Triangles
A triangle containing a right angle is called a right triangle. The right angle is denoted by a small square:A triangle with two equal sides is called isosceles. The angles opposite the equal sides are called the base angles, and they are congruent (equal). A triangle with all three sides equal is called equilateral, and each angle is 60˚. A triangle with no equal sides (and therefore no equal angles) is called scalene:
The altitude to the base of an isosceles or equilateral triangle bisects the base and bisects the vertex angle:
Isosceles:
Equilateral:
The angle sum of a triangle is 180˚:
A. 30 B. 32 C. 40 D. 52 E. 60 

x + 150 = 180 since x and 150 form a straight angle
x = 30 solving for x
z + x + 90 = 180 since the angle sum of a triangle is 180˚
z + 30 + 90 = 180 replacing x with 30
z = 60 solving for z
z = y = 60 since y and z are vertical angles
w + y + 90 = 180 since the angle sum of a triangle is 180˚
w + 60 + 90 = 180 replacing y with 60
w = 30 solving for w
The answer is (A).
x + y > z  
y + z > x  
x + z > y 
Two sides of a triangle measure 4 and 12. Which of the following could equal the length of the third side?
A. 5
B. 7
C. 9
D. 15
E. 20
Each side of a triangle is shorter than the sum of the lengths of the other two sides, and, at the same time, longer than the difference of the two.
Hence, the length of the third side of the triangle in the question is greater than the difference of the other two sides (12 – 4 = 8) and smaller than their sum (12 + 4 = 16).
Since only choices (C) and (D) lie between the values 8 and 16, the answers are (C) and (D).