The sum of the measures of the interior angles of a triangle equals 180º.

This theorem is true for ANY type of triangle (acute, scalene, right, etc.).
If the figure is a triangle, the sum of the measures of its angles equals 180º.

This theorem can be shown to be true by cutting the triangle into three pieces as shown below.
If the triangle's three vertices are rearranged to form a straight line, as shown below,
it can be seen that the three angels form a straight angle which contains 180º.

 Examples:
 1 Find xº.
Solution: 76 + 35 + x = 180
111 + x = 180
x =
69º
 2 Find xº.
Solution: right angle contains 90º (the box)
58 + 90 + x = 180
148 + x = 180
x =
42º
 3 Find xº.
Solution: 26 + 108 + x = 180
134 + x = 180
x =
46º
 4 Find xº.
Solution: Work inside the bottom triangle.
48 + 90 + x = 180
138 + x = 180
x =
42º
 5 m∠ABC=m∠BCD Find m∠ACD.
Solution: mBCD = 56º.
In ΔABC, 85º + 56º + mBCA = 180
mBCA = 39º

mACD = 56º - 39º = 17º
 6 Find m∠B and m∠C.
Solution: mA + mB + mC = 180
38 + x + (x + 2) = 180
40 + 2x = 180
2x = 140
x =
70 = m∠B
x + 2 = 72 = m∠C
 7. The angles in a triangle are represented by (4x - 6)º, (2x + 1)º and (x + 3)º. Is this a right triangle?
Solution: If this is a right triangle, one of the angles must be a right angle, 90º).
(4x - 6) + (2x + 1) + (x + 3) = 180
7x - 2 = 180
7x = 182
x = 26
(4x - 6)º = 98º
(2x + 1)º = 53º
(x + 3)º = 29º
No. None of the angles is 90º.
This triangle is obtuse.
8. The angles in a triangle are in the ratio of 1 : 2 : 3. Find the measure of the angles in the triangle.
Solution:
Represnt the ratio 1 : 2 : 3 as x : 2x : 3x.
x + 2x + 3x = 180
6x = 180
x = 30
The angles are 30º, 60º, and 90º.