Details of Triangle and its Properties – Part – II

 

As we
all know, a triangle is a shape that consists of three sides and three angles.
Taking a look at the triangle’s angles often helps us find out what kind of
triangle we are dealing with.

 

 

Here
are some facts to remember:

 

 

  • The three angles
    in a triangle always add up to 1800
  • The three angles
    of an equilateral triangle are all equal to 600
  • Two angles of an
    isosceles triangle are equal.
  • One angle of a
    right-angled triangle is 900
  • All angles of an
    acute-angled triangle are acute angles, thus smaller than 900
  • One angle of an
    obtuse-angled triangle is obtuse, thus larger than 900 and
    smaller than 1800

 

 

 

 

If two
triangles are congruent they have equal sides, equal areas. 

 

 

Condition
for congruence:

 

 

 

 

1. SAS
condition

 

 

 

 

If two sides and the included angle of one triangle is equal to the
corresponding sides and included
angle of the other triangle, then both triangles are congruent.
AB = DE, BC = EF and ∠B = ∠E,
then ΔABC≈ΔDEF
2. ASA condition

 

 

 

 

If two angles and the included side of one triangle is equal to the
corresponding two angles and    the
included side of the other triangle, then both triangles are congruent.


If ∠A = ∠D, ∠B
= ∠E and AB = DE, then ΔABC≈ΔDEF
3. SSS condition

 

 

 

 

If
three sides of one triangle is equal to the corresponding three sides of other
triangle then both    triangles are
congruent.

 

  If AB = DE, AC = DF and BC = EF, then ΔABC≈ΔDEF

 

 

 

 

4. RHS
condition

 

 

 

 

 

If the
two triangles are right-

angled triangle and hypotenuse and one side of one
triangle is equal    to the
hypotenuse and corresponding side of other triangle, then both triangles are
congruent.

 

if ∠B
=∠E = 90°, AC = DF and AB = DE or BC = EF, then ΔABC≈ΔDEF

 

 

Note:

 

 

 i.
All the congruent triangles are similar but all similar triangles are
not congruent.

 

 

 ii.
The ratio of the areas of two similar triangles is equal to the ratio of
the squares of any two corresponding sides.

 

 

The
following four theorems are most important in solving questions on triangles.
Pythagoras’ theorem: In a right angle triangle the square of the
hypotenuse is equal to the sum of the squares of other two sides.

Pythagorean triplet: There are certain triplets which satisfy the pythagoras’
theorem and are commonly, called pythagorean triplet.


 For example: 3, 4, 5;
5, 12, 13;    24, 10, 26;    24, 7, 25;
15, 8, 17
Appolonious theorem: 

 

 

 

 

 

 

In
triangle ABC, AD is median, which divides BC into two equal parts. Then,

AB2+AC2=2(AD2+BD2)=2(AD2+DC2)

 

 

 

 Stewart
theorem:

 

 

 

 

In
Triangle ABC, AD divides side BC in the ratio m and n. (Here AD need not be
median) then, 

 

 

m.b2+n.c2=a(d2+mn)

 

 

 

 

Mean
proportionality and Mid Point theorem:

 

 

 

 

 

 

In the
first triangle, DE // BC so AD/DB=AE/EC

 

 

In the
second triangle, D and E are mid points of AB and AC respectively. Which
implies, AD/DB=AE/EC=1

 

 

Also, DE=1/2BC