Triangles

The sum of the interior angles of a triangle is always 180 degrees, regardless of the size or shape of the triangle.

The exterior angle of a triangle is supplementary to the opposite interior angle, meaning they add up to 180 degrees.

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

The formula for the area of a triangle is $A = \frac{1}{2}bh$, where $b$ is the length of the base and $h$ is the height (altitude) of the triangle.

The formula for the perimeter of a triangle is $P = a + b + c$, where $a$, $b$, and $c$ are the lengths of the three sides of the triangle.

A right triangle is a triangle that has one right angle (90 degrees).

An equilateral triangle is a triangle that has three equal sides.

An isosceles triangle is a triangle that has two equal sides.

A scalene triangle is a triangle that has no equal sides.

The sine of an angle in a triangle is the ratio of the length of the opposite side to the length of the hypotenuse.

The cosine of an angle in a triangle is the ratio of the length of the adjacent side to the length of the hypotenuse.

The tangent of an angle in a triangle is the ratio of the length of the opposite side to the length of the adjacent side.

The law of sines states that in a triangle, the ratio of the length of a side to the sine of the opposite angle is the same for all three sides.

The law of cosines states that in a triangle, the square of one side is equal to the sum of the squares of the other two sides minus twice the product of those two sides and the cosine of the angle between them.

Using Heron's formula, the area of the triangle is $\sqrt{s(s-a)(s-b)(s-c)}$, where $s = \frac{a+b+c}{2}$. Plugging in the values, we get $s = \frac{5+12+13}{2} = 15$, and the area is $\sqrt{15(15-5)(15-12)(15-13)} = 60$ square units.