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

G H

Figure 7.1(a)

Figure 7.1(b)

Look at Figure 7.1(b)

and The 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 BC Area 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.

Mathematics for Junior High School Year 7 / 279

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

•

• •

280 / Student’s Book – Lines and Angles

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

Mathematics for Junior High School Year 7 / 281

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.

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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

Then check your estimation using your protractor. a.

∠A = . . . .°

∠A is . . . . . . . . . . . angle.

A

B

b. ∠B = . . . .° ∠B is . . . . . . . . . . . angle

c.

C ∠C = . . . .° ∠C is . . . . . . . . . . . . angle.

Mathematics for Junior High School Year 7 / 283

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.

R Q P

T S

284 / Student’s Book – Lines and Angles

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

P K

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

3 4

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.

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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.

286 / Student’s Book – Lines and Angles

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

•

Mathematics for Junior High School Year 7 / 287

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

288 / Student’s Book – Lines and Angles

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

Mathematics for Junior High School Year 7 / 289

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

R Q

P

16. 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. Then check your estimation using your protractor.

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.

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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

C B 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

Mathematics for Junior High School Year 7 / 291

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 “.

292 / Student’s Book – Lines and Angles

32. Determine the measure of ∠AZC and ∠AZD on the figure on the right.

B C

D A

(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 1 3

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

Mathematics for Junior High School Year 7 / 293

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°

294 / Student’s Book – Lines and Angles