Download Angle Pair - Kapitolyo High School

Division of Pasig City and San Juan
SAGAD HIGH SCHOOL
E. Angeles St., Sagad Pasig City
STRATEGIC INTERVENTION MATERIALS
IN
MATHEMATICS III
Prepared and Submitted by:
FEMILA R. PALICAN
EDWIN P. PECO
HELEN U. RAYMUNDO
NENET M. PENARANDA
Head Teacher III - Math
LAZARO P. TORRES
Principal
LIGAYA G. INSIGNE
Education Supervisor I – Secondary Math
Journey to the World of…
What have you observed on the figure
on the board ?
The
figure
formed
a pair
of
angles
What kind of angles
are formed ?
Vertical
Angles are
formed by
intersecting
lines
EXPLORE:
Prepare a whole sheet of paper. You can
make a model of different pairs of angle by
folding a rectangular or square paper, by
folding it lengthwise, cross-wise and
diagonally. The folded corner provides a
convenient model of pair of angles.
From the model figure using a piece
of paper, the broken lines are the folded
crease on the paper. The pairs of angles are
a.
b.
c.
d.
e.
supplementary angles
complementary angles
adjacent angles
linear pair
vertical angles
In geometry, pairs of angles are related to each other in several ways. Some
examples are complementary angles, supplementary angles, vertical angles, adjacent
angles and linear pair.
Two angles are called complementary angles if the sum of their degree measurements equals 90
degrees. One of the complementary angles is said to be the complement of the other.
∠ABC is the complement of ∠CBD
Two angles are called supplementary angles if the sum of their degree measurements equals 180
degrees. One of the supplementary angles is said to be the supplement of the other.
∠ABC is the supplement of ∠CBD
Two pairs of angles are formed by two intersecting lines. Vertical angles are opposite angles in such an
intersection. Vertical angles are equal to each other.
Very often, math questions will require you to work out the values of angles given in diagrams by
applying the relationships between the pairs of angles.
Example 1: Given the diagram below, determine the values of the angles x, y and z.
A
B
O
C
D
Solution:
Step 1: x is a supplement of 65°.
Therefore, x + 65° =180° ⇒ x = 180° – 65° = 115°
Step 2: z and 115° are vertical angles.
Therefore, z = 115°
Step 3: y and 65° are vertical angles.
Therefore, y = 65°
Answer: x = 115°, y = 65° and z = 115°
Two angles which are adjacent and whose non common side are opposite rays and formed a
straight line are called linear pair.
D
∠DEF and ∠DEG are linear pair
25º
F
155º
E
G
Two coplanar angles with a common side and a common vertex but, no common interior points.
∠HIJ and ∠JIK are adjacent angles.
H
j
I
K
1. Theorem 1 - 1 VERTICAL ANGLES – vertical angles are congruent.
∠1 and ∠ 2 are vertical angles, show that ∠1  ∠ 2.
1
3
4
2
* Angle Addition Postulate *
m∠1 + m∠3 = 180, m∠2 + m∠3 = 180
by substitution, m∠1 + m∠3 = m∠2 + m∠3
Subtract m∠3 from each side you will get
m∠1 = m∠2, m∠1  m∠2 or ∠ 1  ∠ 2.
2. Theorem 1 - 2 If two angles are supplements of congruent angles, then the angles
are congruent.
1
2
∠1 and ∠2 are supplementary
∠3 and ∠2 are supplementary
show ∠2  ∠3
∠1 and ∠2 are supplementary, m∠1 + m∠2 = 180
∠3 and ∠2 are supplementary, m∠3 + m∠2 = 180
So m∠1 + m∠2 = m∠3 + m∠3
 m∠1 = m∠3, m∠1  m∠3, ∠1  ∠3
3
3.
Theorem 1 – 3 If two angles are complements of congruent angles, then the two
angles are congruent.
2
1
∠1 and ∠2 are complementary
∠3 and ∠2 are complementary
show ∠2  ∠3
∠1 and ∠2 are complementary, m∠1 + m∠2 = 90
∠3 and ∠2 are complenmentary, m∠3 + m∠2 = 90
So m∠1 + m∠2 = m∠3 + m∠3
 m∠1 = m∠3, m∠1  m∠3, ∠1  ∠3
3
Activity 1
Identify the following pair of
angles
1.
2.
1
2
1
3.
2
4.
30º
150º
50º
40º
b.
Activity # 2
Find the measure of the following angles:
1.
m1 = _______
m2 = _______
2
1
30º
2.
xº = ___________
4xº = __________
5xº = __________
5x
4x
x
3.
1
(x + 10)
3
4
(4x – 35)
x = ___________
m  1 = _________
m  2 = _________
2
m  3 = _________
m  4 = _________
4.
x = ____________
m  1 = __________
(12x – 7)
1
2
(3x - 8)
m  2 = __________
5.
(4x + 5)
1
2
(3x + 8)
m  1 = ____________
m  2 = ____________
c. Activity # 3
SOLVE:
1.
The measure of a supplement of 1 is six times the
measure of a complement of 1. Find the measure of 1, its
supplement, and its complement.
2.
A and B are supplementary angles and
mA = 3x + 12 and mB = 2x – 22. Find the measure of both
angles.
3.
One angle is twice as large as its complement. Find
the measure of both angles.
T
a.
b.
3
1
2
P
4
O
c.
V
W
R
S
Z
T
Q
E
X=
m1 = 2x + 6
m2 = 3x + 10
m3 = x + 20
mADE = 24x
C
B
3
1
2
D
A
a.
x = _____, y = _____
1
(3x + 8 )º
2
3
(5x – 20)º
4
(5x + 4y) º
m1 = ________
m2 = ________
m3 = ________
m4 = ________
b.
m1 = _____________
2
1
25º
m2 = _____________
1.
2.
Activity Card
Activity Card 1
1.
Vertical Angle
2.
Supplementary Angle, Linear Pair
3.
Supplementary Angle
4.
Complementary Angle
5.
Adjacent Angle
Activity Card 3
1.
m  1 = 72º
complement = 18º
supplement = 108º
2..
m A = 126º
m  B = 54º
3.
x, 2(90-x) º
x = 60º ,
complement = 30º
Assessment Card
4.
x=2
m  1 = 10º
m  2 = 16º
m  3 = 22º
m  ADE = 48º
Enrichment Card
a.
x = 14
m  1 = 130º
m  2 = 50º
m  3 = 50º
m  4 = 130º
b.
m  1 = 90º
m  2 = 65º
Activity Card 2
1.
m1=90º
m2=60º
2.
x =18º
4x =72º
5x = 90º
3.
x=15º
m 1=155º
m 3=25º
m 2=155º
m 4=25º
4.
x = 13º
m  1 = 149º
m  2 = 31º
5.
m 1 = 49º
m  2= 41º
Activity 4
a.
 1 and  3,  2 and  4 are supplementary, linear pair and adjacent
b.
 TOQ and  QOV,  QOV and  VOP , VOP and  POT,  POT and  TOQ are all
c.
.
right angles and supplementary angles
 WSZ and  ZST are complementary angles while  RSZ and  ZST are
supplementary
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