Buksis 7.1 new

What are you going
to learn?
to identify the position
of two lines
to define an angle
To estimate the
measure of an angle
to measure an angle
to draw an angle
to determine types of
angles
to describe the angle
formed by two lines
cut by a transversal
line
Key Terms:
• parallel lines
• intersecting lines
• skew lines
• supplementary angles
• complementary
angles
• corresponding angles
• alternate interior
angles
• alternate exterior
angles
• consecutive interior
77..11
Position of Two Lines
Look at Figure 7.1(a) and 7.1(b)
on the right. What is the
relationship between the two
figures?
A
D
B
C
E F
GH
Figure 7.1(a)
Figure 7.1(b)
Look at Figure 7.1(b)
andThe segments DCAB are two parallel segments.
Why? Find other pairs of parallel segments.
l
m
A B
D C
Segments AB and DC are parts of lines l and m, which
are infinite in length as shown above. The lines l and m
can also be denoted as AB and CD .
The segments AB and AD are two intersecting segments.
Why? Find other pairs of intersecting segments.
The segments AB and FG are two skew segments (neither
parallel nor intersecting). Why? Find other pairs of skew
segments.
Mathematics for Junior High School Year 7 / 277

Definition of an Angle
In the elementary school you had learned the definition of
an angle. To remind you, look and do the activity as shown in
Figure 7.2.
Figure 7.2
Draw the position of
the straw on a sheet of
paper.
(b)
Take a straw and then
bend it.
(a)
Figure 7.2(b) is an example of an angle. In Mathematics, an
angle is formed by two rays the initial points of which coincide,
as shown in the following figure.
B
A
C
Legs, BA and BCArea of angle
•
•
Vertex
Figure 7.3
•
Based on Figure 7.3, components of an angle are: two sides
(legs) of the angle, the vertex of the angle, and the area of the
angle. The sides of an angle are two rays forming an angle.
Vertex is the common initial point of both rays. Area of angle is an
area formed by two sides of the angle.
The angle in Figure 7.3 is called angle ABC symbolized by
∠ABC or angle CBA symbolized by ∠CBA or it is sometimes
written as angle B symbolized by ∠B.
278 / Student’s Book – Lines and Angles

Determine the sides, vertex, and write the
name of the angle of the right figure.
PROBLEM 1
PROBLEM 2
Q
P
R
• •
•
S
Q
R
P
•
•
•
•
How many angles are formed in the
figure on the right? Label each angle.
Think and Discuss
Observe the roof of the house.
Write down all components of the roof
which form an angle.
Related to the Real World
More than 3000 years ago, the people of Babylon found out that to
revolve around the sun once on a circle orbit, the earth needed 360
days. They divided the orbit into 360 identical parts. Each part was
named one degree. Thus, one rotation is 360 degrees, symbolized by
360°. Since then, the degree has been used as one of the measurements
of an angle.

Measuring an Angle Using a Protractor
Q R
•Can you measure the angle PQR on
the figure? What device do you use to
measure an angle?
P
••
A device that can be used to measure an angle is called a
protractor as shown in Figure 7.4. below.
There are two groups of scales on a
protractor that can be found on the outer
and the inner part of it. On the outer
part, from the left to the right the scales
0, 10, 20, 30, . . . , 180 can be read and on
the inner part from the left to the right
the scales 180, 170, 160, . . . , 0 can be
read. The intersection between the
horizontal line and the vertical line is
called the centre of the protractor.Figure 7.4
Vertical line
Horizontal line
Centre of protractor
•
To measure the angle PQR above, you can do the following.
1. Put the centre of the protractor on the vertex Q.
2. Coincide the horizontal line of
the protractor, on which the
number 0 is written, on one of
the sides, i.e. on QR.
3. Look at the number on the
protractor coinciding with the
other sides, i.e. side QP which is
coinciding to the line showing
the number 100.
So, the measure of ∠PQR above is 100°.
Estimate the measure of the angles in problems 4 and 5 below
before using your protractor. Then use your protractor to
answer these problems.
Figure 7.5
Q
P
R
•
•
•

