Congruent Figures:
Two figures are said to be congruent to each other, if on placing one over the other, they exactly coincide. Congruent figures are same in size and shape both. Two circles are always congruent to each other. In the congruent triangles, the sides and the angle that coincide by superposition are called . If two line are same in size they are congruent.
In a congruent triangle corresponding sides and corresponding angles are equal. Two figures are congruence if there is an involvement with their vertices such that coupled angles are the comparable to and the comparable sides are interrelated. It is called as word like the one congruence.
SAS Property:
If two sides and the included angle of one triangle are congruent to the corresponding two sides and the included angle of another triangle, then the two triangles are congruence.
S.S.S Property:
If the all three sides of one triangle is equal to the three sides of other triangle, both triangles are know as congruent triangles.
ASA Property:
If two angles and the included side of one triangle are congruence to the corresponding two angles and the included side of another triangle, then the two triangles are congruence.
SAA or AAS Property:
If two angles and a non- included side of one triangle are congruence to the corresponding parts of another triangle, then the triangles are congruence.
R.H.S Property:
In two triangles if one of angles are right angles and hypotenuse and one side of first triangle is equal to hypotenuse and one side of another triangle, triangles are know as congruent triangles.
The L-L Property (The Leg – Leg Property)
If two legs of one right triangle are congruence to the corresponding legs of another right triangle, then the two triangles are congruence.
The L-AA Property (The Leg-Acute Angle Property)
If a leg and an acute angle of one right triangle are congruence to the corresponding leg and an acute angle of another. Then the two triangles are congruence.
The H-AA Property (The Hypotenuse – Acute Angle Property)
If the hypotenuse and an acute angle of one right triangle are congruence to the corresponding hypotenuse and an acute angle of another. Then the two triangles are congruence.
The H – L Property (The Hypotenuse – Leg Property)
If the hypotenuse and leg of one right triangle are congruence to the corresponding hypotenuse and leg of another, then the two triangles are congruence.
Examples of Congruent Figures:
Statement:
In SSS proof, if the corresponding sides of two triangles are proportional, then they are similar.
The proof of the above property is discussed in the form of the following solution.
Solution:
Given:
In `Delta`ABC and `Delta`DEF, we are given that
`(AB)/(DE)` = `(BC)/(EF)` = `(AC)/(DF)`
To prove:
`Delta`ABC ~ `Delta`DEF.
Construction: mark point P on DE and the point Q on DF such that DP = AB, DQ = AC. Join P and Q.
Proof:
Since `(AB)/(DE)` = `(AC)/(DF)` (given)
`(DP)/(DE)` = `(DQ)/(EF)` (construction)
By the converse of Thales theorem, PQ || EF.
`anlge`DPQ = `angle`E (corresponding angles)
`angle`DQP = `angle`F (corresponding angles)
By AAA similarity ( or by AA similarity)
We get `Delta`DPQ ~ `Delta`DEF
`(DP)/(DE)` = `(PQ)/(EF)` `rArr` `(AB)/(DE)` = `(PQ)/(EF)` (given)
But `(AB)/(DE)` = `(BC)/(EF)` (given)
BC = PQ and AB = DP, AC = DQ (construction)
`Delta`ABC `~=` `Delta`DPQ (SSS)
Since `Delta`DPQ ~ `Delta`DEF
`Delta`ABC ~ `Delta`DEF.
Hence, the required result is proved.
Between, if you have problem on these topics greatest integer function, please browse expert math related websites for more help on mathematical induction.
Properties of Congruent Figures
Congruent figures have following properties:
1. The corresponding part of congruent figures are also congruent.
2. The corresponding sides lie opposite to the equal angles and corresponding angles lie opposite to the equal sides.
3. All squares can be similar but not congruent. If squares have equal size (length and width ) they are know as congruent squares.
4. All the rectangles can be similar but Congruent rectangle are equal in size and shape.
Example : Find the missing sides of congruent triangles:
AB = 5cm
AC = 3 cm
EG = 3 Cm
EG = 4 Cm
Find BC and EF?
