Geometry Similarity Congruence

8/8/2019 Geometry Similarity Congruence

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While the lengths of the sides have doubled, the sizes of the angles have not changed.

The angle sum of any triangle is 180, no matter its shape or size. Therefore, the angle

sum cannot increase or decrease if the sides are extended to, say, twice their originallength. That is, doubling the lengths of the sides does not result in a doubling of the

individual angles in the triangle.

1. Properties for similaritiesTwo quadrilaterals are said to be similar if they satisfy the following two properties:

a. The corresponding angles are equalb. The corresponding sides are proportional

2. Exploring the concept1. Figure 1.1 is an enlargement of

Figure 1.2. Use a ruler to findthe length of in each photo.

Write the ratio of the length ofAB in Figure 1 to the length ofAB in Figure 2.

2. Use a protractor to find the measure of in each Figure.3. Write the ratio of size of in Figure 1 to size of in Figure 2.4. Continue finding the measurements in the figures. Find the ratio of the

measurements in Figure 1 to the measurements in Figure 2. Use the same unitsthroughout the activity. Record your results in a table similar to the one shown.

Measurement Figure 1 Figure 2 Ratio

AB ----- ----- -----

AC ----- ----- -----

BC ----- ----- -----

----- ----- -----

Perimeter of Figure ----- ----- -----

3. Drawing Conclusions1. Suppose a segment in Figure 2 has a length of 5 centimeters. Estimate the length

of the corresponding segment in Figure 1.

2. Suppose an angle in Figure 1 has a measure of 35o. Estimate the measure of thecorresponding angle in Figure 2.

A

B

C

Figure 1.1

A

B

C

Figure 1.2


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3. Make some general conclusions about how corresponding lengths,corresponding angles, and corresponding perimeters are related when a figure is

enlarged.

The two polygons in Figure 1.3 below are similar.

Because there is a correspondence between twopolygons such that their corresponding angles are equaland the lengths of corresponding sides are proportional

orABCD is similar toEFGH. The symbol ~ is used to

indicate similarity. So,ABCD~EFGH

Pentagons JKLMNand STUVWbelow

are similar. List all the pairs ofcongruent angles. Write the ratio of the

corresponding sides in a statement ofproportionality.

M

N

JK

LV

W

S T

U

Answer:

Because JKLM and STUVW aresimilar, you can write

VMULTKSJ ,,,

and WN . You can write the

statement of proportionality as follow:

WS

NJ

VW

MN

UV

LM

TU

KL

ST

JK====

Example 1.1

Exercise 1.1

1. The height of a photo is 6 cm, while its width is 6 cm. The photo is then enlarged sothat its height becomes 36 cm. What is its width after enlargement?

2. The height of a ship is 120 cm and the height of its pole is 15 m. If the length of theship in a model is 24 cm, determine the height of the pole

3. The length of the front side of a house is 12 m and its height is 4 m. If the house hasa width of 20 cm in a model, find the height of the house in the model.

A D E H

B

C

FG

EH

DA

GH

CD

FG

BC

EF

AB===

Figure 1.3


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4. A car has the length of 4.5 m and height of 1.2 m. If the car is designed in a modelwith the height of 6 cm, find the length of the car in the model.

5. A war ship with the size of 200 m long and 40 m wide.a. If the width of the ship in a model is 10 cm, calculate the length of the ship in

the model.

b.

If the height of the ships pole in a model is 5 cm, how long is the actual heightof the ships pole?

6. A table with the size of 96 cm 60 cm 84 cm. For a children toy, a model of thetable is made with a length of 6 cm. Calculate:

a. The width and the height of the model.b. The number of linesc. The area of sides in the model and in reality,d. The volume of the model and of the real table.

7. Which of the following shapes has a correspondence to a football field with the sizeof 100 cm 60 cm?

a. A square with the size of 10 m 10 mb. A rectangle with the size of 5 cm 3 cmc. A parallelogram with the size of 10 cm 6 cm, and an angle of 88o

8. Which of the following shapes has a correspondence to a boxing ring with the sizeof 5 cm 5 cm?

a. A carpet with the size of 4 m 4 m.b. A rug with the size of 3 m 3 mc. A book page with the size of 20 cm 15 cm.

9. Which of the following quadrilaterals aredefinitely similar?

a. Two parallelogramsb. Two kitesc. Two rhombusesd. Two squarese. Two isosceles trianglesf. Two equilateral trianglesg. Two hexagons

10.Given two trapeziums as shown on the right. Prove that trapezium ABCD andEFGH are similar.

4. Properties for similarity of two triangles

F

EDB

C

A Q

R

P

Figure 1.5

A D E H

B C

F G

14 cm

10cm

8 cm

12cm

7 cm

5cm

4 cm

6cm


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Consider Figure 1.5 above.

