Module 1 similarity

Module 1
Similarity
What this module is about
This module is about ratio, proportion, and the Basic Proportionality
Theorem and its Converse. In this module, you will learn the meanings of ratio
and proportion. And as you go over the exercises, you will develop skills that you
will need to solve problems especially on triangles in the next module.

What you are expected to learn
This module is designed for you to:
1. apply the fundamental law of proportions
• product of the means is equal to the product of the extremes.
2. apply the definition of proportion of segments to find unknown lengths.
3. illustrate and verify the Basic Proportionality Theorem and its Converse.

How much do you know
1. Express each ratio in simplest form.
a.

12
18

b.

8
12

2. Find the value of x in each proportion.
a.

3
2
=
x
6

b. 4:8 = 8:x

3. State the means and the extremes in each proportion.
a. 3:6 = 6:12

b. 3:4 = 6: 8

4. Write each ratio as a fraction in simplest form.
a. 6 to 30

b. 16 to 48

5. In the figure, points W, X, Y, and Z are collinear. WX = 4, XY = 6 , YZ = 10
●

●

●

●

W

X

Y

Z

Give each ratio in simplest form
a.

wx
xy

b. WX to (XY + YZ)

6. In ∆ ABC, CD = 2, AD = 4, CE = 3, EB = 5
C
D

E

A

B

Find the ratio:
a. CE to BC

b. CD to CA

7. Two consecutive angles of a parallelogram are in the ratio 1:2.
a. What is the measure of the smaller angle?
b. What is the measure of the larger angle?
8. In ∆ABC, DE //AC
B

D

E

A

C

a. Given that BD = 4, DA = 6 and BE = 5, find EC.
b. Given that BE = 4, EC = 5 and BD = 2, find DA.

2

9. In the figure ∠ADE ≅ ∠C
B
E

A

C
D

a. If AE = 2, AB = 8, AD = 3, what is AC?
b. If AD = 4, AC = 6, AE = 3, find AB.
10. In triangle ABC, DE // AB.

C

D

E

A

B

If DC = 6, DA = 8, CE = 2x, EB = 2x + 4
a. What is x?
b. What is CB?

What you will do
Lesson 1
Ratio and Proportion
A ratio is a comparison of two numbers. The ratio of two numbers a and b
where b is not equal to zero can be written in three ways: a :b, a/b and a to b.
A proportion is a statement of equality between two ratios. In the
c
a
proportion a: b = c:d or
=
, a, b, c, and d are called the terms of the
b
d
proportion. In a proportion, the product of the extremes is equal to the product

3

of the means. In the proportion a: b = c:d , the extremes are a and d and the
means are b and c, hence, ad = bc
Example 1
Express the ratio

12
in simplest form.
16

Solution:
Divide the numerator and the denominator by their GCF (Greatest
Common Factor), 4 .
12 ÷ 4 = 3
16 ÷ 4
4
The ratio in simplest form is

3
or 3:4.
4

Example 2
Find the ratio and express your answer in simplest form.
6 hours to 3 days
Solution:
Step 1. Convert 3 days to hours
3 x 24 = 72 ( Since there are 24 hours in one day)
Step 2. Write the ratio in terms of hours
6 hours to 72 hours
6 hours
1
=
72 hours
12
Another solution:
Step 1. Convert 6 hours to days
6
6 ÷ 24 =
24
1
= (Since there are 24 hours in one day)
2
This may be done as follows:
1 day
1
6 hours x
=
of a day
24 hours
4

4

Step 2. Write the ratio in terms of days
1
of a day
1
4
=
÷3
3 days
4
1 1
= x
4 3
1
=
12
Example 3
a. What is AB:BC?
b. What is (AB + BC) CD

3
●
A

5

7

●
B

●
C

●
D

Solution:
a. AB = 3
BC = 5
Hence AB :BC = 3:5
b. AB = 3
BC = 5
CD = 7
Hence (AB + BC) : CD = (3 + 5 ): 7
= 8:7
Example 4
1. a. What is AX:AB?
b. What is AY :YC?

