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Lines
&Angles By - NM Spirit
T- 901-568-0202
Email-
nm.sspirit@gmail.com
Point
An exact location on a plane is
called a point.
Line
Line
segment
Ray
A straight path on a plane,
extending in both directions
with no endpoints, is called a
line.
A part of a line that has two
endpoints and thus has a
definite length is called a line
segment.
A line segment extended
indefinitely in one direction
is called a ray.
Recap Geometrical Terms
RAY: A part of a line, with one endpoint, that continues
without end in one direction
LINE: A straight path extending in both directions with no
endpoints
LINE SEGMENT: A part of a line that includes two
points, called endpoints, and all the points between
them
INTERSECTING LINES: Lines that cross
PARALLEL LINES: Lines that never cross and are always
the same distance apart
Common
endpoint
B C
B
A
Ray BC
Ray BA
Ray BA and BC are two non-collinear rays
When two non-collinear rays join with a common endpoint
(origin) an angle is formed.
What Is An Angle ?
Common endpoint is called the vertex of the angle. B is the
vertex of ABC.
Ray BA and ray BC are called the arms of ABC.
To name an angle, we name any point on one ray, then the
vertex, and then any point on the other ray.
For example: ABC or CBA
We may also name this angle only by the single letter of the
vertex, for example B.
A
B
C
Naming An Angle
An angle divides the points on the plane into three regions:
A
B
C
F
R
P
T
X
Interior And Exterior Of An Angle
• Points lying on the angle
(An angle)
• Points within the angle
(Its interior portion. )
• Points outside the angle
(Its exterior portion. )
Right Angle:
An angle that forms a
square corner
Acute Angle:
An angle less
than a right angle
Obtuse Angle:
An angle greater
than a right angle
Two angles that have the same measure are called congruent
angles.
Congruent angles have the same size and shape.
A
B
C
300
D
E
F
300
D
E
F
300
Congruent Angles
Pairs Of Angles : Types
• Adjacent angles
• Vertically opposite angles
• Complimentary angles
• Supplementary angles
• Linear pairs of angles
Adjacent Angles
Two angles that have a common vertex and a common ray
are called adjacent angles.
C
D
B
A
Common ray
Common vertex
Adjacent Angles ABD and DBC
Adjacent angles do not overlap each other.
D
E
F
A
B
C
ABC and DEF are not adjacent angles
Vertically Opposite Angles
Vertically opposite angles are pairs of angles formed by two
lines intersecting at a point.
APC = BPD
APB = CPD
A
DB
C
P
Four angles are formed at the point of intersection.
Point of intersection ‘P’ is the common vertex of the four
angles.
Vertically opposite angles are congruent.
If the sum of two angles is 900, then they are called
complimentary angles.
600
A
B
C
300
D
E
F
ABC and DEF are complimentary because
600 + 300 = 900
ABC + DEF
Complimentary Angles
If the sum of two angles is 1800 then they are called
supplementary angles.
PQR and ABC are supplementary, because
1000 + 800 = 1800
RQ
P
A
B
C
1000
800
PQR + ABC
Supplementary Angles
Two adjacent supplementary angles are called linear pair
of angles.
A
600 1200
PC D
600 + 1200 = 1800
APC + APD
Linear Pair Of Angles
A line that intersects two or more lines at different points is
called a transversal.
Line L (transversal)
BA
Line M
Line N
DC
P
Q
G
F
Pairs Of Angles Formed by a Transversal
Line M and line N are parallel lines.
Line L intersects line
M and line N at point
P and Q.
Four angles are formed at point P and another four at point
Q by the transversal L.
Eight angles are
formed in all by
the transversal L.
Pairs Of Angles Formed by a Transversal
• Corresponding angles
• Alternate angles
• Interior angles
Corresponding Angles
When two parallel lines are cut by a transversal, pairs of
corresponding angles are formed.
Four pairs of corresponding angles are formed.
Corresponding pairs of angles are congruent.
GPB = PQE
GPA = PQD
BPQ = EQF
APQ = DQF
Line M
BA
Line N
D E
L
P
Q
G
F
Line L
Alternate Angles
Alternate angles are formed on opposite sides of the
transversal and at different intersecting points.
Line M
BA
Line N
D E
L
P
Q
G
F
Line L
BPQ = DQP
APQ = EQP
Pairs of alternate angles are congruent.
Two pairs of alternate angles are formed.
The angles that lie in the area between the two parallel lines
that are cut by a transversal, are called interior angles.
A pair of interior angles lie on the same side of the
transversal.
The measures of interior angles in each pair add up to 1800.
Interior Angles
Line M
BA
Line N
D E
L
P
Q
G
F
Line L
600
1200
1200
600
BPQ + EQP = 1800
APQ + DQP = 1800
Theorems of Lines
and Angles
If a ray stands on a line, then the sum of
the adjacent angles so formed is 180°.
PX Y
QGiven: The ray PQ stands on the line XY.
To Prove: ∠QPX + ∠YPQ = 1800
Construction: Draw PE perpendicular to XY.
