Fatema Shariff, 10 B

SIMILARITY AND CONGRUENCY

Similarity

What is Similarity?

Similar Polygons

Similar Triangles

Area of Similar Triangles

Perimeter of Similar Polygons

Congruency

Similar 3-D Shapes

Similarity

What is similarity?

Similarity is a relation between two shapes.

The word similar means having characteristics in common.

Similar Polygons

Let us take a real life example: you take a photograph of an ancient monument and you want to enlarge the photograph from the original one. When you enlarge the photograph, it is ensured that the shape of the photograph remains the same from the original only the size differs the size changes but the shape remains the same.

For tw o figures to be similar, they have to:

be of the same type, e.g. both rectangles

All the corresponding sides must have the same ratio. This is called the scale factor of the shapes

The corresponding angles of the shapes must be congruent.

When we say congruent we mean equal, equilateral or exactly the same (shape or size).

Corresponding means any pair of angles/sides in similar location.

The symbol used to note that two shapes are similar is ~.

Example 1

Polygon RSTUV is similar to polygon ABCDE.

a) Find the scale factor of polygon RSTUV to polygon ABCDE

b) Find the values of x and y

(xTSVRUy + 218) (453BCEAD)

Answer:

Scale Factor:

Value of x:

Value of y:

Example 2

(1 cm 2 cmAA) (2 cm3 cmB)

The above rectangles are not similar even though their angles are congruent because their corresponding sides are not in the same ratio.

Similar Triangles

To prove the similarity of two triangles, the following conditions must be fulfilled:

Corresponding angles should be equal

Corresponding sides should be in the same ratio

Example 1

Referring to the above figure:

KLM = EFG, KML = EGF and LKM = FEG

Thus, we can say that the triangle KLM and triangle EFG are similar. Symbolically we write the above relation as KLM EFG because

It is important to note that, as in the case of congruency, similar triangles also should be written in the correct correspondence of their vertices i.e. it would be incorrect to write triangle KLM is similar to triangle GFE.Comment by Kals: Good observation

(B ACABC)Example 2

AA Similarity (Angle-Angle)

This theorem states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

By knowing this we also know that equilateral triangles are always similar as in an equilateral triangle, all angles are equal which satisfies the Angle Angle theorem.

(RISNG)Example

Given: ING RNS

GS = 6 in

GN = 3 in

RI = (x + 5) in

IN = (x + 1) in

Find the value of x.

Answer:

Since ING RNS, corresponding sides have a proportional relationship.

We can write the following proportion:

By substitution we have the following:

By simplifying we get the following:

If, then. Therefore,

Then we get,

Therefore

SSS Similarity (Side-Side-Side)

This theorem states that if the measures of the corresponding sides of two triangles are proportional, then the triangles are similar.

(BTLA)

Example

Given: Right triangle BLT with right angle BTL

Altitude TA

AL = 6 cm

TL = 10 cm

Find: BL

Answer:

Right triangle BLT is similar to right triangle TLA as follows:

A BTL TAL (congruent right angles)

A TLB TLB (reflexive property)

The measures of corresponding sides of similar triangles are in proportion.

So

i.e.

SAS Similarity (Side-Angle-Side)

This theorem states that if the measures of two sides of a triangle are proportional to the measures of two corresponding sides of another triangle and the included angles are congruent, then the triangles are similar.

Example

Justify how the below figure is similar by SAS Similarity Theorem.

(50502.5 m4.5 m)

(5 m) (9 m)

Answer:

The triangles are similar by the SAS similarity theorem. The measures of

two sides of one triangle are in proportion with the measures of the

corresponding sides of the other triangle as shown below.

The included angle between the sides of one triangle is congruent to the

included angle between the corresponding sides of the other triangle.

Similarity of triangles is more special than similarity in other polygons and triangles are the only shapes with 3 angles. When there are two triangles, we know that when 2 angles are equal then the triangles are equal but for other polygons if we consider only 2 corresponding angles then it is not necessary for the polygons to be similar as a polygon can have many angles and sides.

Exercise 1 (Similar polygons and triangles)

Find the sides marked with letters:

1)

(EDBAC662fe3)

(ZYYYXW345vw)

2)

3) From rectangle ABCD a square is cut off to leave the rectangle BCEF. Rectangle BCEF is similar to ABCD. Find x and hence state the ratio of the sides of rectangle ABCD. ABCD is called the Golden Rectangle and is an important shape in architecture.

(AB6DCEFx11)

4) A tree of height 4 m casts a shadow of length 6.5 m. Find the height of a house casting a shadow 26 m long.

(ACBcba )Area of similar triangles

(ABCcab)

Here you will notice that, as mentioned earlier, the area of the similar figures is equal to the square of their corresponding sides.Comment by Kals: You have missed the word ratio.

