Notes on Congurency and Similarities | Grade 8 > Compulsory Maths > Geometry | KULLABS.COM

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### Congruent Triangles

The triangles having same size and shape are called congruent triangles. Two triangles are congruent when the three sides and three angles of one triangle have the measurements as three sides and three angles of another triangle. The symbol for congruent is ≅.

In the following figure, ΔABC and ΔPQR are congruent. We denote this as ΔABC ≅ ΔPQR. #### Postulate and Theorems for Congruent Triangles

Postulate (SAS)

If two sides and the angle between them in one triangle are congruent to the corresponding parts in another triangle, then the triangles are congruent. In the given figure,

AB ≅ PQ Sides (S)

∠B ≅ ∠Q Angle (A)

BC ≅ QR Side (S)

Therefore, ΔABC ≅ ΔPQR

Theorem (ASA)

A unique triangle is formed by two angles and the included side.

Therefore, if two angles and the included side of one triangle are congruent to two angles and the included side of the another triangle, then the triangles are congruent. In the figure,

∠B ≅ ∠E Angle (A)

BC ≅ EF Side (S)

∠C ≅ ∠F Angle (A)

Therefore, ΔABC ≅ ΔDEF

Theorem ( AAS)

A unique triangle is formed by two angles and non-included side. Therefore, if two angles and the side opposite to one of them in a triangle are congruent to the corresponding parts in another triangle, then the triangles are congruent. In the figure,

∠A ≅ ∠X Angle (A)

∠C ≅ ∠Z Angle (A)

BC ≅ YZ Side (S)

Therefore, ΔABC ≅ ΔXYZ

Theorem (SSS)

A unique triangle is formed by specifying three sides of a triangle, where the longest side (if there is one) is less than the sum of the two shorter sides.

Therefore, if their sides of a triangle are congruent to three sides of another triangle, then the triangles are congruent. In the figure

AB ≅ PQ Sides (S)

BC ≅ QR Sides (S)

CA ≅ RP Sides (S)

Therefore, ΔABC ≅ ΔPQR

### Similar Triangles

#### Methods of providing triangles similar

1. If the corresponding sides of a triangle are proportion to another triangle then the triangles are similar.
Example If∠A ≅ ∠D and ∠B ≅ ∠E, Then ΔABC ∼ ΔDEF

2. If the corresponding angle of a triangle is congruent to another triangle then, the triangles are similar.
Example If $$\frac{AB}{DE}$$ = $$\frac{BC}{EF}$$ = $$\frac{AC}{DF}$$, then ΔABC ∼ ΔDEF

3. Conversing first and the second method we can prove triangle similar as their sides being proportional and angles congruent.
Example ∠A ≅ ∠D and $$\frac{AB}{DE}$$ = $$\frac{AC}{DF}$$ then ΔABC ∼ ΔDEF

#### In case of overlapping triangles

When the lines are parallel in a triangle, then they intersect each other which divides the sides of a triangle proportionally.

Verification: In ΔPQR and ΔSPT

 Statements Reasons ST⁄⁄QR Given $$\angle$$PST $$\cong$$ $$\angle$$QSR Corresponding angles ΔPQR $$\cong$$ ΔSPT Common Angle P $$\frac{PS}{SQ}$$ = $$\frac{PT}{TR}$$ ST⁄⁄QR

Example

Given the following triangles, find the length of x. Solution:

The triangles are similar by AA rule.So, the ratio of lengths are equal.

$$\frac{6}{3}$$ = $$\frac{10}{x}$$

or, 6x = 30

or, x = $$\frac{30}{6}$$

$$\therefore$$ x = 5 cm

• There is three easy way to prove similarity. If two pairs of corresponding  angles in a pair of triangles are congruent, then the triangles are similar.
• When the three angle pairs are all equal, the  three pairs of the side must  be proportion.
• When triangles are congruent and one triangle is placed on the top of other sides and angles that are in the same position are called corresponding parts.
• Congruent and similar shapes can make calculations and design work easier.
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#### Click on the questions below to reveal the answers

Solution:

Here, AB=1.6 cm DE=1.9 cm

BC=1.3 cm EF=1.4 cm

CA=1.9 cm DF=1.9 cm

$$\angle$$A=45o$$\angle$$D=40o

$$\angle$$B=100o$$\angle$$E=70o

$$\angle$$C=35o$$\angle$$F=70o

The angles arms of the triangle is not equal so it is not congurent.

Solution:

Here given, congurent triangle is PQ and LM, QR and MN, PR and LN. So that, congurent angles is $$\angle$$P and(\angle\)L,(\angle\)R and(\angle\)N,(\angle\)Q and(\angle\)M.

Solution:

Here given, congurent triangle is XY and AB, YZ and BC, XZ and AC. So that, congurent angles is $$\angle$$X and(\angle\)A,(\angle\)Y and(\angle\)B,(\angle\)Z and(\angle\)C.

