A triangle is equal to 180°. The sum of angles in each of the six types of triangles (obtuse, scalene, equilateral, isosceles, acute, and right) is 180°. This can be demonstrated by calculating the sum of angles in equilateral triangle, isosceles triangle, and right triangle.
Each internal angle in an equilateral triangle is equal to 60°. Therefore, the sum of the three angles in an equilateral triangle is 180°.
In an equilateral triangle,
S = 60° + 60° + 60°
Let S be the sum of the angles in any type of triangle. If the three angles in a triangle are <A, <B, and <C,
S = A + B + C
In the case of a right triangle, there is a right triangle facing the hypotenuse and two other angles facing the opposite and adjacent sides. If the angle facing the opposite side is A and the angle facing the adjacent side is B, the sum of the two angles is 90°.
In a right triangle, if right angle = C,
A + B = 90°
S = A + B + C
= A + B + 90°
= 90° + 90°
Isosceles triangle has two equal internal angles called base angles and two equal sides. If one of the base angles is 40° the other base angle is 40° and the sum of the base angles is 80°. The third angle is derived by subtracting the sum of the base angles from 180°.
Let the third angle in an isosceles triangle be C and the base angles A and B.
C = 180° – (A + B)
Substituting 40° for A and B,
C = 180° – (40° + 40°)
= 180 ° – 80°
In an isosceles right triangle, one of the angles is 90° and the remaining two angles are 45° each. If C is angle 90°, A = 45°, and B = 45°.
In an isosceles right triangle,
S = 90° + 45° + 45°