LEARN MATH STEP BY STEP THROUGH VERY EASY PROCESS

GEOMETRY-TRIANGLE-CONGRUENCE OF TWO TRIANGLE

__Congruency of two Triangles –__

**To prove that two triangles
are congruent, we need not prove that the six elements of one are equal to the
corresponding six elements of the other. Any of the following four conditions
is enough to prove the congruency of two triangles.**

__Side-Angle-Side (S-A-S) Condition –__

**If any two sides and the included
angle of one triangle are considered to be equal to the corresponding two sides
of the triangle and the included angle of the other triangle then the two
triangles are will be called congruent.**

**In, ∆ ABC and ∆ DEF,**

**So, AB = DE, BC = EF, and ∠ABC
= ∠DEF,**

** ∆ ABC ≅ ∆ DEF**

__Angle-Side-Angle
(A-S-A) Condition -__

**If two angles and the included
side of a triangle are considered to be equal to the corresponding two angles
and the included side of the other triangle then the given two triangles are
congruent.**

**In, ∆ ABC and ∆ DEF, if ∠ABC = ∠DEF, ∠ACB = ∠DEF, and BC = EF, then ∆ ABC ≅ ∆ DEF**

**If two angles of ∆ ABC are equal to two
angles of ∆ DEF then the third angles of ∆ ABC will be equal to the third angle
of ∆ DEF because the sum of the three angles of any triangle is 180⁰. Thus we
can say that two triangles are congruent if any two angles and one side of one
triangle are equal to two angles and the corresponding side of the other
triangle. This condition for congruence is denoted by Angle- Angle–Side
(A-A-S)**

__Side-Side-Side
(S-S-S) Condition –__

**If three sides of one given triangle are
equal to three sides of another given triangle then the obtained two triangles
are considered to be congruent. In ∆ ABC and ∆ DEF, AB = DE, BC = EF and CA =
FD. **

**So,
∆ ABC ≅ ∆ DEF**

__Right Angle
Hypotenuse-Side (R-H-S) Condition –__

**If the hypotenuse and one side of a
right-angled of the given triangle are respectively equal to the hypotenuse and
the corresponding side of the another right-angled of the given triangle then
the two triangles are considered to be congruent.**

**In ∆ ABC and ∆ DEF, ∠ABC = ∠DEF = 90⁰,**

**Hypotenuse AC = hypotenuse DF
and AB = DE,**

**So,
∆ ABC ≅ ∆ DEF**