Angles
Let’s prove that sum of angles in a triangle is always 1800
Looking on a figure above and remembering what we learn about parallel lines we can tell:
If the line 
is parallel to the opposite side of the triangle. The anles marked a are equal, and so ar the angles marked c.
We also know that angles that make up a straight line have a sum of 1800 , so a + b + c = 1800
The angles inside the triangle are also a, b and c.
Сonsequence
Because of any polygon with n sides can be divided into n-2 triangles the sum of the anles in any polygon with n sides is 180(n-2)0
Angle-Side relationship in Triangles
Largest angle of a triangle is always across from the longest side, and vice versa. Likewise, the smallest angle is always across from the shortest side
An isosceles triangle is a triangle with two equal sides. If two sides in triangle are equal, then the angles across from those sides are equal, too, and vice versa.
The triangle inequality
The sum of any two sides of triangle is always greater than the third side. This mean that the length of any side of a triangle must be between the sum and the difference of the other two sides.
12-10 < AB < 12+10
2 < AB < 22
The External anlge theorem
The extended side of a triangle forms an external angle with the adjacent side. The external angle of a triangle is equal to the sum of the two “remote interior” angles.
Notice that this follows from our angle theorem:
a + b + x = 180 and c + x = 180, therefore, a + b =c
Problem 1
In the figure above, if AB = BD, then what would be x ?
Solution
So if AB = BD then ∠BAD = ∠BDA, in the triangle ABD : ∠BAD + ∠BDA +50 = 180 therefore ∠BAD + ∠BDA = 130 2*∠BAD = 130, so ∠BAD = 65
In triangle ACD : ∠A +∠C +90 = 180 so ∠A + x = 90, remembering that ∠A = 65 will get : x= 25
Problem 2
In the figure above, if AD = DB = DC, then what would be x+y ?
Solution
In ∆ ABD : AD = DB ⤇ ∠BAD = ∠ABD = x, using that sum of anlges in triangle = 180, will get: 2x + 100 = 180 ⤇ 2x = 80 ⤇ x= 40;
Analogically, in ∆ BDC : BD = DC ⤇ ∠BCD = ∠DBC = y, ∠BDC= 180 – 100 = 80, 2y + 80 = 180 ⤇ 2y = 100 ⤇ y = 50
So x + y = 50 +40 = 90.
