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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:

The angle sum of a triangle is 180˚:


In the figure, w =

  1. 30
  2. 32
  3. 40
  4. 52
  5. 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).
The area of a triangle is , where b is the base and h is the height. Sometimes the base must be extended in order to draw the altitude, as in the third drawing immediately below:

In a triangle, the longer side is opposite the larger angle, and vice versa:


Pythagorean Theorem (right triangles only): The square of the hypotenuse is equal to the sum of the squares of the legs.


Pythagorean triples: The numbers 3, 4, and 5 can always represent the sides of a right triangle and they appear very often: 52  = 32  + 42. Another, but less common, Pythagorean Triple is 5, 12, 13: 132  = 52  + 122.


Two triangles are similar (same shape and usually different sizes) if their corresponding angles are equal. If two triangles are similar, their corresponding sides are proportional:

If two angles of a triangle are congruent to two angles of another triangle, the triangles are similar.
In the figure, the large and small triangles are similar because both contain a right angle and they share A.

Two triangles are congruent (identical) if they have the same size and shape.
In a triangle, an exterior angle is equal to the sum of its remote interior angles and is therefore greater than either of them:
e = a + b  and  e > a  and  e > b


In a triangle, the sum of the lengths of any two sides is greater than the length of the remaining side:
x + y > z
y + z > x
x + z > y

In the figure, what is the value of x ?

  1. 30
  2. 32
  3. 35
  4. 40
  5. 47

Since 2x + 60 is an exterior angle, it is equal to the sum of the remote interior angles.


That is, 2x + 60 = x + 90. Solving for x gives x = 30.


The answer is (A).

In a 30˚–60˚–90˚ triangle, the sides have the following relationships:

In general for x

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