Congruence and Similariry

MATERIAL

Congruency & Similarity

PENYUSUN
Ronald Naradus Simanjuntak

NIM : 2022084679

Daftar Isi

Daftar Isi ......................................................................................................................................... 2
Kata Pengantar ............................................................................................................................3
Petunjuk Penggunaan modul ............................................................................................. 4
KD, IPK dan Tujuan Pembelajaran .......................................................................... 5
Map Concept .................................................................................................................................. 6
Materi 1 : Congruency

Properties of Congruent Figures ..................................................................... 7
Identifying congruent Figures ............................................................................... 10
Congruent Triangles .................................................................................................... 11
Proving Congruent Triangles................................................................................. 13
Applications of Congruency....................................................................................... 14

Materi 2 : Similarity
Similar Shapes ................................................................................................................... 15
Enlargement using lengths......................................................................................... 17
Perimeter, Area and Volume................................................................................ 19
Similar Triangles............................................................................................................... 21
Proving Similar Triangles.......................................................................................... 22
Applications of Similarity............................................................................................ 23

Formative Test ............................................................................................................................ 25
Daftar Pustaka .......................................................................................................................... 27

2

Kata Pengantar



Salam dan Bahagia



Puji syukur kehadirat Tuhan yang senantiasa melimpahkan segala rahmat dan

kasih-Nya sehingga penyusun dapat menyelesaikan modul ini.



Modul ini disusun untuk memenuhi kebutuhan peserta pendidikan dan pelatihan
Diklat dalam rangka sertifikasi guru profesional di bidang guru Matematika.



Teknik penyajian yang diangkat dilakukan secara terpadu tanpa pemilihan
berdasarkan jenjang pendidikan. Cara ini diharapkan bisa meminimalisir terjadinya

pengulangan topik berdasarkan jenjang pendidikan.



Pembahasan modul ini dimulai dengan menjelaskan tujuan yang akan dicapai.
Kelebihan modul ini, Anda bisa melihat keterpaduan ilmu matematika.



Pembahasan yang akan disampaikan pun disertai dengan soal-soal yang dapat
digunakan untuk mengukur tingkat ketercapaian dan ketuntasan.



Penyusun menyadari bahwa di dalam pembuatan modul masih banyak kekurangan,
untuk itu penyusun sangat membuka saran dan kritik yang sifatnya membangun.

Mudah-mudahan modul ini memberikan manfaat.




Tangerang , 15 November 2022


Penulis

3

Petunjuk Penggunaan Modul

Untuk mempelajari modul ini hal-hal yang perlu dilakukan oleh
peserta didik adalahvsebagai berikut.

Baca pendahuluan modul untuk mengetahui arah
pengembangan modul.
Membaca kompetensi dasar dan tujuan yang ingin dicapai
melalui modul.
Agar memperoleh gambaran yang utuh mengenai modul,
maka pengguna perlu
membaca dan memahami peta konsep.
Mempelajari modul secara berurutan agar memperoleh
pemahaman yang utuh.
Memahami contoh-contoh soal yang ada, dan
mengerjakan semua soal latihan yang ada.
Jika dalam mengerjakan soal menemui kesulitan,
kembalilah mempelajari materi yang terkait.
Ikuti semua tahapan dan petunjuk yang ada pada modul
ini.
Mempersiapkan alat tulis untuk mengerjakan soal-soal
latihan.
Selamat belajar menggunakan modul ini, semoga
bermanfaat.

