Pairs of
Angles and
Lines
Erra C. Reyes
Math Teacher
At the end of today’s lesson, you
should be able to:
derive relationships of geometric
figures using measurements by
inductive reasoning;
supplementary angles,
complementary angles, congruent
angles, vertical angles, adjacent
angles, linear pairs, perpendicular
lines, and parallel lines.
Find my
Partner
Choose the pairs of angles which can form
a right angle and a straight angle.
13o
47o 54o D
B
A
167o 43o 126o
C F E
Find my
Partner
Choose the pairs of angles which can form
a right angle and a straight angle.
Pairs of angles Pairs of angles
which form a which form a
right angle straight angle
13o
D ∠A and ∠ B and ∠
54 o
47o
B ∠F E
∠C and
A ∠D
167o 126o
43o
C F E
Pairs of
Angles
Pairs of
Angles
ADJACENT ANGLES
COMPLEMENTARY ANGLES
SUPPLEMENTARY ANGLES
LINEAR PAIR
VERTICAL ANGLES
CONGRUENT ANGLES
Adjacent
Angles
Adjacent angles are two distinct
angles with common vertex and a
A
common side.
D ∠1 and ∠2 are adjacent
angles with a common
common side
vertex B and a common
1
2
side
B C
common vertex
Adjacent
Angles
Other examples:
Common side
Common vertex
Adjacent or
Not?
I
R
Is each of the given pairs of
angles adjacent? If not,
why?
E
∠FDR and ∠RDI
F D
∠NDE and ∠FDR
∠EDN and ∠RDF
N ∠FDN and ∠NDE
Complementary
Angles
Complementary angles are two
angles whose sum of measures
equals 90o.
∠A and ∠B are
complementary
57o 33o
angles since their
A B sum is equal to 90o
m∠A + m∠B = 90o
57o + 33o = 90o
Complementary
Angles
Note: Not all complementary angles have to be
adjacent. As long as the sum of the two angles
is 90o, then they are complementary.
Supplementary
Angles
Supplementary angles are two
angles whose measures have the
sum of 180o.
∠C and ∠D are
supplementary
126o 54o
C D
angles since their
m∠C + m∠D = 180o
sum is equal to 180o
126o + 54o = 180o
Supplementary
Angles
Note: Not all supplementary angles have to be
adjacent. As long as the sum of the two angles
is 180o, then they are supplementary.
Please Complete
Me!
Find the measures of a, b, c and d if possible
following the given conditions.
Measure of
Measure of Angle Measure of Supplement
Complement
85o a 95o
38o 52o b
105o c 75o
x 90 - x d
Please Complete
Me!
Find the measures of a, b, c and d if possible
following the given conditions.
Measure of
Measure of Angle Measure of Supplement
Complement
85o a 95o
The measure of the complement a is 5o
because 90o - 85o = 5o
Please Complete
Me!
Find the measures of a, b, c and d if possible
following the given conditions.
Measure of
Measure of Angle Measure of Supplement
Complement
38o 52o b
The measure of the supplement b is 142o
because 180o - 38o = 142o
Please Complete
Me!
Find the measures of a, b, c and d if possible
following the given conditions.
Measure of
Measure of Angle Measure of Supplement
Complement
105o c 75o
The angle measuring 105o has no complement
since 105o is greater than 90o.
Please Complete
Me!
Find the measures of a, b, c and d if possible
following the given conditions.
Measure of
Measure of Angle Measure of Supplement
Complement
x 90 - x d
The answer is 180o – x because the
supplement of an angle is 180o minus the
given angle represented by x.
Please Complete
Me!
Find the measures of a, b, c and d if possible
following the given conditions.
Measure of
Measure of Angle Measure of Supplement
Complement
85o 5o 95o
38o 52o 142o
105o none 75o
x 90o - x 180o - x
Linear Pair
Linear pair is composed of two adjacent
angles whose measures have the sum of
180o. These are adjacent angles that form
a straight line. ∠GEH and ∠HEF are
H
adjacent angles.
m∠GEH + m∠HEF = 180o
126o 54o 126o + 54o = 180o
G E F
Therefore, ∠GEH and ∠HEF
are linear pair.
Linear Pair
Without identifying the
measures, it is obvious that
G E F the sum is 180o since a
straight angle was formed.
Vertical Angles
Vertical angles are two nonadjacent angles
formed by two intersecting lines.
E
H ∠1 and ∠2 are vertical
angles since they were
2 formed by 2 intersecting
1
F lines and
R
A
Vertical Angles
Vertical angles are congruent.