Perimeter of similar polygons

Suppose that a triangle with sides a, b, and c has been scaled by sto get a similar triangle with corresponding sides A, B, and C.Thus, and and.

Let's investigate the relationship between the perimeters of these two triangles:

perimeter of original triangle=

Perimeter of scaled triangle

Thus, the perimeter ends up being scaled by the same factor that scales the sides.A similar calculation shows that this result is indeed true for polygons in general: The ratio of the perimeters of two similar polygons is equal to the ratio of the corresponding sides.

Exercise 2 (Areas and perimeter of similar polygons and triangles)

Find the unknown area (question 1 & 2):

1)

(8 cm16 cmA) (3 cm 6 cm18 cm)

2) (58 cm5 cm20 cm10 cm3 cmA)

Find the unknown length marked for the following pairs of similar shapes (question 3 & 4):

3)

(3 cm4 cm9 cmz)

4) (y36 cm6 cm4 cm)

5) The isosceles triangles below are similar.

(GHIFJE1668X)

a) What is the similarity of triangle EFG to triangle HIJ? Simplify your ratio.

b) What is the value of x? Use mathematics to explain how you determined your answer. Use words, symbols, or both in your explanation.

c) What is the perimeter of each triangle?

d) What is ? Simplify your ratio.

Congruency

Figures that are congruent are equal in both size and shape.

Congruent shapes can be similar but similar shapes cannot be congruent.

In our daily life, we come across many objects with same shape and size. Objects, which have the same shape and size, are called congruent objects. The relation of two objects being congruent is called congruence. The term congruent means that two or more things are the exact same size and shape.

In the above figure, the quadrilaterals ABCD and EFGH are exactly equal with their corresponding angles and sides, hence they are congruent.

The corresponding parts of congruent triangles are equal. It can be shown that the converse is also true i.e. if there is a correspondence between the vertices of the two triangles so that the corresponding parts of two triangles are equal, then the two triangles are congruent.

Example

Find out if the two triangles XYZ and KLM, below, are congruent.

Answer :

If triangle KLM covers the triangle XYZ exactly, we say that the two triangles are congruent, otherwise not. The matching parts of the triangle are also called the corresponding parts of the two triangles.

Since triangle XYZ is congruent to triangle KLM then their corresponding parts are congruent. If two figures are congruent, then their corresponding parts are congruent too. It is important to note that triangle XYZ is congruent to triangle KLM and we can say that triangle KLM is congruent to XYZ since vertex X corresponds to vertex K, vertex Y corresponds to vertex L, and vertex Z corresponds to vertex M.

XYZ KLM

Exercise 3 (Congruency)

1) (D) (E)Identify pairs of congruent shapes below:

(B) (C)

(A)

(F) (G)

Similar 3-D Shapes

Similar solids have the exactly same shape but not necessarily the same size. To determine whether two solids are similar or not, you can compare the measurements of the corresponding linear measurements.

In two similar polyhedral, the corresponding faces are similar and all the corresponding edges are proportional.

We can state whether the two solids are similar by getting the ratio of 1:1 for the ratio of the corresponding measurements.

For two solids to be congruent:

The corresponding angles have to be congruent

The corresponding edges have to be congruent

The areas of the corresponding faces have to be equal

The volumes have to be equal

If two solids are similar with a scale factor of, then the surface areas will have a ratio of and the volumes will have a ratio of.

(R)

(r)

Example 1

(V = ?6)

(V = 504)

Relationship between the surface area the sides and the volume the sides

Exercise 4 (Volumes of 3-D Shapes)

1) Find the unknown volume V.

(3.1V) (886.2)

2) Two similar cylindrical tins have base radii of 6 cm and 8 cm respectively. If the capacity of the larger tin is 252 cm, find the capacity of the small tin.

3) Find the height y of the cylinder.

(270y)

(710)

4) (VRadius: 12 cm)

(4.5Radius: 1.2 cm)

Answers

Exercise 1 (page 7)

1) e = 9 cm, f = 4.5 cm

2) v = 5 cm, w = 6 cm

3) 0.618; 1.678:1

4) 16 m

Exercise 2 (page 9)

1) 128 cm

2) 14.5 cm

3) 4.5 cm

4) 18 cm

5) a)

b) x = 3

c)

d) or 2

Exercise 3 (page 11)

1) A and G, B and E

Exercise 4 (page 14)

1) V = 11 cm

2) V = 106.3 cm

3) y = 21 cm

4) v = 4500 cm

GRADE X - ASSESSMENT - SIMILARITY AND CONGRUENCY

Writing skill

Drawing skill

Formatting skill

Polygon

Triangle

3D shapes

Questions

Examples

Total

3

2

3

2

2

3

3

2

20

creativity flow neatness

pictures labeling

page paragraphs math equations

always-similar polygons appln

theorems applns

perimeter vol, SA

3D shapes congruency variety

ex+ans

3

2

3

1

2

3

2

2

18

15