9

Mathe
matics
Quarter 3 – Module 6:
Triangle Similarity
 What I Need to Know

This module was specifically developed and designed to provide you fun
and meaningful learning experience, with your own time and pace.

After going through this module, you are expected to:

● Illustrates similarity of figures (M9GE-IIIg-1)

● Prove the conditions for similarity of triangles. (M9GE-IIIg-h-1)
● Solve problems and prove statements by means of the AA Similarity
Postulate, SAS Similarity Theorem, SSS Similarity Theorem, Right Triangle
Similarity Theorem and Special Right Triangle Theorem (M9GE-IIIg-h-1)

What I Know

Find out how much you already know about the module. Write the letter
that you think is the best answer to each question on a sheet of paper. Answer all
items. After taking and checking this short test, take note of the items that you
were not able to answer correctly and look for the right answers as you go through
this module.

1. ∆𝐴𝐵𝐶~∆𝑃𝑄𝑅 based from the figure below. Which of the following will support
the similarity statement?
B Q
a. AA Similarity Postulate
8 10 16 20
b. SAS Similarity Theorem A C P R
6 12
c. SSS Similarity Theorem

d. ASA Similarity Postulate

2. Which theorem or postulate proves that the triangles below are similar?
a. AA Similarity Postulate G T

b. SSS Similarity Theorem 6mm 18mm

55°
c. SAS Similarity Theorem E 8mm O 55°
R I
24mm
d. ASA Similarity Postulate

3. Which theorem proves that ∆𝐸𝐷𝐶~∆𝐸𝐴𝐵?

a. SAS ~ Theorem b. SSS ~ Theorem

 c. ASA ~ Theorem d. AA ~ Postulate

4. If two angles of one triangle are congruent to 2 angles of another triangle,

then the triangles are similar by _________
a. SSS Theorem c. ASA Postulate
b. AA Postulate d. SAS Theorem

5. Which pairs of triangles below are similar?

a. d and f
b. d and g

c. f and g
d. No triangles are similar.

6. Which of the conditions below, prove that the two triangles are similar?
a. SAS ~ Theorem 5{

b. SSS~Theorem
15
c. ASA~ Postulate
d. AA~ Postulate

7. The lengths of the corresponding sides of two similar triangles are 5 cm and
20 cm. Which of the following is the ratio of the corresponding sides?
a. 1: 2 b. 1: 4 c. 3:4 d. 2:3

For items 8 & 9, let ∆𝐴𝐵𝐶 and ∆𝑅𝑆𝑇 be similar

triangles.

8. Find the value of 𝑥.

. a. 20 b. 18 c. 16 d. 15

9. What must be the value of y?

a. 20 b. 18 c. 16 d. 15

10. In ∆𝐴𝐵𝐶, ̅𝐷𝐸̅̅̅ ∥ ̅𝐴𝐶̅̅̅. If |𝐵𝐷| = 12 𝑐𝑚, |𝐵𝐶| = 30 𝑐𝑚, |𝐴𝐶| = 35 𝑐𝑚 and |𝐴𝐷| =
3 𝑐𝑚, then which of the following must be
the length of ̅𝐷𝐸̅̅̅?
a. 16 cm B
b. 20 cm
c. 24 cm D E
d. 28 cm A C
What’s In

Directions: Identify the given figures whether they are similar or not. Write
SHEEESH if the figures are similar otherwise write BOOGSH, then justify your
answer.

1. 2. _

3. 4. _
 Lesson 1 Similarity of Triangles

What’s New

Similarity
In this lesson, you will learn that there are triangles and other polygons that have
the same shape but do not necessarily have the same size. The illustrative
example below will give you an idea on how we can say that the given figures are
similar

If you will observe, parallelogram LAKE and parallelogram BIRD have the same
shape. When you pair the corresponding vertices, the angles coincide. It shows
that their corresponding angles are congruent: ∠A ≅∠I; ∠K≅ ∠D ; ∠E ≅ ∠R ;
and ∠L ≅ ∠B.

Another thing is that the ratios of the measure of the lengths of their
corresponding sides are equal.

BI 4 ID 6 DR 4 RB 6 1
Thus, in BIRD to LAKE, = = = = = = = = . Here, the scale
LA 8 AK 12 KE 8 EL 12 2
1
factor k is
. We could also turn it around as LAKE to BIRD where
2
LA 8 AK 12 KE 8 EL 12
= = = = = = = =2. Now here, the scale factor k is 2.
BI 4 ID 6 DR 4 RB 6
Based on the illustrative example, two polygons are similar (the symbol is ∼) if
their vertices can be paired so that corresponding angles are congruent and the
lengths of their corresponding sides are proportional.

