Chapter 1 ( Basic Concepts in Geometry )rey castro
Chapter 1 Basic Concepts in Geometry
1.1 Points, Lines and Planes
1.2 Line Segment
1.3 Rays and Angles
1.4 Some Special Angles
1.5 Angles Made By A Transversal
1.6 Transversal Across Two Parallel Lines
1.7 Conditions For Parallelism
Chapter 1 ( Basic Concepts in Geometry )rey castro
Chapter 1 Basic Concepts in Geometry
1.1 Points, Lines and Planes
1.2 Line Segment
1.3 Rays and Angles
1.4 Some Special Angles
1.5 Angles Made By A Transversal
1.6 Transversal Across Two Parallel Lines
1.7 Conditions For Parallelism
1.5 Complementary and Supplementary Angles Dee Black
Some slides lifted from: http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=4&ved=0CEsQFjAD&url=http%3A%2F%2Fdionmath.wikispaces.com%2Ffile%2Fview%2F2.3%2BComplementary%2Band%2BSuppl.%2BAngles.ppt&ei=_wVFUbzHCa-o4AP9ooGwBQ&usg=AFQjCNF-KDyDx_yiVaUuMJMdM6yOJqHASQ&sig2=wH2TZ9xGxsHgtc4cCnn2QQ&bvm=bv.43828540,d.dmg&cad=rja
1.5 Complementary and Supplementary Angles Dee Black
Some slides lifted from: http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=4&ved=0CEsQFjAD&url=http%3A%2F%2Fdionmath.wikispaces.com%2Ffile%2Fview%2F2.3%2BComplementary%2Band%2BSuppl.%2BAngles.ppt&ei=_wVFUbzHCa-o4AP9ooGwBQ&usg=AFQjCNF-KDyDx_yiVaUuMJMdM6yOJqHASQ&sig2=wH2TZ9xGxsHgtc4cCnn2QQ&bvm=bv.43828540,d.dmg&cad=rja
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2. Contents
• Recap the terms
• Angles in daily life
• What is an angle?
• Naming an angle
• Interior and exterior of an angle
• Measurement of angle
• Types of angle: Right angle
Obtuse angle
Acute angle
Straight angle
• Test Yourself - 1
• Congruent angles
• Pairs of angles: Types
• Test Yourself - 2
• Pairs of angles formed
by a transversal
• Test Yourself - 3
3. 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
4. If we look around us, we will see angles
everywhere.
Angles In Daily Life
5. 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 .
Ray BA and ray BC are called the arms of ABC.
6. Fact: We can also think of an angle formed by rotating one
ray away from its initial position.
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: ÐABC or ÐCBA
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. Angles are accurately measured in
degrees.
Protractor is used to measure and
draw angles.
Measurement Of An Angle
10. There are four main types of angles.
Straight angle
Right
angle
Acute
angle
Obtuse
angle
A
B C
A
B C
A
B C
B
A C
Types Of Angles
11. Right angle: An angle whose measure
is 90 degrees.
Right Angle Acute Angle
Straight Angle Obtuse Angle
19. A
B
C
D
E
F
P
Q R
Which of the angles below is a right angle, less than a right
angle and greater than a right angle?
Right angle
Greater than a right angle
Less than a right angle
1. 2.
3.
20. 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
21. Pairs Of Angles : Types
• Adjacent angles
• Vertically opposite angles
• Complimentary angles
• Supplementary angles
• Linear pairs of angles
22. 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
23. Vertically Opposite Angles
Vertically opposite angles are pairs of angles
formed by two lines intersecting at a point.
APC = BPD
APB = CPD
A
D
B
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.
24. 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
25. 700
D
E
F
300
p
Q
R
If the sum of two angles is more than
900 or less than 900, then they not
complimentary angles.
DEF and PQR are not complimentary because
700 + 300 = 1000
DEF + PQR
26. If the sum of two angles is 1800 then they
are called supplementary angles.
PQR and ABC are supplementary, because
1000 + 800 = 1800
R
Q
P
A
B
C
1000
800
PQR + ABC
Supplementary Angles
27. If the sum of two angles is more than 1800
or less than 1800, then they are not
supplementary angles.
DEF and PQR are not supplementary because
ABC + DEF
1100 + 800 = 1900
D
E
F
800
C
B
A
1100
28. Two adjacent supplementary angles
are called linear pair of angles.
A
600 1200
P
C D
600 + 1200 = 1800
APC + APD
Linear Pair Of Angles
29. Name the adjacent angles and linear pair of angles in the
given figure:
Adjacent angles:
ABD and DBC
ABE and DBA
Linear pair of angles:
EBA, ABC C
D
B
A
E
600
300
900
EBD, DBC
C
D
B
A
E
600
300
900
30. Name the vertically opposite angles and adjacent angles in
the given figure:
A
D
B
C
P
Vertically opposite angles: APC and BPD
APB and CPD
Adjacent angles: APC and CPD
31. A line that intersects two or more lines at different points is
called a transversal.
Line L (transversal)
B
A
Line M
Line N
D
C
P
Q
G
F
Pairs Of Angles Formed by a Transversal
Line M and line N are parallel lines.
Line L intersects line M an
med at point P and another f angles our at point Q by the
transversal L.
Eight angles are formed in all by the transversal L.
32. Pairs Of Angles Formed by a Transversal
• Corresponding angles
• Alternate angles
• Interior angles
33. 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
B
A
Line N
D E
L
P
Q
G
F
Line L
34. Alternate Angles
Alternate angles are formed on opposite sides of the
transversal and at different intersecting points.
Line M
B
A
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.
35. 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
B
A
Line N
D E
L
P
Q
G
F
Line L
600
1200
1200
600
BPQ + EQP = 1800
APQ + DQP = 1800
36. Name the pairs of the following angles formed by a
transversal.
Line M
B
A
Line N
D E
P
Q
G
F
Line L