Determine the measure of the angles below.
a. b.
Look at a given figure of a dinosaur.
The position of the dinosaur on the
ground makes an angle. Measure
the angle formed by the dinosaur
with the ground.
Estimate the measure of each angle below. Then check your
estimation using your protractor.
1. 2.
3. 4.
PROBLEM 4
B •
A•
Related to the Real World
•
C
K•
•
M
L •
PROBLEM 5
PROBLEM 6

The sum of
the measure
of two angles
can be
written as an
angle having
similar size.
Look at the figure on the right.
If ∠APB = (11x−5)° , ∠BPC =
(7x)°, and ∠APC = 85°,
determine the value of x and the
degree of ∠APB.
PROBLEM 7
B •
A•
•
C
P •
PROBLEM 8
M
• K
•
• L
Estimate angle MLK as shown
by the arrow.
Then check your estimation
using your protractor.
Types of Angles
A
B C D
Figure 7.7
P
Q
R
Figure 7.6
Look at Figures 7.6 dan 7.7.

Work in Groups
Estimate the measure of each angle:
∠ABC, ∠BCA, ∠PQR dan ∠BCD.
Then check your estimation using your protractor.
The measure of ∠ABC = . . . . , ∠BCA = . . . . , ∠PQR = . . . . , and
∠BCD = . . . ..
∠ABC is an example of acute angle, ∠BCA is a right angle, ∠PQR
is an obtuse angle and ∠BCD is a straight angle.
• An angle sizes between 0° and 90° is called an acute angle.
• An angle sizes 90° is called a right angle.
• An angle sizes between 90° and 180° is called an obtuse
angle.
• An angle sizes 180° is called a straight angle.
PROBLEM 9
Estimate the measure of ∠A, ∠B, and ∠C
a.
∠A = . . . .°
∠A is . . . . . . . . . . . angle.
A
B
b.
∠B = . . . .°
∠B is . . . . . . . . . . . angle
c.
C ∠C = . . . .°
∠C is . . . . . . . . . . . . angle.

Figure 7.8
A
B C
D
O
Look at the fence design on
the right and do the following
activities.
Activities
(1) Estimate the sizes of ∠ABO, ∠OBC, ∠COD and ∠DOA.
(2) Measure the sizes of ∠ABO, ∠OBC, ∠COD and ∠DOA using
your protractor.
(3) Find the sum of ∠ABO and ∠OBC. How much is it?
(4) Find the sum of ∠COD and ∠DOA. How much is it?
(5) Find two angles whose sum is 90°.
(6) Find two angles whose sum is 180°.
Two angles having the total sum of 90° are called complementary
angles. One of the angles is the complement of the other angle.
Two angles having the total sum of 180° are called supplementary
angles. One of the angles is the supplement of the other angle.
If ∠P = 42° and ∠Q is the complementary of ∠P, determine
the measure of ∠Q.
PROBLEM 10
PROBLEM 11
If ∠PQS = 90°, ∠SQT = (x+28)° and ∠TQR =
(6x −15)°, determine the measure of ∠SQT,
∠TQR and mention the complementary angle.
RQP
T
S

CHECKING FOR UNDERSTANDING
Look at the map on the
right. Line AB is a straight
side of the street. Find a pair
of two supplementary
angles.
A
B
Figure 7.9
PROBLEM 12
PK
L
M
If ∠KPL = (2x)° and ∠LPM = (3x)°,
determine the value of x.
Properties of Angles Formed by Two Lines Intersected by a
Transversal
If Figure 7.9 is made into a sketch,
it will look like the figure on the right.
Line l and m are intersected by line
l
m
c
1 2
34
5 6
7 8
B
A
AB . There are 8 angles formed, i.e.
∠1, ∠2, ∠3, ∠4, ∠5, ∠6, ∠7, and ∠8. All
angles form pairs of angles as the
following.
a. Corresponding angles, i.e. ∠1 and ∠5.
Mention other corresponding angles.
b. Alternate interior angles, i.e. ∠3 and ∠5.
Mention other alternate interior angles.
c. Alternate exterior angles, i.e. ∠1 and ∠8.
Mention other alternate exterior angles.
d. Interior angles on the same side of the transversal, i.e. ∠4 and ∠5.
Mention other consecutive interior angles.
e. Exterior angles on the same side of the transversal, i.e. ∠1 and ∠7.
Mention other consecutive exterior angles.
f. Vertical angles, i.e. ∠1 and ∠3.