To indicate that parallelogram LAKE is similar to parallelogram BIRD, you can

write LAKE ∼ BIRD. If you use this notation, write the corresponding vertices on
the same order.

Proves the Conditions for Similarity of a Triangle – SAS, SSS

Lesson 2
and AA Similarity Theorem

What is It

In this lesson, we are only going to focus on the similarity of two triangles. We will
apply our prior knowledge on the definition of similar polygons to understand the
postulates and theorems in proving the similarity of triangles.
To prove the similarity of two triangles using the definition of similarity, we
must establish that the three corresponding angles are congruent and that the
three ratios of the lengths of corresponding sides are equal.
.
Example

Illustration
Given: △MNP ⟷ △XYZ , ∠M ≅ ∠X,
∠N ≅ ∠Y and ∠P ≅ ∠Z
Prove: △MNP ∼ △XYZ

Proof:

Statement Reason

Construct 𝐴𝐵 ̅̅̅̅̅, such that 𝐴𝑁̅̅̅̅ ≅ 𝑋𝑌̅̅̅̅; By construction

𝑁𝐵̅̅̅̅ ≅ 𝑌Z
∠N ≅ ∠Y Given
△ANB ≅ △XYZ SAS Congruence Theorem
 ∠NAB ≅ ∠X, ∠NBA ≅ ∠Z Corresponding parts of congruent
triangle are congruent. (CPCTC)
∠NAB ≅ ∠M, ∠NBA ≅ ∠P Transitive Property
m∠MNP = m∠ANB Reflexive Property
∠MNP ≅ ∠ANB Definition of congruent angles
If two lines are cut by a transversal,
𝐴𝐵̅̅̅̅‖ 𝑀𝑃 the corresponding angles are
congruent and the two lines are
parallel.
N𝐴 = NB Basic Proportionality Theorem
𝑁𝑀 NP
NA = YX ; NB = YZ Congruent segments have equal
measures.
△ANB ∼ △MNP Definition of similar triangles
∴ △MNP ∼ △XYZ Transitive Property

Here are the other Triangle Similarity Theorems.

1.1 SAS Similarity Theorem
If an angle of one triangle is congruent to an angle of another triangle and if
the lengths of the sides including these angles are proportional, then the triangles
are similar.

Illustration
YX YZ
Given:△XYZ ⟷ △ABC, ∠Y ≅ ∠B and =
BA BC
Prove: △XYZ ∼ △ABC

Proof:

Statements Reasons
Draw such that
and By construction
.
Y B Given
XYZ MBN SAS Congruence Theorem
and CPCTC
Given
By substitution

Converse of Basic Proportionality

MN ‖ AC
Theorem
If two parallel lines are cut by a
BMN BAC and BNM BCA transversal, corresponding angles are
congruent.
B B Reflexive Property
ABC MBN AAA Similarity Theorem
XYZ ABC Transitive Property
Example 1
Show that the triangles ABC and ADE in the figure on
the right are similar.

Solution
● ∠BAC ≅ ∠DAE by Reflexive Property
● Since the measures of the lengths of the sides are given, calculate the ratios
AB 21 7 AC 7
of the corresponding sides. = = and =
AD 6 2 AE 2
● The length of the corresponding side of the two triangles are proportional
and the included angles are congruent, therefore, ∠ABC ∼ ∠ADE.

Example 2

Given 𝐴𝐸 = BE , prove that △BEA ∼ △CED.

𝐷𝐸 CE

Solution

Proof:

Statements Reasons
AE = BE Given or by hypothesis
DE CE
BEA and CED are vertical angles. Definition of vertical angles
BEA CED Vertical angles are congruent.
BEA CED SAS Similarity Theorem

1.2 SSS Similarity Theorem

If the three corresponding sides of two triangles are proportional, then the two
triangles are similar.

Illustration
Given: △ARM ⟷ △LEG, 𝐴𝑅 = RM = AM
𝐿𝐸 EG LG
Prove: △ARM ∼ △LEG

Proof:

Statements Reasons
Draw XY such that XE = AR AND EY =
RM By construction
Congruent segments have equal
XE = AR and EY = RM measures.
 AR = RM = AM Given
LE EG LG
XE = EY By substitution
LE EG
E E Reflexive Property
LEG XEY SAS Similarity Theorem
XY = XE Definition of similar triangles
LG LE
By substitution (XE = AR)

XY = LG ; AM = LG Multiplication Property
XY = AM Transitive Property
ARM SSS Congruece Theorem
ARM LEG Transitive Property

Example 1
Show that the triangles MNP and QRS in the figure below are similar.

