Lines And Angles
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description
Geometry, Lines and Angles
Transcript of Lines And Angles
PointAn 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
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An exact position or location
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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
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INTERSECTING LINES: Lines that cross
PARALLEL LINES: Lines that never cross and are always the same distance apart
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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 ÐABC.
Ray BA and ray BC are called the arms of ABC.
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To name an angle, we name any point on one ray, then the vertex, and then any point on the other ray.
For example: ÐABC or ÐCBA
We may also name this angle only by the single letter of the vertex, for example ÐB.
A
BC
Naming An Angle
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An angle divides the points on the plane into three regions:
A
BC
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. )
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The figure formed when two rays share the same endpoint
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
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Two angles that have the same measure are called congruent angles.
Congruent angles have the same size and shape.
A
BC
300
D
EF
300
D
EF
300
Congruent Angles
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Pairs Of Angles : Types
• Adjacent angles
• Vertically opposite angles
• Complimentary angles
• Supplementary angles
• Linear pairs of angles
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Adjacent AnglesTwo 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
EF
A
B
C
ABC and DEF are not adjacent angles
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Vertically Opposite AnglesVertically opposite angles are pairs of angles formed by two lines intersecting at a point.
ÐAPC = ÐBPD
ÐAPB = ÐCPD
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.
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If the sum of two angles is 900, then they are called complimentary angles.
600
A
BC
300
D
EF
ÐABC and ÐDEF are complimentary because
600 + 300 = 900
ÐABC + ÐDEF
Complimentary Angles
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If the sum of two angles is 1800 then they are called supplementary angles.
ÐPQR and ÐABC are supplementary, because
1000 + 800 = 1800
RQ
PA
BC
1000 800
ÐPQR + ÐABC
Supplementary Angles
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Two adjacent supplementary angles are called linear pair of angles.
A
600 1200
PC D
600 + 1200 = 1800
ÐAPC + ÐAPD
Linear Pair Of Angles
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A line that intersects two or more lines at different points is
called a transversal.
Line L (transversal)
BALine M
Line NDC
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.
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Pairs Of Angles Formed by a Transversal
• Corresponding angles
• Alternate angles
• Interior angles
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Corresponding AnglesWhen 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.
ÐGPB = ÐPQE
ÐGPA = ÐPQD
ÐBPQ = ÐEQF
ÐAPQ = ÐDQF
Line MBA
Line ND E
L
P
Q
G
F
Line L
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Alternate AnglesAlternate angles are formed on opposite sides of the transversal and at different intersecting points.
Line MBA
Line ND E
L
P
Q
G
F
Line L
ÐBPQ = ÐDQP
ÐAPQ = ÐEQP
Pairs of alternate angles are congruent.
Two pairs of alternate angles are formed.
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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 MBA
Line ND E
L
P
Q
G
F
Line L
6001200
1200600
ÐBPQ + ÐEQP = 1800
ÐAPQ + ÐDQP = 1800
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Theorems of Lines and Angles
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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
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Vertically opposite angles are equal in measure
To Prove : ÐA = Ð C and ÐB = ÐD
Ð A + Ð B = 1800 ………………. Straight line
‘l’
Ð B + Ð C = 1800 ……………… Straight line
‘m’
Þ Ð A + Ð B = Ð B + Ð C
Þ ÐA = ÐC
Similarly ÐB = ÐD
A
BC
D
lm
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.
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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 = ∠ QRCSimilarly, ∠ AQR = ∠ QRD. Hence the theorem is proved
BA
C D
p
S
Q
R
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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 anglesBut, ∠ BQR = ∠ QRC ---------------- (2) Given from (1) and (2),
∠ PQA = ∠ QRCBut they are corresponding angles.So, AB || CD
BA
C D
P
S
Q
R
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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 NM Spirit
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