Mention other vertical angles. What are the measures of a pair of
vertical angles?
CHECKING YOUR UNDERSTANDING
1. In the figure above, what is the name of ∠1 and ∠2?
2. ∠1 and ∠2 are called adjacent angles, why?
3. Are two adjacent angles always supplementary? Explain.
4. Are two adjacent angles always complementary? Explain.
5. Are two supplementary angles always adjacent? Explain.
6. Are two complementary angles always adjacent? Explain.
7. Measure all pairs of opposite angles. What is your conclusion?
Explain your answer.
8. What angles are formed if two lines are cut by a transversal
line?
Exploration
1. Draw two parallel lines intersected by a transversale.
2. Mark all of the eight angles number 1, 2, 3, 4, 5, 6, 7, and 8. Measure them.
3. Based on the angles measured, investigate the pairs of corresponding
angles, alternate interior angles, alternate exterior angles, interior angles on
the same side of the transversal, exterior angles on the same side of the
transversal, vertical angles.
4. Make a prediction based on the result above about the angles formed if two
parallel lines are cut by a transversal line.

1. Look at the figure below.
a. Mention the parallel
lines.
b. Mention the
intersecting lines.
p
2. For each of the following angles, determine the sides, vertex,
and write the name of the angle.
a. b. c. d.
3. Mention all of the angles formed in the following figures.
a. b.
c. d.
r
s
q
m n
I
J
H
•
•
L
K
M
••
U
U
S
M
•
•
T
G
•
•
•O
R
S
Q
P
E
D
C
B
A
• •
•
•
W
F
S
Q
•
••
E
V
R
D
•

4. Critical Thinking. In each of the figures below, there are
lines which are not linear and coinciding on the same vertex.
2 rays 3 rays 4 rays 5 rays
. . . angles . . . angles . . . angles . . . angles
a. Determine the number of angles formed in each of the
above figures and write your answer in the blanks
above.
b. Do you notice a certain pattern of the number of angles?
How many angles are formed if there are 7 rays?
c. Write a formula showing the number of angles formed
if there are n rays.
5. Estimate the measure of each of the following angles. Then,
check your estimation using your protractor.
Write your answer in the nearest whole number.
a. b. c.
d. e. f.
6. Determine the measure of the smallest angle formed by the
long and short clock hands at:
a. 02.00 b. 04.00
c. 02.30 d. 03.30

7. Look at the figure on the left.
Determine the measure of the following
angles.
a. ∠APB b. ∠APD c. ∠BPC
d. ∠BPD e. ∠DPC f. ∠DPB
•
• C
•D
P
A •
• B
Use the figure on the right to answer
problems 8 to 12. •S
•
• Q
P •
X
•
T
• R
8. If ∠SXT = (3x–4)°,
∠RXS = (2x+5)° and
∠RXT = 111°, determine
the measure of ∠RXS.
9. If ∠PXQ = (2x)° and ∠QXT = (5x – 23)°, determine the
measure of ∠QXT.
10. If ∠QXR = (x+10)°, ∠QXS = (4x–1)° and ∠RXS = 91°,
determine the measure of ∠QXS.
11. If ∠QXR = (3x+5)°, ∠QXP = (2x−5)° and ∠RXP = (x+50)°,
determine the measure of ∠RXT.
12. If ∠TXS = (x+4)°, ∠SXR = (3x+4)° and ∠RXP=(2x+4)°,
determine the measure of ∠PXS.
O
M
•
N
L
•K
•
•
•
13. If ∠KOM = 80° , ∠LON = 95°
and ∠KON = 120° , determine
the measure of ∠LOM.
P X
Q
R S
•
T
•
•
•
•
U
•
14. Estimate the measure each of the
following angles. Then check your
estimation using your protractor.
a. ∠PXU c. ∠QXT
b. ∠SXQ d. ∠TXR