Solution
● Since the measures of the lengths of the sides are given, calculate the ratios
of the corresponding sides. =2

● The ratios of the lengths of the three corresponding sides of the two
triangles are equal, thus △MNP ∼△QRS.

Example 2
a. Given: △CAR and △PET. State the proportions that must be true if △CAR ∼△PET
by SSS Similarity.
b. Given the statement that shows the proportionality of the three corresponding
sides of the two triangles. DO = ON = DN , name the two similar triangles.
KE EY KY
Solution
a. 𝐶𝐴 = AR = CR
𝑃𝐸 ET PT
b. △DON and △KEY

1.3 AA Similarity Theorem

If two angles of one triangle are congruent respectively to two angles of another
triangle, then the triangles are similar.

Illustration
Given:∠A ≅∠O ; ∠C ≅∠D
Prove:△CAT ∼△DOG

Proof:
Statements Reasons
∠A ≅∠O ; ∠C ≅∠D Given
m∠A ≅ m∠O ; m∠C ≅ m∠D Definition of Congruent Angles
m∠A + m∠C = m∠O ≅ m∠D Addition Property
m∠A + m∠C + m∠T= 180 The sum of the measures of the
m∠O + m∠D + m∠G = 180 interior angles of a triangle is 180.
m∠A + m∠C + m∠T= m∠O + m∠D + Transitive Property
m∠G
m∠T = m∠G Addition Property
∠T = ∠G Definition of congruent angles
△CAT ≅ △DOG AAA Similarity Theorem

Example1
Given: 𝑈𝑉̅ || BC
Prove: △ABC ≅ △AUV by AA Similarity Theorem

Solution

Proof:

Statements Reasons
Given
If two parallel lines are cut by a
∠𝐴𝑈𝑉 ≅ ∠ABC transversal, corresponding angles are
congruent.
m∠BAC ≅ 𝑚∠UAV Reflexive Property
∠BAC ≅ ∠UAV Definition of congruent angles
AA Similarity Theorem

Example 2
Given: 𝐴𝐵 || 𝐷𝐶. Name at least two pairs of corresponding
angles that are congruent to prove that △AOB ∼ △DOC by
AA Similarity Theorem.

Solution
● If 𝐴𝐵 and 𝐷𝐶̅ are parallel and cut by transversal AD, then the congruent
alternate interior angles are ∠BAO and ∠CDO.
● If 𝐴𝐵 and 𝐷𝐶̅ are parallel and cut by transversal BC, then the congruent
alternate interior angles are ∠ABO and ∠DCO.
● Vertical angles are congruent, hence ∠AOB and ∠DOC are congruent.
1.4 Right Triangle Similarity Theorem
In a right triangle, the altitude to the hypotenuse divides the triangle into
similar triangles, each similar to the original triangle.

Illustration
Given: △GRA is a right triangle with ∠GRA as right angle,
𝐺𝐴̅ as the hypotenuse and 𝑅𝑌̅ is the altitude to the
hypotenuse of △GRA.

Prove: △GRA ∼ △RYG ∼ △RYA

Proof:

Statements Reasons
GRA is a right triangle with ∠GRA as
right angle, ̅𝐺𝐴̅̅̅ as the hypotenuse Given
and 𝑅𝑌̅̅̅̅ as the altitude to the
hypotenuse of △GRA.
𝑅𝑌̅̅̅̅ ⊥ ̅𝐺𝐴̅̅̅ Definition of Altitude
∠RYG and ∠RYA are right angles Definition of Perpendicular Lines
∠RYG ≅ ∠RYA ≅ ∠GRA Definition of Right Angles
∠YGR ≅ ∠RGA ; ∠YAR ≅ ∠RAG Reflexive Property
∴ △GRA ∼ △RYG ∼ △RYA AA Similarity Theorem

Example
● Using the figure on the right, name the three similar triangles.
● Write the proportions that exist among corresponding parts of similar
triangles.
Solution
● △ABC, △ACH and △CBH
● 𝐴𝐵 = BC = AC = AC = CH = AH = AB = BC = AC
AC CH AH CB BH CH CB BH CH

1.5 Special Right Triangle Theorem

We have two theorems under the special triangle:

1.5.1 The Isosceles Right Triangle Theorem or the 45°-45°-90° Right

Triangle Theorem
● The length of the hypotenuse of a 45°-45°-90°triangle is √2 times the
length of the leg and each leg is √2/2 times the hypotenuse.

