... (Construction) ... GPB ~ PQD ... (Corresponding angles) ... (1) Line l line n ...

(Given)Line q is a transversal ... (Construction) ... GPB ~ QRF...

(Corresponding angles) ...(2) From (1) and (2), PQD ~ QRF.But, this is a pair of corresponding angles formed by

transversal q intersecting line m and line n.... line m line n ... (Corresponding angles test for

parallel lines )

A B

C

Q

P

D

FE R

M

G



Corollary 2 : If a line coplanar with two parallel lines is perpendicular to one of them, then it is also perpendicular to the other.Given: Line l // line m. Line q is the transversal intersecting them in points P and Q respectively. Line q line 1. To prove: Line q line m. Proof: Line l line m and line q is the

transversal. ... (Given) EPB ~ EQD... (Corresponding angles) (1) transversal. ... (Given) EPB ~ EQD... (Corresponding angles) (1)

EPB = 90° ... (Given) (2)

From (1) and (2), EQD = 90°.... line q line m.

A B

C

Q

P

D

F

E

l

m

q



Corollary 3 : If a line is perpendicular to two coplanar lines, then those two lines are parallel to each other. Given: Line q 1. line I and line q line mTo prove: Line I II line m . Proof: Line q line I ... (Given) (1) ... EPB=90° [From(l)] (2)

Line q line m (Given) (3)... EQD = 90° [From (3)] (4) From (2) and (4), EPB= EQD.But these are corresponding angles . Line l line m ... (Corresponding angles test for parallel lines) (i) If a transversal A B

C

Q

P

D

F

E



(i) If a transversal intersects two parallel lines, then state the relation between alternate angles. Ans. Both the pairs of alternate angle are congruent. (ii) If each of the two lines are parallel to the third line, then what is the relation between them? Ans. The three lines are parallel to one another. (iii) If a transversal intersects two parallel lines and the correspond ing sides of two angles are parallel, then what is the relation between these angles? Ans. The two angles are congruent.

Exercise 1.4.



In the figure, line 1 II line m and line p is the transversal. If r = 20°, then find a: b. [Figure is not to exact measurement.] Solution: r = a

'" (Vertically opposite angles) '" (Given)

a + b = 180° 20° + b = 180° b = 160° . 20° 1

: b = 20° . 160° = - = - = 1 . 8 r = 20° a = 20°

Line [II line m and line p is the transversal. '" (Given) ... (Interior angles)

.. , b = 180° - 20°

rayat shikshan sanstha,karmaveer vidyaprabodhini,madhya vibhag,satara. std-9 th sub- maths...

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