A
4
X

2
Y

7

3.5

B

C

Solution
a. AX = 4
AB = AX + X B
= 4+7
= 11
Hence AX : AB = 4:11

5

b. AY = 2
YC = 3.5
Hence AY:YC = 2:3.5
3.5 can be written as 3

Therefore,

But

5
1
7
or 3 or .
10
2
2

2
AY
=
7
YC
2

7
7
4
2
can be written as 2 ÷
which is equal to 2 x or
7
2
2
7
2

Hence

AY 4
=
or AY:YC = 4:7
YC 7

Example 5
State the means and the extremes in the following statement.
3:7 = 6:14
Solution:
The means are 7 and 6 and the extremes are 3 and 14.
Example 6
Determine whether each pair of ratios forms a proportion.
a.

4 6
,
5 8

b.

4 8
,
7 14

Solution:
a. A proportion is an equality of two ratios

4 6
=
5 8
4:5 = 6:8

6

The product of the means is equal to the product of the extremes
5(6) =4(8)
30 = 32 This is a false statement
Hence the two ratios do not form a proportion.
b. A proportion is an equality of two ratios
c.
4
8
=
7
14
4:7 = 8:14
The product of the means is equal to the product of the extremes
7(8) = 4(14)
56 = 56 This is a true statement
Hence the two ratios form a proportion.
Example 7
Find the value of x.
3 = x
10
30
Solution:
Step 1. Rewrite in the ratio
3:10 = x:30
Step 2. Find the products of the means and the extremes. Then solve for x.
Remember, the product of the means is equal to the product of the
extremes.
10(x) = 3(30)
10x = 90
x=9
Example 8
The measures of two complementary angles are in the ratio 1:2. Find the
measure of each angle.

7

Solution:
Representation:
Let x = the measure one angle
90 –x = the measure of its complement
Proportion:

1
x
=
90 − x
2
x : (90-x) = 1:2
1(90 –x) = 2(x)
90-x = 2x
-x –2x = - 90
-3x = - 90
x = 30 measure of one angle
90 – x = 60 measure of its complement
Example 9
The measure of two supplementary angles are in the ratio 2:3. Find the
Solution:
Representation:
Let x = the measure of one angle
180 – x = the measure of its supplement
Proportion:

x
=
180 − x

2
3

x:(180-x) = 2:3
2(180-x) = 3(x)
360 – 2x = 3x
-2x –3x = -360

8

-5x = - 360
x = 72 the measure of one angle
180 – x = 108 the measure of its supplement
Example 10
In ∆ABC below, m∠C = 90. The measures of ∠ A and ∠B are in the ratio
4:5. Find the measures of ∠ A and ∠B.
B

C

A

Solution:
Representation:
Let x = measure of ∠ A
90-x = measure of ∠B (Because, ∠A and ∠B are acute angles)
Proportion:
x:90-x = 4:5
4(90-x) = 5x
360-4x = 5x
-4x – 5x = -360
-9x = -360
x = 40 measure of ∠A
90-x = 50 measure of ∠B
Example 11
In the figure, ∠ABD and ∠CBD form a linear pair. If the measures of ∠ABD
and ∠CBD are in the ratio 7 to 3, what is the measure of ∠CBD?

D ●
●
A

B

9

●
C

Solution:
Let x be the measure of ∠CBD.
180-x be the measure of ∠ABD
[Remember that if two angles form a linear pair they are supplementary]
Proportion:
m ∠ ABD = 7
m ∠ CBD 3
Substitute 180-x for m ∠ ABD and x for m ∠ CBD

7
180 − x
=
x
3
(180-x):x = 7:3
7x = 3(180-x)
7x = 540 – 3x
7x + 3x = 540
10 x = 540
x = 54 measure of ∠CBD.
180 – x = 126 measure of ∠ABD.
Example 12
In the figure, EB ⊥ EH and ET is in
the interior of ∠ BEH. If the
measures of ∠BET and ∠HET
are in the ratio 1:5, what is
the measure of ∠HET?

