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.

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:

are some facts to remember:

- The three angles

in a triangle always add up to 180^{0} - The three angles

of an equilateral triangle are all equal to 60^{0} - Two angles of an

isosceles triangle are equal. - One angle of a

right-angled triangle is 90^{0} - All angles of an

acute-angled triangle are acute angles, thus smaller than 90^{0} - One angle of an

obtuse-angled triangle is obtuse, thus larger than 90^{0}and

smaller than 180^{0}

If two

triangles are congruent they have equal sides, equal areas.

triangles are congruent they have equal sides, equal areas.

**Condition**

for congruence:

for congruence:

**1. SAS**

condition

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

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.

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

condition

If the

two triangles are right-

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

=∠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.

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 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.

hypotenuse is equal to the sum of the squares of other two sides.

following four theorems are most important in solving questions on triangles.

**Pythagoras’ theorem:**In a right angle triangle the square of thehypotenuse 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:**

AB^{2}+AC^{2}=2(AD^{2}+BD^{2})=2(AD^{2}+DC^{2})

**Stewart**

theorem:

theorem:

m.b

^{2}+n.c^{2}=a(d^{2}+mn)

**Mean**

proportionality and Mid Point theorem:

proportionality and Mid Point theorem:

In the

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

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

second triangle, D and E are mid points of AB and AC respectively. Which

implies, AD/DB=AE/EC=1

Also, DE=1/2BC