4

KD, IPK dan Tujuan Pembelajaran

Kompetensi Dasar

3.6. Menjelaskan dan menentukan kesebangunan dan kekongruenan
antarbangun datar.

4.6. Menyelesaikan masalah yang berkaitan dengan kesebangunan dan
kekongruenan antarbangun datar

Indikator Pencapaian Kompetensi

3.6.1 Menemukan syarat-syarat dua bangun segi banyak yang kongruen.
3.6.2 Menyimpulkan pembuktian dua segitiga kongruen
3.6.3 Menemukan syarat dua bangun segi banyak yang sebangun
3.6.4 Menyimpulkan pembuktian dua segitiga sebangun
4.6.1 Memecahkan masalah sehari-hari berdasarkan hasil pengamatan yang

terkait penerapan konsep kekongruenan bangun datar segi banyak.
4.6.2 Memecahkan masalah sehari-hari berdasarkan hasil pengamatan yang

terkait penerapan konsep kesebangunan bangun datar segi banyak

Tujuan pembelajaran

Setelah mengamati masalah di LKPD peserta didik dapat menemukan syarat
dua bangun segi banyak yang kongruen.
Setelah mencermati video pembuktian kekongruenan lewat Interactive Flat
Panel (IFP) peserta didik dapat membuat kesimpulan teorema -teorema
segitiga kongruen
Setelah mencermati masalah di bahan pembelajaran siswa dapat menemukan
syarat dua bangun segi banyak yang sebangun
Setelah mencermati video lewat Interactive Flat Panel (IFP) peserta didik
dapat menyimpulkan teorema dua segitiga sebangun
Setelah diskusi kelompok peserta didik dapat menyelesaikan masalah sehari-
hari yang terkait penerapan konsep kekongruenan bangun datar segi banyak.
Setelah diskusi kelompok peserta didik dapat masalah sehari-hari terkait
penerapan konsep kesebangunan bangun datar segi banyak

5

Map Concept

Congruency and Similarity

Congruency Similarity

Congruent Congruent Similar Similar
figures Triangles figures figures

6

Congruency

Properties of congruent Figures
Investigation

Figure 1 : Show four pairs of scissors

Figure 1
What can we say about the shape, size, orientation and position of the
pairs of scissors?
if we cut out the pair of scissors and stack them up, what will we
observe?
The pair of scissor in A can be move to look like sciccors in B by
translation and rotation 90 degrees clockwise. How can we move the pair
of scissoor A to look like scissors C and D?

7

Congruency

Properties of congruent Figures

Investigation

From the investigation we observe that :

Two figures are congruent if they have exactly the same shape and size.
They can be mapped onto each other under translation, rotation and
reflection.

Thinking Are they congruent?
time Explain your answer

Figure 2

Congruence is a property of geometrical figures. The two pairs of scissors in
Fig. 2 are congruent because they have exactly the same shape and size.
Fig. 3 shows some patterns that are formed by congruent figures. These are
known as tessellations, which can be found in many real-life objects

Figure 3
Another real-life application of congruence is photocopying as the photocopied
document is of the same shape and size as the original document. The concept
of congruence also plays an important role in the manufacturing sector. The
congruence pen refills allows us to our pens when they run dry.

8

Congruency

Class
Discussion

Congruence in the Real World
1.Look around your classroom or school. Find at least 3 different sets of
congruent objects.
2.Tessellations, like those shown in Fig. 3, can be found on floor tiles. What
are some other objects that exhibit tessellations?
3.Discuss with your classmates other real-life applications of congruence.

Worked Which shapes are congruent?
Example

Solutions Congruent shapes

A, H, N

B, O, L

K, M

P, Q 9

Congruency

Identifying Congruent Shapes

D D' C'
C

A B A' B'

Figure 4

Figure 4 shows two congruent quadrilaterals ABCD and A'B'C'D'. The vertex
A corresponding to vertex A' because they have the same angle. Similarly, the
vertices that correspond to B, C and D are B', C', D' respectively.

The symbol ' ' means congruent to, Thus for two quadrilaterals in figure 4,
we have ABCD A'B'C'D'

The corresponding angles and sider of congruent figures are equal

Worked C W
Example 120o 55
Z
B o X
6 cm

A 10 cm

D Y

Given that ABCD is congruent to XYZW, copy and complete each

of the following :
(1) BCD = YZW = ....... (3) AB = ............. = .............. cm

(2) .................. = XWZ = ....... (4) ......... = YZ = ............. cm

10

Congruency

Congruent Triangles

Triangles are congruent when they have
exactly the same three sides and exactly the same three

angles.