B●

E

T●
●
H

Solution:
Let x = measure of ∠ BET
90-x = measure of ∠ HET
[Remember that perpendicular rays form a right angle and
the measure of a right angle is 90.]
Proportion:
m∠ BET = 1
m∠HET 5

10

Substitute x for m∠BET and 90-x for m∠HET

1
x
=
90 − x
5
x : (90-x) =1:5
5x = 1(90 – x)
5x = 90 – x
5x + x = 90
x = 15 measure of ∠BET
90-x = 75 measure of ∠HET

Try this out
Set A
Find the ratio of each of the following. Use the colon and write your answers in
simplest form.
1. 14 cm to 28 cm
2. 8 hours to 12 hours
3. 16 days to 36 days
4. 42 dm to 63 dm
5. 12 inches to 36 inches
6. 24 feet to 36 feet
7. 12 inches to 24 inches
8. 6 inches to 2 feet
9. 3 feet to 24 inches
10. 8 hours to 1 day
11. 14 days to 2 weeks
12. 3 weeks to 24 days
13. 5 minutes to 60 seconds
14. 5 centimeters to 2 decimeters
15. 4 months to 1 year
16. 3 feet to 12 inches
17. 3 hours to 72 minutes
18. 2 weeks to 8 days
19. 2 centuries to 300 years
20. 200 years to 3 centuries
Set B.
Find the missing number

11

1.

2
n
=
4
10

2. n = 2
6 3
3. 3 = 5
n
10
4. 6 =
10
5. n =
7

3
n
2
5

6. 4 =
8

n
7

7. 7 = 5
n
2.5
8. 3 = 4
3.6 n
9. 2n = 5
3
2
10. 6 = 3n
5
4
11. 3n = 6
7 14
12. 4 = 2
5n 10
13. 3 = n
2
7
14. 3n = 3
5
2
15. 8 = 2
5 n

12

16. 6n = 9
8
2
17. 5n = 6
9
4
18. 5 = 12
4
n
19. 15 = n
3 4
20. 6 = 2
n 5
Set C
1. The measures of two supplementary angles are in the ratio 1:2. Find the
2. The measures of two supplementary angles are in the ratio 1: 4. Find the
measure of each angle
3. The measures of two complementary angles are in the ratio 2:3. Find the
4. The measures of two complementary angles are in the ratio 3:7. Find the
5. a. What is the ratio of AB to AC?
b. What is AE:ED?
A
5

6

B

E

5

6

C

D

6. a. What is BE:CD?
b. What is AB :BC?

D
E

8
4

A

C

13

3

B

7. a. What is AB:BC?
b. What is AB: (BC + CD)

x
y
b. What is x + y
y+z

5
4

●
A

5
●
B

8
●
C

●
D

8. a. What is

●

3

●

X

6

7

●

Y

●

Z

Determine whether each pair of ratios forms a proportion.
9. 6 , 7
7 8
10. 8, 16
9 18
11. 6, 18
9 27
12. 7 , 42
6 36
13. 11, 22
12 24
14. 13, 7
9 6
15. 22, 4
11 2
16. 14, 12
7 6
17. The acute angles in a right triangle are in the ratio 3: 6. Find the measure
of the larger of the two angles.
18. The acute angles in a right triangle are in the ratio 2 to seven. What is
the measure of the smaller of the two angles.
19. In the figure at the right, ∠1 and ∠2
form a linear pair. If the measures of
∠1 and ∠2 are in the ratio 2 to 8. Find
the measures ∠1 and ∠2.