DA = (Corresponding angles)

EB = (Corresponding angles)

FC = (The other two angles are equal)

Therefore, the corresponding angles are equal.

Similarly,

AB : DE = 4 : 6 = 2 : 3

AC : DF = 4 : 6 = 2 : 3

BC : EF= 4 : 6 = 2 : 3

Therefore, the ratio of the corresponding sides is equal.

Now we can conclude that

If the corresponding angles of two triangles are equal in size the corresponding

sides are proportional or the ratio of the corresponding sides is equal. Hence if the

corresponding angles of the two triangles are equal then the two triangles aresimilar.

By the similar method, we can prove that ABCand PQR above are similar.

Consider also the illustration below

A B

C

P Q

R

A is corresponding to P , B is corresponding to Q , C is corresponding to

R and PA = , QB = and RC =


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Example 1.2

In ABCand DEFgiven that o60=BAC , o40=ABC o60=FDE , ando

60=

DEF . Explain why the two triangles are similar? Then mention thesides which are proportional!

Answer:

Consider both triangles below:

ED

F

A

C

B

70o30o

70o 80o

It is seen from the figure thato70== EDFBAC o30== EFDABC o80== DEFACB

Hence, ABC and DEF are similar,

since the corresponding angles are equalin size. The proportional sides are

DE

AC

EF

BC

DF

AB==

Example 1.3

A

C

B

4cm

3cm

5cm

P

Q

R

15cm

9cm

12cm

In ABC and DEF given that o60=BAC , o40=ABC o60=FDE , ando60=DEF . Explain why the two triangles are similar? Then mention the sides

which are proportional!

Are ABCand PQR similar?Answer:

3

1

15

5==

PQ

AB,

3

1

9

3==

QR

BCand

3

1

12

4==

PR

AC

All sides are proportional, so ABCand

PQR are similar. The corresponding

angles which are equal areA = P , B = Q , C = R .

Exercise 1.2

1. Which of the following figure is pairs of similar figures and explain why they aresimilar


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A

C

B

12cm

15cm

9cm

D

EF

60o 50o70o 60oK L

M

3cm

4cm

5cm

PQ

R

70o

50o

X

YZ 8cm

6cm

10cm

S

T

U

2. Given triangles ABCand PQR, where o45=A o65=B o45=P ando65=R

a. Are ABCand PQR similar?b. Write down the pairs of corresponding sides which are proportional

3. Given triangles XYZand UVW, where o50=X o75=Y o50=U ando55=W

a. Are ABCand PQR similar?b. Write down the pairs of corresponding sides which are proportional

4. Check whether the following measures of triangles are similar to a triangles withsides 5cm, 12 cm and 13 cm or not?a. 4 cm, 7.5 cm, and 8.5 cmb. 2 cm, 4.8 cm, and 5.4 cmc. 15 cm, 36 cm, and 39 cm

5. Given triangles ABC and DEF DA = , EB = and FC = , lineAPand DQ are bisectors of angles A and D , respectively, such that they interest

BCat P andEFat Q. Prove thata. ABC and DEFare similarb.

EF

BC

DQ

AP=

5. Finding the unknown sides of similar trianglesWe have known that if two triangles have pairs of corresponding angles then both

triangles are similar. This also means that the ratio of the corresponding angles is equal.

By using this property we can find the unknown length of a side using this ratio.


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Example 1.4

Given triangles

Q

R

P

6cm 18cm

A B

C

12 cm8 cm

xy

a. Prove that ABCand PQR. are similarb. Find the corresponding sides which are similarc. Determine the value forx andyAnswer:

a. Since A = P , B = Q , C = R , then ABC andPQR. are similar

b.PR

AC

QR

BC

PQ

AB==

c. The scale factor is3

2

12

8= , so x = the scale factor 18

12183

2==x cm

Similarly, 962

312factorscalethe === yy cm


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Example 1.5

In the figure below, AB = 4 cm, AF= 5 cm, BF= 3 cm, andBC= 6 cm.

a. Prove thatABFand

ADC.b. Determine the corresponding sides which are proportional

c. Determine the length ofAD and DC

Answer:

a. Consider ABFand ADC.o90== ADCABF

CADBAF = (coincide)

ACDAFB = (the other two angles are

equal)

Hence,ABFand

ADCare similar.

b.AC

AF

DC

BF

AD

AB==

c. 810

54=== AD

ADAC

AF

AD

ABcm

610

53=== DC

DCAC

AF

DC

BFcm

F

BA

5cm

3c

m

4 cm

D

C6 cm

E

Exercise 1.3

1. Calculate the length of the labeled sides in the following pairs of similar triangles.

2

3

6

a

6

4

1

a. b.

1.6

c.

c

b4.8

1.2

d.

46

23

18

x

5

7

15

y

e.

12

18

25

x

f.

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Geometry Similarity Congruence