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

•  Click here for Basic Triangle Part - I

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