14

1

2

20. In the figure below, BA ⊥ BC. BD is in the interior of ∠ABC. If the
measures of ∠ABD and ∠CBD are in the ratio 7 to 2. Find the measure
of ∠ABD.
A
D
B

C

Lesson 2
The Basic Proportionality Theorem and Its Converse
The Basic Proportionality Theorem: If a line is parallel to one side of a
triangle and intersects the other two sides in distinct points, then it divides the
two sides proportionally.
Illustration:

A
E

F

B

C

In the figure, if EF is parallel to BC and intersects AB and AC at points
E and F respectively, then
a. AE = AF
EB
FC
b. AE = AF
AB
AC
c. EB = FC
AB
AC

15

The Converse of the Proportionality Theorem: If a line divides the two
sides of a triangle proportionally, then the line is parallel to the third line.
Illustration:
A
E

F

B

C

If in ∆ABC, AE = AF, then EF // BC.
EB
FC
You can verify the Basic Proportionality Theorem by doing the following
activity. In this activity you need a ruler, a pencil and a protractor.
1. Draw ∆ABC such AB = 5 cm and AC = 10 cm.
2. Draw point E on side AB such that it is 2 cm away from vertex B.
3. Draw a line parallel to BC passing thru point E intersecting AC at point F.
Illustration:
A
E
●

F
●

2
B

C

4. Verify whether line EF is really parallel to BC by measuring ∠AEF and
∠ABC. Recall: If two lines are cut by a transversal and a pair of
corresponding angles are congruent, then the lines are parallel.
5. Find the lengths of AE, AF and FC.
6. Is AE = AF ?
EB FC
Is AE = AF ?
AB AC
Is EB = FC ?
AB AC

16

You can verify the Converse of the Basic Proportionality Theorem by
doing the following activity.
1. Draw ∆ABC such that AB = 7 cm and AC = 14 cm.
2. Draw point E on side AB such that it is 3 cm away from vertex A. What is
the length of EB?
3. Draw point F on side AC such that it is 6 cm away from vertex A. What is
the length of FC?
4. Draw a line passing through points E and F. Notice that line EF divides
sides AB ad AC proportionally.

3
6
=
4
8
Illustration:
A
3

6

E●
4

●F
8

B

C

5. What can you say about line EF? Is it parallel to BC?
Verify by measuring ∠AEF and ∠ABC.
Example 1
In ∆ABC, DE // BC. If AD = 6, DB = 8 and EC = 12, find AE.
A
6

x

D

E

8

12

B

C

Solution:
Let x = AE
x = 6
12
8
x = 3
12
4
x:12 = 3:4
12(3) = 4(x)
4x = 12(3) (by Symmetric Property of Equality)

17

4x = 36
x=9
Example 2
A
In ∆ACD, BE //CD If AB = 8, BC = 4, AE = 6, find AD and ED
Solution:

8

6

B
Let x = AD
AB = AE
AC AD

E

4
C

D

8 = 6
8 +4
x
8:12 = 6:x
12(6) = 8(x)
8x = 72
x=9
AD = 9
Hence: ED =AD – AE
=x–6
=3
Example 3

A
2x

3x

D

E

12
B

C

In ∆ABC, DE // BC, AD =2x, AB = 20, AE = 3x and EC = 12. Find AD
Solution:
Step 1.

AD = AE
AB
AC
2x
3x
=
20
3x + 12
3x
x
=
10
3x + 12

18

x(3x + 12) = 10(3x)
3x2 + 12x = 30x
3x2 + 12x – 30x = 0
3x2 – 18x = 0
3x(x – 6) = 0
3x = 0
x=0

by factoring

x–6=0
x=6

Equating both factors to 0.

Step 2. Substitute 6 for x in 2x
2x = 2(6)
= 12
Hence AD = 12

Try this out
Set A
In the figure, DE // BC.
A
D

E

B

C

True or False
1. AD = AE
DB
EC
.

