**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 answer is (A).

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.

c^{2} = a^{2} + b^{2}

**Pythagorean triples:** The numbers 3, 4, and 5 can always represent the sides of a right triangle and they appear very often: 5^{2} = 3^{2} + 4^{2}. Another, but less common, Pythagorean Triple is 5, 12, 13: 13^{2} = 5^{2} + 12^{2}.

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

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