Since ∠1 and ∠2 are
E
vertical angles then they
H are congruent angles.
Congruent is denoted by
2
1 this symbol .
F
∠1
R
A ∠2
Congruent
Angles
Two angles which have the same
measure are congruent angles.
∠R and ∠Q are congruent
angles.
In symbols, ∠R ∠Q.
Vertical Angles
Vertical angles are congruent.
Since ∠1 and ∠2 are
E
H vertical angles then
they are congruent
2 75o
angles.
75o1
F
If m∠1 = 75o then m∠2 =
A
R
75o .
Vertical Angles
Vertical angles congruent.
E Since ∠HFE and
H ∠AFR are vertical
107o angles then they are
2 congruent angles.
1
F
107o If m∠HFE = 107o
A
R then m∠AFR = 107o .
Name Me
Please!
Name the pairs of vertical angles in the
given figure.
∠AOB and ∠DOC
are vertical angles
∠BOC and ∠AOD
are vertical angles
Name Me
Please!
Name the pairs of vertical angles in the
given figure.
∠1 and ∠3 are
vertical angles
∠2 and ∠4 are
vertical angles
My angle
measures…
Given the figure at
the left, if m∠2 = 76o,
what is m∠4?
m∠4 =
76o
My angle
measures…
If m∠AOB = 126o,
what is m∠DOC?
m∠DOC =
126o
My angle
measures…
Given the figure below, what is the measure of ∠EBC
and ∠DBC?
E
A
133o
47o 47o
B
133o
D C
My angle
measures…
Given the figure below what is the measure of ∠EBC,
∠ABE and ∠DBC?
E
A
144o
36o 36o
B
144o
D C
More on Angle
Pairs
Given this figure, give what is being asked.
I
R Name the pair of vertical angles
45o 45o E
F D
∠FDR and
N
∠NDE
More on Angle
Pairs
Given this figure, give what is being asked.
I
R
45o 45o E
F D ∠RDI and ∠IDE
N
More on Angle
Pairs
Given this figure, give what is being asked.
I
R Name the pairs of congruent
angles
45o 45o E
F D
∠FDR and ∠IDE
N ∠FDR and
∠NDE
More on Angle
Pairs
Given this figure, give what is being asked.
I
R
Name pairs of angles which
formed a linear pair
45o 45o E ∠FDR and ∠RDE
F D
∠FDR and ∠FDN
N
∠RDI and ∠IDN
More on Angle
Pairs
Given this figure, give what is being asked.
I
R Find the measure of the
following angles:
45o 45o E 1. ∠RDI
F D 2. ∠NDE
3. ∠FDN
N
More on Angle
Pairs
Given this figure, give what is being asked.
I 1. m∠RDI
R
m∠FDR + m∠IDE + m∠RDI =
180o 45o + 45o + m∠RDI = 180o
45o 45o E
D
90o + m∠RDI = 180o
F
m∠RDI = 180o - 90o
N
m∠RDI = 90o
More on Angle
Pairs
Given this figure, give what is being asked.
I 2. m∠NDE
R
∠NDE and ∠FDR are vertical
45o 45o E
angles.
F D
m∠FDR = 45o
N therefore, m∠NDE = 45o
More on Angle
Pairs
Given this figure, give what is being asked.
I 3. m∠FDN
R ∠FDR and ∠ FDN formed
linear pair
45o 45o E
m∠FDR + m∠ FDN = 180o
F D
45o + m∠ FDN = 180o
N
m∠ FDN = 180o - 45o
m∠ FDN = 135o
Pairs of
Lines
Pairs of
Lines
INTERSECTING LINES
PERPENDICULAR LINES
PARALLEL LINES
Intersecting
Lines
Intersecting lines are lines that meet at a
point. When two lines intersect, they
define angles at the point of intersection.
E
G
R
T
Perpendicular
Lines
Perpendicular lines are lines that intersect
at one point and form a 90o angle.
90o
H A K
N
Perpendicular
Lines
Some of the examples of real – world objects
that suggest intersecting lines or
perpendicular lines are the following:
cross road signage hands of a
intersecting road clock kite’s
skeleton
Parallel Lines
Parallel lines are lines that never intersect.
The distance between the two lines is fixed
and the two lines are going in the same
direction.
A B
C D
Parallel Lines
Looking around, you can see things that
represent a parallel lines.
Railway tracks Fence Design Opposite sides
of blackboard
Who am I?
Identify pairs of lines as intersecting,
parallel or perpendicular.
parallel lines
perpendicular lines
intersecting lines