6. BD = AE
AD
EC

2. AD = AE
AB
AC

7. DB = AB
EC
AC

3. AD = EC
DB
AE

8. AE = EC
AD
DB

4. AD = AC
AB
AE

9. AD = AE
EC
DB

5. DB = EC
AB
AC

10. AD = DB
AE
EC

19

B
In the figure, AB // ED.
D
11. CD = CA
BD AE
12. BC = AC
CD
AE
A

E

C

13. AE = BC
EC BD
14. AE = CD
AC BD
15. CD = CB
AB AC
Set B
In ∆ABC , DE // BC.
A
D

E

B

C

1 If AD = 1, DB = 2, and EC = 4, find AE.
2. If AD = 2, DB = 4, and AE = 1, find EC.
3. If BD = 6, AD = 4, and AE = 5, find EC.
4. If BD = 5, AD = 6, and CE = 9. what is AE?
5. If AD = 2, AB = 8, and AC = 10 what is AE?
In ∆ ABC, DE // BC

C
E

A

D

B

20

6. If AD = 4, AB = 12 and AC = 15, what is AE?
7. If AD = 6, DB = 2 and AC = 10 what is AE?
8. If AB = 12, AE = 4 and EC = 6, what is AD?
9. If AD = 2.5, DB = 4 and EC = 6, find AE.
10. If DB = 5, AE = 3.5 and EC = 7, what is AD?
In ∆CAT, AC // GD.

A
G

C

D

T

11. If CD = 10, DT = 12, AG = 8, what is GT?
12. If CT = 10, CD = 4, AT = 7, what is AG?
13. If AG = 3, TG = 6, DT = 8, what is CD?
14. If AG = 4, GT = 12, CD = 6, what is DT?
15. If AT = 12, AG = 6, CT = 16, what is CD?
Set C

A

D

E

B
1.
2.
3.
4.
5.

C

If AD = x + 1, DB = 2, AE = 10 and EC = 5, what is x?
If AD = x + 2, DB = 4, AE = 6 and EC =8, what is x?
If AD = 4, DB = 6, AE = x + 3 and EC = 7.5, what is x?
If AD = 2, DB = 6, AE = x + 2 and EC = 9, find x
If AD = 5, DB = x + 2, AE = 4 and EC = 4.8, what is x?

In ∆ BAC, BC // DE

C
E

B

D

A

21

6. If AD = 6, AE = 9, and AC = 21, what is BD?
7. If AD = 4, BD = 5 and AE = 6, what is EC?
8. If DB = 6, AE = 6 and EC = 9 what is AD?
9. If AD = x, AB = 5, AE = 2x and EC = 4, what is AD?
10. If AD = x, AB = 5, AE = 2x and EC = 6, what is AD?
In ∆ABC, EF// BC.

C
F

A

E

B

11. If AE = x, EB = x + 10, AF = 4 and FC = 6, what is x?
12. If AE = 6, EB = 8, AF = y and FC = 2y –2, what is y?
13. If AE = 5, AB = x +2, AF = 10 and AC = 3x, what is x?
14. If AE = x + 4, EB = 12, AF = x + 5 and FC = 14, what is x?
15. If AE = 6, EB = 4y –1, AF = 2 and FC = 3, what is y?

Let’s summarize
1. A ratio is a quotient of two numbers .
2. A proportion is an equality of two ratios.
3. The Basic Proportionality Theorem:
If a line is parallel to one side of a triangle and intersects the other two
sides in distinct points, then it divides the two sides proportionally.
4. The Converse of the Basic Proportionality Theorem:
If a line divides the two sides of a triangle proportionally, then the line is
parallel to the third side

What have you learned
1. The value of n in 4:5 = n:20 is
A. 14
B. 15

C. 16
D. 17

2. The value of x in 2x = 3
6
2
A. 3.5

C. 5.5

22

B. 4.5

D. 6.5

18
as a fraction in lowest terms
30
3
C.
5

3. Write the ratio
A.

9
16

B.

15
9

D.

5
3

4. Which of the following proportions is true?
A. 2: 3 = 4:5
C. 4: 8 = 2: 4
B. 5:7 = 3: 2
D. 4 :7 = 7: 4
5. Two consecutive angles of a parallelogram are in the ratio 2:3. What is
the measure of the smaller angle?
A. 36
C. 108
B. 72
D. 90
A
6. In ∆ ABC, XY // BC.
X

Y

B

C

Given that AX = 5, XB = 6, and AY = 8, what is YC?
A. 9
C. 7.6
B. 7
D. 9.6
7. In the figure below, What is AB:(BC + CD)?
3
●
A

7
●
B

2
●
C

●
D

A.