Illustration
Given:△ABC is a 45°-45°-90° triangle.
Prove: c = a√2

Proof:
 △ABC is a 45°-45°-90° triangle. Using the Pythagorean Theorem
(a² + b² = c²), a² + a² = c². Simplifying, it follows that c² = 2a², c = √2𝑎², and c= a√2.

1.5.2 The 30°-60°-90° Right Triangle Theorem

● In a 30°-60°-90° triangle, the length of the hypotenuse is 2 times the
length of the shorter leg and the length of the longer leg is √3 times
the length of the shorter leg. On the other hand, the shorter leg is ½
the hypotenuse or √3/3 times the longer leg.

Illustration
Given:△ABC is a 30°-60°-90° triangle.
Prove: c = 2a and b = a√3

Proof:
Draw △ADC so that △ABC ≅ △ADC. m∠BAC + m∠DAC = m∠BAD = 60°. m∠B =
m∠D = m∠BAD = 60°. This shows that △ABD is equiangular, and hence,
equilateral. It follows that c = 2a. Using Pythagorean Theorem, a2 + b2 = (2a)2 =
4a². When simplified, b² = 3a² or b = a√3.

What’s More

Direction: Fill in the blanks.

A.

Given: ∆EAT~∆RUN

1. ET↔ .
2. UN↔ .
3. AE↔ .
4. ∠U↔ .
5. ∠N↔ .

B.

Given: AB 丄 CD and CB ≌ BD
Prove: ∆ABC ≌ ∆ABD

Statements Reasons

1. (1) Given

2. ∠ABC & ∠ABD are right angles Definition of 丄

 3. AB ≌ AB (2)

4. ∆ABC ≌ ∆ABD (3)

What I have learned

Directions: Below is an exit ticket. Complete the table below by writing a good
definition of the different theorems in similarities of triangles.
Theorems Definition

SSS Similarity (a)

Theorem

AA Similarity Theorem (b)

SAS Similarity (c)

Theorem

45-45-90 Right (d)

Triangle Theorem

30-60-90 Right (e)

Triangle Theorem

Right Triangle (f)

Similarity Theorem
 What I can do

Directions: Prove that “Triangles similar to the same triangles are similar” by
supplying the missing statements and reasons of the proof below.

Given: ∆ABC ~∆XYZ and ∆DEF ~∆XYZ.

Prove: ∆ABC ~𝐷𝐸F

Statements Reasons

1. ∆ABC ~∆XYZ and ∆DEF ~∆XYZ. 1.

2. ∠𝐴 ≅ ∠𝑋, ∠𝐵 ≅ ∠𝑌, ∠𝐶 ≅ ∠𝑍, ∠𝐷 ≅ 2.

∠𝑋, ∠𝐸 ≅ ∠𝑌, ∠𝐹 ≅ ∠Z

3. 3. Transitive property

4. ∆ABC ∆ABD 4.

Assessment

Directions: Prove and solve the following.

1. If AB ∥ DE ,Prove ∆ABC ~∆EDC

2. If 𝑆𝑇 ∥ 𝐴𝐶̅, show that ∆𝐵𝑆𝑇 ~ ∆𝐵𝐴𝐶

3. Prove that if two isosceles triangles have congruent vertex angles, then they are
similar.

AC BC
4. Given: =
PR QR

Prove: ∆ABC ~ ∆PQR

Additional Activities

A.Directions: Analyze the figure below. Fill in the blanks to satisfy the statements.
∆ 𝐒𝐄𝐓 ~ ∆ 𝐀𝐈𝐃

1. DI↔ .
2. TS↔ .
3. AD↔ .
4. ∠I↔ .
5. ∠D↔ .

B.Direction: Do what is required involving similarity of figures.

Given: ∆𝐴𝐵𝐶~∆𝐷𝐸𝐹

1. Which side corresponds to BC?

2. If m∠𝐷 = 38°, what is m ∠𝐴?
3. What is the value of x?
4. What is the value of y?
5. What is the scale factor of ∆𝐴𝐵𝐶~∆𝐷𝐸𝐹?

Answer Key