Name: ............................ Date: ............................

Similar Figures

1. Rectangles ABCD and EFGH are similar.

a. Find the value of x.b. Find the value of y.c. po = ……………..

d.

2. Dimas made a wooden frame. The

outer edge was 120 cm long and 80 cm wide, and the width of the frame was 10 cm, as shown in the

right figure.

The length of the inner part of the frame is … cmThe width of the inner part of the frame is … cm

Are the outer part and the inner part of the frame similar?

Mathematics for Junior High School Grade 9 / 1

1.1.

11

A

B

CDH G

E

F

1512

7.56

x 4.8

y18

150opo

3.Are two rectangles always similar? To answer this question, do the following activities:a. Draw two rectangles that are not similar. Write their dimensions .

b. Draw two similar rectangles. Write their dimensions.

2 / Student’s Worksheet – Similarity and Congruency

1.1.

22

Name: ............................ Date: ............................

Mathematical Modeling

Using a ruler and a protractor, draw DEF with D = 35, F = 80, and DF = 4cm draw TRS with T = 35, S = 80, and ST = 7cm measure the lengths of EF, ED, RS, and RT

determine the proportions of .

Write down the results that you have obtained in the following table.

Side lengths of DEF

Side lengths of RST

Ratio Value

EF ED RS RT

Are DEF and TRS similar?

Do the results above demonstrate this: if in two triangles the corresponding

angles are the same size, then their corresponding sides are proportional?

What does this mean?

Use a ruler and a protractor.

Draw triangle ABC with AB = 8 cm, BC = 6 cm, and AC = 7 cm.

Mathematics for Junior High School Grade 9 / 3

Draw triangle PQR with PQ = 4 cm, QR = 3 cm, and PR = 3.5 cm. Measure A, B, C, P, Q, R. Is A = P, B = Q, C = R?

Are ABC and PQR similar?Do the results above demonstrate this: if in two triangles the corresponding sides are proportional, their corresponding angles are the same size? What does this mean?

4 / Student’s Worksheet – Similarity and Congruency

Name: ............................ Date: ............................

Similar Triangles

1. a.

10

ABC and PQR are similar because ………………………………….… …………...................................................................................................…………...................................................................................................

b. Z W 80o 80o

70o

30o U V X Y

XYZ and UVW are similar because ……………………………….… …………...................................................................................................…………...................................................................................................

2. a. ABC and PQR are similar becauseA = RB = QC = P

Mathematics for Junior High School Grade 9 / 5

1.1.

33

R

P

10

Q8

C

3BA

4

Therefore, the proportion of the corresponding sides is

b. XYZ and UVW are similar because .

Therefore, the angles of the same size are = = =

6 / Student’s Worksheet – Similarity and Congruency

Name: ............................ Date: ............................

Congruent Triangles

1.Look at the following figures.

a. Copy ABC on a piece of cardboard.

b. Cut out the new triangle, and label each angle as shown in Figure 1.

c. Set the new ABC to find which triangle is congruent to it.

d. Which triangle is congruent to the new ABC?

e. Follow steps a to d for PQR to determine which triangle is congruent to

PQR.

Mathematics for Junior High School Grade 9 / 7

Figure 1

1.41.4

2. Look at the following figure.

a. If AHI is reflected to line HI, then it exactly coincides with … Therefore, AHI …

b. If GHI is reflected to line GI, then the triangle coincides with … Therefore, GHI …

c. Fill in the blank spaces:AHI … …

d. If GFI is moved along and in the direction of FU, then GFI coincides with …Therefore, GFI ………..

e. Based on the results in c and d above, we get:AHI … … …

Using Figure 2, answer problems 3 and 4 below:

3. Mention the transformation (which can be a combination) to make:

a. CID coincide with CZD....................................................................................

........................................................................................................

....................b. CID coincide with CZD

........................................................................................................

8 / Student’s Worksheet – Similarity and Congruency

Figure 2

........................................................................................................

c. DZE coincide with YZE....................................................................................

........................................................................................................

....................d. GFP coincide with EYR

........................................................................................................

........................................................................................................e. CDZ coincide with EDI

............................................................................................................

....................................................................................

....................f. BCI coincide with DEI

........................................................................................................

....................................................................................

....................g. FEP coincide with QRE

........................................................................................................