3
7

C.

3
14

B.

10
9

D.

1
3

8. Which of the following pairs of ratios forms a proportion?
3 5
6 8
A. ,
C. ,
5 9
7 9

23

B.

5 10
,
,
7 14

D.

9. In ∆ ABC, DE // AB.

5 6
,
6 5

C
D

E

A

B

If DC = x, AD = 4, CE = x + 1 and EB = 8, what is x?
A. 4
C. 2
B. 3
D. 1
10. In the figure, DE // AC.

B
D

E

A

C

If BE =2, BC = 6 and BD = 3, what is BA?
A. 8
B. 9

C. 5
D. 12

11. Write the ratio 2 hours : 1 day as a fraction in lowest terms
A. 2:1
C. 1:12
B. 1:2
D. 12:1
12. Which of the following proportions is false?
A. 2:7 =8:28
B. 3:5 = 9:15

C. 4:7 = 8:14
D. 5:6 = 15:24

13. In ∆ABC, DE // BC. Which of the following proportions is false?
B
D

A

E

24

C

A. AD = AE
DB EC

C. EC = DB
AC AB

B. AB = AC
AD AE

D. AD = EC
AB AC

14. in the figure. Points A, B, and C are collinear. The ratio of ∠ABD to ∠CBD
is 5:1 . What is m∠CBD?
D●
●
A

●
B

A. 150
B. 30

●
C
C. 50
D. 100

15. ∆ ABC is a right triangle. If the ratio of ∠A to ∠B is 2:3, what is m∠ B?
B

C

A

A. 18
B. 27

C. 20
D. 36

25

Answer Key
How much do you know
2
3
2
b.
3

1. a.

2. a. x = 4
b. x = 16
3. The means are 6 and 6
The extremes are 3 and 12

1
5
1
b.
3

4. a.

2
3
1
b.
4

5. a.

3
8
1
b.
3

6. a.

7. a. 60
b. 120
8. a. 7.5
b. 2.5
9. a. AC = 12
b. AB = 4.5
10. a. x = 4
b. CB = 28

26

Try this out
Lesson 1
Set A
1. 1:2
2. 2:3
3. 4:9
4. 2:3
5. 1:3
6. 2:3
7. 1:2
8. 1:4
9. 3:2
10. 1:3
11. 1:1
12. 7:8
13. 5:1
14. 1:4
15. 1:3
16. 3:1
17. 5:2
18. 7:4
19. 2:3
20. 2:3

Set B
1. n = 5
2. n = 4
3. n = 6
4. n = 5
5. n = 2.8
6. n = 3.5
7. n = 3.5
8. n = 4.8
9. n = 3.75
10. n = 1.6
11. n = 1
12. n = 4
13. n = 10.5
14. n = 2.5
15. n = 1.25
16. n = n = 6
17. n = 2.7
18. n = 9.6
19. n = 20
20. n = 40

Set C
1.
2.
3.
4.
5.

Set B
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.

Set C
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.

60 and 120
36 and 144
36 and 54
27 and 63
a.1:2
b. 1:1
6. a. 1:2
b. 3:5
7. a. 4:5
b. 4:13
8. a. 1:2
b. 9:13
9. No
10. Yes
11. Yes
12. Yes
13. Yes
14. No
15. Yes
16. Yes
17.60

Lesson 2
Set A
1. True
2. True
3. False
4. False
5. True
6. False
7. True
8. True
9. False
10.True
11.False
12. False
13.False
14.False
15.False

2
2
7.5
10.8
2.5
5
7.5
4.8
3.75
2.5
9.6
2.8
4
18
8

27

3
1
2
1
4
8
7.5
4
3
2
20
3
4
2
2.5

What have you learned
1. C
2. B
3. C
4. C
5. B
6. D
7. D
8. B
9. D
10. B
11. C
12. D
13. D
14. B
15. D

28

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