Proof: ∠QPX = ∠QPE + ∠EPX
= ∠QPE + 90° ………………….. (i)
∠YPQ = ∠YPE − ∠QPE
= 90° − ∠QPE …………………. (ii)
(i) + (ii)
⇒ ∠QPX + ∠YPQ = (∠QPE + 90°) + (90° − ∠QPE)
∠QPX + ∠YPQ = 1800
Thus the theorem is proved
E
Vertically opposite angles are equal in
measure
To Prove :  =  C and B = D
A +  B = 1800 ………………. Straight line ‘l’
 B + C = 1800 ……………… Straight line ‘m’
  +  =  B + C
 A = C
Similarly B = D
A
B
C
D
l
m
Proof:
Given: ‘l’ and ‘m’ be two lines intersecting at O
as. They lead to two pairs of vertically
opposite angles, namely,
(i) ∠ A and ∠ C (ii) ∠ B and ∠ D.
If a transversal intersects two lines such that a
pair of corresponding angles is equal, then the
two lines are parallel to each other.
Given: Transversal PS intersects parallel lines
AB and CD at points Q and R respectively.
To Prove:
∠ BQR = ∠ QRC and ∠ AQR = ∠ QRD
Proof :
∠ PQA = ∠ QRC ------------- (1) Corresponding angles
∠ PQA = ∠ BQR -------------- (2) Vertically opposite angles
from (1) and (2)
∠ BQR = ∠ QRC
Similarly, ∠ AQR = ∠ QRD. Hence the theorem is proved
BA
C D
p
S
Q
R
If a transversal intersects two lines such that a
pair of alternate interior angles is equal, then the
two lines are parallel.
Given: The transversal PS intersects lines AB and CD
at points Q and R respectively
where ∠ BQR = ∠ QRC.
To Prove: AB || CD
Proof:
∠ BQR = ∠ PQA --------- (1)
Vertically opposite angles
But, ∠ BQR = ∠ QRC ---------------- (2) Given
from (1) and (2),
∠ PQA = ∠ QRC
But they are corresponding angles.
So, AB || CD
BA
C D
P
S
Q
R
Lines which are parallel to the same line are
parallel to each other.
Given: line m || line l and line n || line l.
To Prove: l || n || m
Construction:
BA
C D
P
S
R
l
n
m
E F
Let us draw a line t transversal
for the lines, l, m and n
t
Proof:
It is given that line m || line l and
line n || line l.
∠ PQA = ∠ CRQ -------- (1)
∠PQA = ∠ ESR ------- (2)
(Corresponding angles theorem)
∠ CRQ = ∠ ESR
Q
But, ∠ CRQ and ∠ESR are corresponding angles and they are
equal. Therefore, We can say that
Line m || Line n
Call us for more information
91-901-568-0202

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Lines And Angles

  • 1. Lines &Angles By - NM Spirit T- 901-568-0202 Email- nm.sspirit@gmail.com
  • 2. Point An exact location on a plane is called a point. Line Line segment Ray A straight path on a plane, extending in both directions with no endpoints, is called a line. A part of a line that has two endpoints and thus has a definite length is called a line segment. A line segment extended indefinitely in one direction is called a ray. Recap Geometrical Terms
  • 3.
  • 4. RAY: A part of a line, with one endpoint, that continues without end in one direction LINE: A straight path extending in both directions with no endpoints LINE SEGMENT: A part of a line that includes two points, called endpoints, and all the points between them
  • 5. INTERSECTING LINES: Lines that cross PARALLEL LINES: Lines that never cross and are always the same distance apart
  • 6. Common endpoint B C B A Ray BC Ray BA Ray BA and BC are two non-collinear rays When two non-collinear rays join with a common endpoint (origin) an angle is formed. What Is An Angle ? Common endpoint is called the vertex of the angle. B is the vertex of ABC. Ray BA and ray BC are called the arms of ABC.
  • 7. To name an angle, we name any point on one ray, then the vertex, and then any point on the other ray. For example: ABC or CBA We may also name this angle only by the single letter of the vertex, for example B. A B C Naming An Angle
  • 8. An angle divides the points on the plane into three regions: A B C F R P T X Interior And Exterior Of An Angle • Points lying on the angle (An angle) • Points within the angle (Its interior portion. ) • Points outside the angle (Its exterior portion. )
  • 9. Right Angle: An angle that forms a square corner Acute Angle: An angle less than a right angle Obtuse Angle: An angle greater than a right angle
  • 10. Two angles that have the same measure are called congruent angles. Congruent angles have the same size and shape. A B C 300 D E F 300 D E F 300 Congruent Angles
  • 11. Pairs Of Angles : Types • Adjacent angles • Vertically opposite angles • Complimentary angles • Supplementary angles • Linear pairs of angles
  • 12. Adjacent Angles Two angles that have a common vertex and a common ray are called adjacent angles. C D B A Common ray Common vertex Adjacent Angles ABD and DBC Adjacent angles do not overlap each other. D E F A B C ABC and DEF are not adjacent angles
  • 13. Vertically Opposite Angles Vertically opposite angles are pairs of angles formed by two lines intersecting at a point. APC = BPD APB = CPD A DB C P Four angles are formed at the point of intersection. Point of intersection ‘P’ is the common vertex of the four angles. Vertically opposite angles are congruent.