Same Sides

is congruent to and also to

because they all have exactly the same sides.
But:

is NOT congruent to
because the two triangles do not have exactly the same sides.

11

Congruency

Same Angles
Does this also work with angles? Not always!
Two triangles with the same angles might be congruent:

is congruent to

only because they are the same size
But they might NOT be congruent because of different sizes:

is NOT congruent to

because, even though all angles match, one is larger than the other.
So just having the same angles is no guarantee they are congruent.

12

Congruency

Proving Congruent Triangles
Investigation

SSS Congruence Test SAS Congruence Test

SAA/ AAS Congruence Test ASA Congruence Test

13

Congruency

Applications of Congruency

Worked
Example

A private resort
installed 2 additional
triangular pools,
as an added attraction
to their clients.
Triangular pool ABS is congruent with the triangular pool PRD. Two sides of
the triangular pool ABS measure 30 ft. and 40 ft. Find perimeter of the
triangular pool PRD?

Solutions

1 feet = 30, 48 cm 14

Congruency

Applications of Congruency

Worked
Example
Karim took six congruent blocks and glued
them together to make the L-shaped solid
shown. The blocks are cubes with each edge
measuring 10 cm. If Karim paints all the
surfaces of this new figure, how many square
centimetres will be painted?

Solutions

15

SIMILARITY

Similar Shapes

Two shapes are said to be similar if they are exactly the same
shape, but are different in size.
Scaling all the lengths of the original shape can create a similar
shape. This means they have been enlarged or shortened in the
same proportions. We call this the scale factor.

Which of the following shapes are similar?

16

SIMILARITY

Enlargement using lengths
We can find similar
shapes by multiplying the lengths of a shape by the

scale factor:

New length = original length × scale factor.
Enlargement of scale factor 2
4 cm × 2 = 8 cm
2 cm × 2 = 4 cm
Two rectangles. One is 4 cm by 2 cm, the other is 2 cm by 1 cm
Enlargement of scale factor ½
4 cm × ½ = 2 cm
2 cm × ½ = 1 cm

17

SIMILARITY

Enlargement using lengths
Boxes of matches a
re sold in three different sizes: small, medium and large.

The medium box is an enlargement of the small box using a scale factor 2.
1.What are the dimensions of the medium box?
2.What scale factor enlargement is the large box of the small box?
3.What scale factor enlargement is the large box of the medium box?

1. Scale factor 2 2. Scale factor = new ÷ old:

length = 3 cm × 2 = 6 cm, Small to large scale factor

width = 1 cm × 2 = 2 cm, = 14 cm ÷ 4 cm

height = 4 cm × 2 = 8 cm. = 3.5

Medium box is 6 cm by 2 cm by 8 cm.

3. Scale factor = new ÷ old 18
Medium to large scale factor
= 14 ÷ 8
= 1.7

SIMILARITY

Perimeter, area and volume

When we enlarge a shape by a scale factor, the length of each edge and the
perimeter are multiplied by the scale factor.

When we enlarge a shape by a scale factor, the area of the shape is multiplied
by the square of the scale factor.

19

SIMILARITY

Perimeter, area and volume

When we enlarge a shape by a scale factor, the volume of the shape is
multiplied by the cube of the scale factor.

Exercise 1

1. Two similar posters have areas of 24 cm 2 and 384 cm 2. If the smaller
poster has a perimeter of 20 cm, calculate the perimeter of the larger
poster.
2.Two similar tanks are filled with water. One has a capacity of 30 m3,
the other a capacity of 240 cm2 . Calculate the scale factor for the
lengths of the tanks and the scale factor for the surface areas.

20

SIMILARITY

Similar triangles

Two triangles are similar if the angles are the same size or the corresponding
sides are in the same ratio. Either of these conditions will prove two triangles
are similar.

