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:

$h=s\frac{\sqrt{3}}{2}$

a + b + c = 180°
The angle sum of a triangle is 180°:

Example:
In the figure, w = ?

(A) 30
(B) 32
(C) 40
(D) 52
(E) 60

since x and 150 form a straight angle
x + 150 = 180
solving for x
x = 30
since the angle sum of a triangle is 180°
z + x + 90 = 180
replacing x with 30
z + 30 + 90 = 180
solving for z
z = 60
since y and z are vertical angles
z = y = 60
since the angle sum of a triangle is 180°
w + y + 90 = 180
replacing y with 60
w + 60 + 90 = 180
solving for w
w = 30

The area of a triangle is $\frac{1}{2}$bh, 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:

A =
$\frac{1}{2}$bh

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

50° is larger than 30°, so side b is longer than side a.

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

c2 = a2 + b2

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:

$\frac{a}{d}=\frac{b}{e}=\frac{c}{f}$

If two angles of a triangle are congruent to two angles of another triangle, the triangles are similar.

In the figure above, 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

Example: Two sides of a triangle measure 4 and 12. Which one of the following could equal the length of the third side?

(You can select more than one answer.)

(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).

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