....................................................................................

...................h. GFP coincide with XWP

........................................................................................................

....................................................................................

....................i. IFE coincide with ZMN

............................................................................................................

............................................................................................................

j. OPX coincide with YNR....................................................................................

............................................................................................................................

Mathematics for Junior High School Grade 9 / 9

4.Mention three triangles which are congruent with:

a. FEP : 1) ........... 2) ............. 3) ...............

b. ACI : 1) ........... 2) ............. 3) ...............

c. GEP : 1) ........... 2) ............. 3) ...............d. ACG : 1) ........... 2) ............. 3) ...............e. ALX : 1) ........... 2) ............. 3) ...............

5. Continue the tiling below to complete the grid paper.

10 / Student’s Worksheet – Similarity and Congruency

Figure 1

Name: ............................ Date: ............................

Properties and Conditions

of Two Congruent Triangles

1. Look at the following figures. ABC PQR.

a. Copy ABC on a piece of cardboard.

b. Cut out the triangle, and then label the angles as shown in Figure 1.c. Set the new ABC so that point A coincides with P, and point B

coincides with Q. Then fill in the following blank spaces:(i). Point A coincides with point …, B with … , and C with …(ii). Line segment AB coincides with PQ, so that AB = PQ(iii). Line segment AC coincides with …, so that AC = …(iv). Line segment BC coincides with …, so that BC = …

d. The sides that coincide with each other are called corresponding sides.

AB and … are corresponding sides.AC and … are corresponding sides.BC and … are corresponding sides.

e. The angles that coincide with each other are called corresponding angles.

Mathematics for Junior High School Grade 9 / 11

C

BA

R

QP

1.51.5

CAB and … are corresponding angles.ABC and … are corresponding angles.ACB and … are corresponding angles.

f. Your answers to e) and f) above show that

the length of the corresponding sides are ………. and the size of the corresponding angles are ………..

2. In Figure 2 below, the sides of ABC correspond to those of PQR, so that AB = PQ, AC = PR and BC = QR. P

A

R Q Q

C B

Figure 2

a. Copy ABC on a cardboard.

b. Cut out the triangle on the cardboard, and label the angles as shown in Figure 2.

c. Can you set the new ABC so that it coincides with PQR?d. What is your conclusion?

Therefore, if the three sides of ABC are the same as those of PQR, then ABC PQR.

3. Look at the following pairs of congruent triangles. By measuring them,

find the corresponding sides of the same length and the corresponding angles of the same size.

a. P Q

12 / Student’s Worksheet – Similarity and Congruency

R J

I

H

b. K L E

D

M F

4. Is STR NMP? If so, determine the values of a, b and c.

Mathematics for Junior High School Grade 9 / 13

T

S

R

8

k

767O

63O

wO

P

N

M

8k

b

a O

c O

wO

1.61.6

Name: ............................ Date: ............................

Application of Congruence

1. Look at Figure 1.

Figure 1

Given ABD = CBD, show that AB = CB.

Fill in the following blank spaces:

According to the requirement (. . . , . . . , . . . ), ABD . . .

14 / Student’s Worksheet – Similarity and Congruency

Statement

a. BAD = BCD

b. ABD = CBD

c. BAD + ABD = BCD +

....

d. BAD+ ABD+ ADB =

BCD+ CBD+ . .. = 1800

e. ADB = . . .

f. BD = BD

a. Given (in Figure 1)

b. .............

c. ..............

d. The sum of the

three angles in a

triangle is 180°.

e. clear

f. ...

A D C

B

and ... are corresponding sides, so that AB = . . .

2. Look at Figure 2.

ABC is an isosceles triangle. B is the vertex of ABC.

ABD = CBD. BD is the angle bisector of B.

Show that:

a. AD = DC

b. BD is perpendicular to AC.

To solve the two problems above, fill in the following blank spaces.

Figure 2

a) Look at ABD and CBD.1). AB = CB2). ABD = . . .3). BD = BD4). According to the requirement ( . . . , . . . , . . . ), ABD

. . .5). and ... are corresponding sides, so that AD = . . .

b) Based on the result in (a), it can be concluded that AD = ...

It means that is the segment bisector of ABC.Since is the segment bisector of ABC, then . . . . . .

Mathematics for Junior High School Grade 9 / 15

B

CAD