Sol :
Both triangles are congruent ( Given )
So All the corresponding sides are equal
BC = FG = 4cm
EF = AB = 5cm
Two figures are said to be congruent to each other, if on placing one over the other, they exactly coincide. Congruent figures are same in size and shape both. Two circles are always congruent to each other. In the congruent triangles, the sides and the angle that coincide by superposition are called . If two line are same in size they are congruent.
In a congruent triangle corresponding sides and corresponding angles are equal. Two figures are congruence if there is an involvement with their vertices such that coupled angles are the comparable to and the comparable sides are interrelated. It is called as word like the one congruence.
SAS Property:
If two sides and the included angle of one triangle are congruent to the corresponding two sides and the included angle of another triangle, then the two triangles are congruence.
S.S.S Property:
If the all three sides of one triangle is equal to the three sides of other triangle, both triangles are know as congruent triangles.
ASA Property:
If two angles and the included side of one triangle are congruence to the corresponding two angles and the included side of another triangle, then the two triangles are congruence.
SAA or AAS Property:
If two angles and a non- included side of one triangle are congruence to the corresponding parts of another triangle, then the triangles are congruence.
R.H.S Property:
In two triangles if one of angles are right angles and hypotenuse and one side of first triangle is equal to hypotenuse and one side of another triangle, triangles are know as congruent triangles.
The L-L Property (The Leg – Leg Property)
If two legs of one right triangle are congruence to the corresponding legs of another right triangle, then the two triangles are congruence.
The L-AA Property (The Leg-Acute Angle Property)
If a leg and an acute angle of one right triangle are congruence to the corresponding leg and an acute angle of another. Then the two triangles are congruence.
The H-AA Property (The Hypotenuse – Acute Angle Property)
If the hypotenuse and an acute angle of one right triangle are congruence to the corresponding hypotenuse and an acute angle of another. Then the two triangles are congruence.
The H – L Property (The Hypotenuse – Leg Property)
If the hypotenuse and leg of one right triangle are congruence to the corresponding hypotenuse and leg of another, then the two triangles are congruence.
Examples of Congruent Figures:
In SSS proof, if the corresponding sides of two triangles are proportional, then they are similar.
The proof of the above property is discussed in the form of the following solution.
Solution:
Given:
In `Delta`ABC and `Delta`DEF, we are given that
`(AB)/(DE)` = `(BC)/(EF)` = `(AC)/(DF)`
To prove:
`Delta`ABC ~ `Delta`DEF.
Construction: mark point P on DE and the point Q on DF such that DP = AB, DQ = AC. Join P and Q.
Proof:
Since `(AB)/(DE)` = `(AC)/(DF)` (given)
`(DP)/(DE)` = `(DQ)/(EF)` (construction)
By the converse of Thales theorem, PQ || EF.
`anlge`DPQ = `angle`E (corresponding angles)
`angle`DQP = `angle`F (corresponding angles)
By AAA similarity ( or by AA similarity)
We get `Delta`DPQ ~ `Delta`DEF
`(DP)/(DE)` = `(PQ)/(EF)` `rArr` `(AB)/(DE)` = `(PQ)/(EF)` (given)
But `(AB)/(DE)` = `(BC)/(EF)` (given)
BC = PQ and AB = DP, AC = DQ (construction)
`Delta`ABC `~=` `Delta`DPQ (SSS)
Since `Delta`DPQ ~ `Delta`DEF
`Delta`ABC ~ `Delta`DEF.
Hence, the required result is proved.
Between, if you have problem on these topics greatest integer function, please browse expert math related websites for more help on mathematical induction.
Properties of Congruent Figures
Congruent figures have following properties:
1. The corresponding part of congruent figures are also congruent.
2. The corresponding sides lie opposite to the equal angles and corresponding angles lie opposite to the equal sides.
3. All squares can be similar but not congruent. If squares have equal size (length and width ) they are know as congruent squares.
4. All the rectangles can be similar but Congruent rectangle are equal in size and shape.
Example : Find the missing sides of congruent triangles:
AB = 5cm
AC = 3 cm
EG = 3 Cm
EG = 4 Cm
Find BC and EF?
Both triangles are congruent ( Given )
So All the corresponding sides are equal
BC = FG = 4cm
EF = AB = 5cm
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