Solution:

Here, $$\angle$$A=52o $$\angle$$=88o $$\angle$$C=40o

$$\angle$$P=52o $$\angle\Q=88o \(\angle$$R=40o

AB=1 cm BC=1.2 cm CA=2 cm

PQ=1.3 cm QR=1.6 cm PR=2.6 cm

$$\angle$$A= $$\angle$$P, $$\angle$$B= $$\angle$$Q and $$\angle$$C= $$\angle$$R

$$\frac{AB}{PQ}$$=$$\frac{1.0}{1.3}$$,$$\frac{BC}{QR}$$,$$\frac{1.0}{1.3}$$,=$$\frac{CA}{RP}$$=$$\frac{1.0}{1.3}$$

Hence, given triangle is similar triangle.

Solution:

Here, $$\angle$$ A=40o$$\angle$$ B=70o$$\angle$$ C=70o

$$\angle$$ P=41o$$\angle$$ Q=62o$$\angle$$ R=77o

AB=1.5 cm BC=1 cm CA=1.5 cm

PQ=1.1 cm QR=1 cm RP=1.6 cm

Here,$$\angle$$A $$\neq$$$$\angle$$P,$$\angle$$B$$\neq$$$$\angle$$Q and $$\angle$$C$$\neq$$ $$\angle$$R

Hence, given triangle is not similar triangle.

Solution:

Here, $$\angle$$A=38o $$\angle$$B=90o $$\angle$$C=52o

$$\angle$$P=38o $$\angle$$Q=90o $$\angle$$r=52o

AB=1.6 cm BC=1.2 cm

CA=2 cm PQ=0.8 cm

QR=0.6 cm PR=1 cm

$$\angle$$A= $$\angle$$P, $$\angle$$B= $$\angle$$Q and $$\angle$$C= $$\angle$$R

and $$\frac{AB}{PQ}$$=$$\frac{1.6}{0.8}$$=2, $$\frac{BC}{QR}$$=$$\frac{1.2}{0.6}$$=2, $$\frac{CA}{RP}$$=$$\frac{2}{1}$$=2

Hence, given triangle is similar triangle.

From the given figure,

∠STU ≅ ∠SVW and TU ≅ VW

Here, ∠TSU and ∠VSW are vertical angles. Since vertical angles are congruent,

∠TSU ≅ ∠VSW.

Finally, put the three congruency statements in order. ∠STU is between ∠TSU and TU, and ∠SVW is between ∠VSW and VW in the diagram.

∠TSU ≅ ∠VSW (Angle)

∠STU ≅ ∠SVW (Angle)

TU ≅ VW (Side)

Hence, the given triangles are congurent as it forms AAS theorem.

From the given figure,

BC ≅ BH and ∠BCF≅∠BHG.

Here, ∠CBF and ∠GBH are vertical angles. Since vertical angles are congruent,

∠CBF ≅ ∠GBH.

Finally, put the three congruency statements in order. BC is between ∠BCF and ∠CBF, and BH is between ∠BHG and ∠GBH in the diagram.

∠BCF ≅ ∠BHG Angle

BC ≅ BH Side

∠CBF ≅ ∠GBH Angle

Hence, the congruent sides and angles form ASA. The triangles are congruent by the ASA Theorem.

From the figure,

∠XWY≅∠YWZ and ∠WXY≅∠WZY.

Here, the triangles share WY. By the reflexive property of congruence, WY ≅ WY.

Finally, put the three congruency statements in order. ∠WXY is between ∠XWY and WY, and ∠WZY is between ∠YWZ and WY in the diagram.

∠XWY ≅ ∠YWZ Angle

∠WXY ≅ ∠WZY Angle

WY ≅ WY Side

Hence, the congruent sides and angles form AAS. The triangles are congruent by the AAS Theorem.

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• ### Which of the following triangles are always similar?

Right triangles
Isosceles triangles
Similar triangles
Equilateral triangles

60cm
35cm
30cm
45cm

32.5m
33m
35m
37m

22cm, 45cm
22cm, 45cm
20cm, 48cm
25cm, 50cm

5cm
6cm
7cm
4cm
• ### By which postulate, the given two triangles ABC and DEF are congruent? Which is the corresponding sides of AC? `

SAS postulate, EF.
ASA Theorem, EF
AAS Theorem, EF
SSS Theorem, EF
• ### In the given figure, if triangle PQR is congruent to triangle XYZ, find the value of two unknown angles and the value of y. (angle)Q=45o, (angle)X=60o, Y= 6cm
(angle)Q=30o, (angle)X=45o, Y=8cm
(angle)Q=60o, (angle)X=45o, Y=7cm
(angle)Q= 60o, (angle)X= 55o, Y= 5cm

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