- What is Congruence
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- Triangle Congruence
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- Properties of Congurent trianngles
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- Different Criterion for Congruence of the triangles
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- How to Prove the congurence of Two Triangle
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- Inequality of Triangle
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- Important defination for Triangles
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- Solved Examples

- In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.
- A combination of rigid motions, namely a translation, a rotation, and a reflection is also permitted in Congruence. Translation means sliding,Rotation means turning and Reflection means Flipping. If two figures can be overlapped using translation,rotation or reflection then they are congruence
- It has come from Latin word congruere "agree, correspond with"

- We need to learn congruence in order to identify and describe many Congruent object in real life. It may be some sculpture, some old monuments,
- In Pyramid, all the three triangular surface are Congruent.AROUND 2600 B.C., construction of the Great Pyramid of Khufu began. It took the ancient Egyptians about 30 years to transform 6.5 million tons of stone into a pyramid with a square base and four congruent triangular faces
- We need several identical shirts button ,various factory part and many numerous objects. All these need the concept of Congruence
- Symbol used is ≅
- Two angles are congruent if they are equal
- Two circle are congruent if they have equal radii
- Two squares are congruent if the sides are equal
- Two line segments are congruent if they have equal length
- Two triangles are congruent if three sides and three angles of one triangle is congruent to the corresponding sides and angles of the other

- Corresponding sides are equal $AB=DE \; , \; BC=EF \; ,\; AC=DF$
- Corresponding angles are equal

- We write this as

$ \Delta ABC \cong \Delta DEF$ - The above six equalities are between the corresponding parts of the two congruent triangles. In short form this is called C.P.C.T
- We should keep the letters in correct order on both sides
- We have already studied that two triangles are Congruent when all the sides and all the angles are equal. But we don't need to prove all these while solving the Problem

- We just need to prove the congruence using the different criterion like SSS,ASA,SAS,RHS,AAS

- Don't use AAA

- Use the theorem learn in previous Geometry chapter like vertically opposite angles,alternate interior angles,corresponding angles

- Write down the corresponding angles and corresponding sides carefully

- We need to be careful with the labelling when our Triangles are in different positions

## Congruence

Two Geometric figure are said to be congruence if they are exactly same size and shape

## Triangle Congruence

## Properties of Congruent triangles

1)**Reflexive Property of Congruent Triangle**: Every Triangle is Congruent to itself

$ \Delta ABC \cong \Delta ABC$

2)**Commutative or symmetric property of Congruent Triangle**:

if $ \Delta ABC \cong \Delta DEF$ then $ \Delta DEF \cong \Delta ABC$

3)**Transitive Property Of Congruent Triangle**

if $ \Delta ABC \cong \Delta DEF$ and $ \Delta DEF \cong \Delta LKM$ then $\Delta ABC \cong \Delta LKM$

Since Congruence relation satisfy Reflexive ,symmetric and Transitive ,it is an equivalence relation

## Different Criterion for Congruence of the triangles

### SAS Congruence (AXIOM)

### ASA Congruence (Theorem)

### AAS Congruence (Theorem)

### SSS Congruence (Theorem)

### RHS Congruence (Theorem)

**Caveat**

1)__AAA( Angle Angle Angle) is not the right condition to Prove congruence.__

The below two equilateral triangles have same angles 60^{0}.But they are not congruent

2)__SSA and ASS is also not the right condition to Prove congruence.__

The below figure explain why it is not the right condition

## How to Prove the congruence of Two Triangle

## Inequality of Triangle

1) In a triangle angle opposite to longer side is larger

2) In a triangle side opposite to larger angle is larger

3) The sum of any two sides of the triangle is greater than the third side

In triangle ABC

$AB +BC > AC$

4) Difference of any two sides of triangle is less than the third side

$AC-AB < BC$ where $AC > AB$

## Important definition for Triangles

### Median

A line Segment joining the corner of the triangle to the mid point of the opposite side of the triangle. All the median of the triangle passes through the same point

### Centroid

Point of intersection of the three median of the triangle is called the centroid of the triangle. The centroid divides each of the median in the ration of 2:1

### Altitude

A line Segment from the corner of the triangle and perpendicular to the opposite side of the triangle. All the altitude of the triangle passes through the same point

### Ortho-centre

Point of intersection of the three altitude of the triangle is called the orthocenter of the triangle.

### In-center

All the angle bisector of the triangle passes through same point .It is called in-center.

### Circumcenter

The perpendicular bisector of the sides of the triangles passes through same point. It is called circumcenter

## Solved Example

**Question 1**

Here $LMN \cong PQR$

1)What is the measure of the angles P,M,R,N?

2)Write down the corresponding Congruent sides?

**Solution**:

By Congruence, All the corresponding angles and corresponding sides are equal

$LMN \cong PQR$

Steps to find the All the corresponding angles and corresponding sides

1) For angles, The First alphabet on right hand side corresponds to First alphabet on Left hand side,Similarly second and third alphabet

So $\angle L= \angle P$

$ \angle M=\angle Q$

$\angle N=\angle R$

Now $\angle L=105^0$

$\angle Q=45$, So $\angle M=45^0$

$ \angle P=105^0$

Now $\angle R$ can be found from the theorem

"Sum of the all the angles in a triangle is 180^{0}"

So $\angle R=180-105-45=30^0$

$\angle N=30^0$

2) For sides,The first two alphabet on right hand side corresponds to First two alphabet on Left hand side,Similarly 2,3 and 3,1

$LM=PQ$

$MN=QR$

$LN=PR$

**Question 2**

Please check if these triangle are congruent and then provide the congruent relation

**Solution**:

Here Two angles and side between them is equal,So by ASA Criterion, these triangle are congruent

The congruency can be written as by simply equating the angles

$ \Delta ABC \cong \Delta RPQ$

### Quiz Time

Class 9 Maths Class 9 Science