  • 14. If the sum of two angles is 900, then they are called complimentary angles. 600 A B C 300 D E F ABC and DEF are complimentary because 600 + 300 = 900 ABC + DEF Complimentary Angles
  • 15. If the sum of two angles is 1800 then they are called supplementary angles. PQR and ABC are supplementary, because 1000 + 800 = 1800 RQ P A B C 1000 800 PQR + ABC Supplementary Angles
  • 16. Two adjacent supplementary angles are called linear pair of angles. A 600 1200 PC D 600 + 1200 = 1800 APC + APD Linear Pair Of Angles
  • 17. A line that intersects two or more lines at different points is called a transversal. Line L (transversal) BA Line M Line N DC P Q G F Pairs Of Angles Formed by a Transversal Line M and line N are parallel lines. Line L intersects line M and line N at point P and Q. Four angles are formed at point P and another four at point Q by the transversal L. Eight angles are formed in all by the transversal L.
  • 18. Pairs Of Angles Formed by a Transversal • Corresponding angles • Alternate angles • Interior angles
  • 19. Corresponding Angles When two parallel lines are cut by a transversal, pairs of corresponding angles are formed. Four pairs of corresponding angles are formed. Corresponding pairs of angles are congruent. GPB = PQE GPA = PQD BPQ = EQF APQ = DQF Line M BA Line N D E L P Q G F Line L
  • 20. Alternate Angles Alternate angles are formed on opposite sides of the transversal and at different intersecting points. Line M BA Line N D E L P Q G F Line L BPQ = DQP APQ = EQP Pairs of alternate angles are congruent. Two pairs of alternate angles are formed.
  • 21. The angles that lie in the area between the two parallel lines that are cut by a transversal, are called interior angles. A pair of interior angles lie on the same side of the transversal. The measures of interior angles in each pair add up to 1800. Interior Angles Line M BA Line N D E L P Q G F Line L 600 1200 1200 600 BPQ + EQP = 1800 APQ + DQP = 1800
  • 23. If a ray stands on a line, then the sum of the adjacent angles so formed is 180°. PX Y QGiven: The ray PQ stands on the line XY. To Prove: ∠QPX + ∠YPQ = 1800 Construction: Draw PE perpendicular to XY. Proof: ∠QPX = ∠QPE + ∠EPX = ∠QPE + 90° ………………….. (i) ∠YPQ = ∠YPE − ∠QPE = 90° − ∠QPE …………………. (ii) (i) + (ii) ⇒ ∠QPX + ∠YPQ = (∠QPE + 90°) + (90° − ∠QPE) ∠QPX + ∠YPQ = 1800 Thus the theorem is proved E
  • 24. Vertically opposite angles are equal in measure To Prove :  =  C and B = D A +  B = 1800 ………………. Straight line ‘l’  B + C = 1800 ……………… Straight line ‘m’   +  =  B + C  A = C Similarly B = D A B C D l m Proof: Given: ‘l’ and ‘m’ be two lines intersecting at O as. They lead to two pairs of vertically opposite angles, namely, (i) ∠ A and ∠ C (ii) ∠ B and ∠ D.
  • 25. If a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines are parallel to each other. Given: Transversal PS intersects parallel lines AB and CD at points Q and R respectively. To Prove: ∠ BQR = ∠ QRC and ∠ AQR = ∠ QRD Proof : ∠ PQA = ∠ QRC ------------- (1) Corresponding angles ∠ PQA = ∠ BQR -------------- (2) Vertically opposite angles from (1) and (2) ∠ BQR = ∠ QRC Similarly, ∠ AQR = ∠ QRD. Hence the theorem is proved BA C D p S Q R
  • 26. If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the two lines are parallel. Given: The transversal PS intersects lines AB and CD at points Q and R respectively where ∠ BQR = ∠ QRC. To Prove: AB || CD Proof: ∠ BQR = ∠ PQA --------- (1) Vertically opposite angles But, ∠ BQR = ∠ QRC ---------------- (2) Given from (1) and (2), ∠ PQA = ∠ QRC But they are corresponding angles. So, AB || CD BA C D P S Q R
  • 27. Lines which are parallel to the same line are parallel to each other. Given: line m || line l and line n || line l. To Prove: l || n || m Construction: BA C D P S R l n m E F Let us draw a line t transversal for the lines, l, m and n t Proof: It is given that line m || line l and line n || line l. ∠ PQA = ∠ CRQ -------- (1) ∠PQA = ∠ ESR ------- (2) (Corresponding angles theorem) ∠ CRQ = ∠ ESR Q But, ∠ CRQ and ∠ESR are corresponding angles and they are equal. Therefore, We can say that Line m || Line n
  • 28. Call us for more information 91-901-568-0202