15. a. When do the long and short clock hands form a 90° angle?
b. When do the long and short clock hands form a 180°
angle?
A
B
C
RQ
P16. Estimate the measure each of
angles ∠BAC and ∠PQR in
the right figure.
. Then check your estimation
using your protractor.
17. Critical Thinking. Look at the roofs of both houses in the
following figures.
P
Q
R
A
C
B
The angles formed by the roofs are ∠ABC dan ∠PQR.
a. Estimate the measure of each of angles ∠ABC and ∠PQR.
b. Which angle is bigger?
c. On which roof does rain fall faster?
d. What is your conclusion about the relationship between
the measure of the angle of the roof and the speed of the
rain to fall from it?
18. Answer the following questions and give examples.
a. Do two acute angles always have the same measure? Give your
reason.
b. Do two right angles always have the same measure? Give your
reason.
c. Do two obtuse angles always have the same measure? Give
your reason.
d. Do two straight angles always have the same measure? Give
your reason.

19. a. Without measuring, mention the type of the angles below.
i) ii) iii)
iv) v) vi)
b. Match your answer by measuring the angles using a protractor.
Is there any wrong answer?
20. Critical Thinking. If ABC is a triangle, which of the following is
impossible to happen? Give your reason.
a. ∠A is an obtuse angle.
b. ∠A is acute, ∠B is acute and ∠C is acute.
c. ∠B is a right angle, ∠A is obtuse.
d. ∠A is a right angle.
21. Look at the figure on the
right, then mention the
type of each of the
following angles.
a. ∠MAL c. ∠LBC e. ∠ABC g. ∠LMA
b. ∠ALC d. ∠BLN f. ∠ACN h. ∠CNM
N
M
L
CB
A
22. Mention the complementary and supplementary angles of
each of the following angles.
a. 38° b. 66° c. 80°
d. 54° e. 12° f. 90°
23. Look at the figure on the right. Mention:
a. pairs of complementary angles
b. pairs of supplementary angles.
L
K
N
P
J
M

24. Given ∠A is the complementary of ∠B.
If ∠A = (7x+4)° and ∠B = (4x+9),
determine:
a. the value of x b. the measure of ∠A and ∠B.
25. Given ∠P is the supplementary of ∠Q. If ∠P = (6x+4)° and
∠Q = (10x)°, determine:
a. the value of x b. ∠P and ∠Q.
26. Determine the measure of two complementary angles whose
measure difference is 12°.
27. Angles A and B are two complementary angles, and so are
angles ∠C and ∠D. If ∠A = (2x+3)° , ∠B = (y−2)° , ∠C =
(2x−y)° and ∠D = (x−1)°, determine:
a. the value of x b. the value of y
c. the measure of ∠A d. the measure of ∠B
e. the measure of ∠C f. the measure of ∠D
28. There is an angle 60° smaller than the multiple of 3 of its
complement. Determine the measure of the angle.
29. There is an angle 5° smaller than the multiple of 4 of its
supplement. Determine the measure of the angle.
30. Critical Thinking
a. Why are ∠1 and ∠3
complementary?
Explain.
b. Explain also why ∠2
and ∠4 are
complementary.
1
4
3
2
31. Critical Thinking. Investigate whether the statement below is
true!
“The measure of an acute angle equals to the difference
between its supplement and twice of its complement “.

32. Determine the measure of
∠AZC and ∠AZD on the
figure on the right.
BC
DA
(5x – 22)° (3x + 16)°
Z
L
K
N
P
J
M
33. Look at the figure on the left. Mention the
pairs of vertical angles.
34. Look at the figure on the right.
Mention the kind of the following
pairs of angle.
5
13
4
2
6
a. ∠1 and ∠2. b. ∠4 and ∠5.
c. ∠3 and ∠6. d. ∠4 and ∠6.
1
2
110°
3
4
6
5
35. Estimate the measure of ∠1, ∠2, ∠3, ∠4,
∠5, and ∠6 on the figure on the left. Then
check your estimation using your
protractor.
36. Look at the figure on the right.
Mention the kind of the
following pairs of angle.
a. ∠1 and ∠2.
b. ∠3 and ∠4.
c. ∠5 and ∠6.
5
1
3
4
2
6

Determine the measure of ∠1 and ∠2 on the figure below.
State your reason.
1 2
110°37. 38.
1 2
80° 70°
Algebra. Determine the value of x on the figure below.
39. 40.
x°
30°
(2x)°
x°

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