Triangle B is an enlargement of
triangle A by a scale factor of 2.
Each length in triangle B is twice as
long as in triangle A.
The two triangles are similar.

21

SIMILARITY

Example
State whether the two triangles are similar. Give a reason to support your
answer.

Yes, they are similar. The two lengths have been increased by a scale
factor of 2. The corresponding angle is the same.

Prove similar triangles

AA Similarity SAS Similarity

SSS Similarity 22

SIMILARITY

Applications of Similarity

Worked
Example

A ramp is built enable wheel-chair access to a building that is 24 cm above
ground level. The ramp has a constant slope of 2 in 15, which means that for
every 15 cm horizontally its rises 2 cm. Calculate the length of the base of the
ramp.

∆ ∆In DEC and ABC,

<DCE = <ACB (A)
<DEC = <ABC (A)

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t156C1eh511/6n55eA1/C(8+(5(1B5E1A1a01525+/Cb=B)x+Bo=x)C=vC)>C=eE1t8/1r50Bi+aCxncgmles
LeSnog, tlehnogfthbAoa=pfspetr==hooe(xf3i1m8br2aa1a34.5





smt209eep90lyoc7+fm6=15.87r1a6mm) (.p18is021.+821 4m2. )
23

SIMILARITY

Applications of Similarity

Your
turn

In the diagram below, a large flagpole stands outside of an office building.
Marquis realizes that when he looks up from the ground, 60m away from the
flagpole, that the top of the flagpole and the top of the building line up. If the
flagpole is 35m tall, and Marquis is 170m from the building, how tall is the
building?

1. Are the triangles in the diagram similar? Explain
2. Determine the height of the building using what you know about scale factors
3. Determine the height of the building using ratios between similar figure.
4. Determine the height of the building using ratios within similar figures

24

Formative Test

1) Which of the following is NOT true about similar figures?

A. Similar figures always have the same shape.
B. Similar figures always have the same size.
C. Similar figures always have corresponding angles that are equal
D. Similar figures always have corresponding sides that are proportional.

2) A regular hexagonal gazebo floor has got meters length. All will be covered
with congruent equilateral triangles with a side length. How many congruent
equilateral triangles are needed?
A. 36
B. 72
C. 96
D. 144

3) The picture shows a photo with dimensions 12 cm x 15 cm is stuck on the
piece cartoon. If the remaining size in the bottom is 3 cm and right, left ,top
of the cartoon is x cm. If the photo and cartoon is similar, the value of x is...
A. 1 cm
B. 2 cm
C. 3 cm
D. 4 cm

25

Formative Test

4) photo with a base of 25 cm and a height of 40 cm is attached to a piece
of cardboard so that the width of the remaining cardboard to the left and
right of the photo is 5 cm, while the rest of the top is cm longer than the
left rest of the cardboard. The cardboard under the photo is used to
write the name. If the photo and cardboard are similar, the maximum area
of cardboard that can be used to write the name is....
A. 210 m2
B. 245 m2
C. 350 m2
D. 385 m2

5) The length of tree's shadow is 16 m. At the same time and location, the
length of flagpole's shadow is 6 m. If the flagpole is 4.5 metres height, then
the tree's height is ....
A. 8 m
B. 9 m
C. 10 m
D. 12 m

26

Daftar Pustaka

Mathematics Book 3A BPK PENABUR, Eka Susanto
Dr Joseph Yeo, 2015, New Syllabus Mathematics 7th Edition Book Part 3
Dr Joseph Yeo, 2015, New Syllabus Mathematics 7th Edition Book Part 2
https://www.mathsisfun.com/geometry/triangles-congruent.html
https://www.youtube.com/watch?
v=_pedzAC3zCY&t=9s&pp=ugMICgJpZBABGAE%3D
https://study.com/skill/practice/solving-word-problem-involving-congruent-triangles-
questions.html
https://www.bbc.co.uk/bitesize/guides/z2w3cwx/revision/1

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