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Converse of the Pythagorean Theorem

If a^2 + b^2 = c^2, then the triangle is a right triangle, where a and b are the lengths of the legs and c is the length of the hypotenuse of the triangle.

Pythagorean Inequality Theorems

If c^2 > a^2 + b^2, then the triangle is obtuse and

Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

θ

A greek symbol used to represent an angle

sine of θ

sin(θ) =length of opposite side/length of hypotenuse

cosine of θ

cos(θ)=length of adjacent side/length of hypotenuse

tangent of θ

tan(θ)=length of opposite side/length of adjacent side

angle of depression

The angle below horizontal that an observer must look to see an object that is lower than the observer. Note: The angle of depression is congruent to the angle of elevation due to the alternate interior angles theorem

angle of elevation

The angle above horizontal that an observer must look to see an object that is higher than the observer. Note: The angle of elevation is congruent to the angle of depression due to the alternate interior angles theorem

Complementary Angles

The sum of the measure of two angles is 90 degrees.

Opposite side

In a right triangle, the side of the triangle opposite the vertex of one acute angle is called the opposite side, it is the leg not included to form the angle.

Similar triangles

Triangles are similar if they have the same shape but not necessarily the same size. Triangles whose corresponding angles are congruent are similar. Corresponding sides of similar triangles are all proportional.

For each pair of complementary angles in a right triangle,

the sine of one angle is equal to the cosine of its complement.

A right triangle

is a triangle that includes one right, or 90-degree, angle.

three sides

Like any triangle, a right triangle has

three angles

A triangle also has

Two of the sides

come together at the right angle

Hypotenuse

in a right triangle, the side opposite the right angle

Angle C

is always the right angle.

the angles of a Right Triangle

are labeled with upper-case letters A, B, and C.

P = a + b + c

The perimeter of a right triangle is equal to the sum of the lengths of the three sides. Using the conventional labels: In words: Perimeter (P) is equal to side a plus side b plus side c.

A = ½ab

The area of a right triangle is equal to one-half the product of side a and side b. Or in mathematical terms:

The Pythagorean Theorem

For any right triangle, the lengths of the two sides and the hypotenuse are related by an important mathematical statement called the

c^2 = a^2 + b^2

This formal statement of the Pythagorean Theorem relates the lengths of the three sides of a right triangle:

The hypotenuse

is always the longest of the three sides of a right triangle

The length of the hypotenuse

is always less than the sum of the two other sides

the length of the third side

Given the lengths of two sides, you can calculate

side c — the hypotenuse

If you are given the lengths of sides a and b, you can determine the length of

you can determine the length to the remaining side

If you are given the length of the hypotenuse and one other side (a or b)

a, b = the lengths of the other two sides

By solving the Pythagorean Theorem for c, you have an equation for determining the length of the hypotenuse: