Chapter 4 - Triangles & Diagonals

The Angles of a Triangle

1. The sum of the three angles of a triangle equals 180 2. Angles correspond to their opposite sides. This means that the largest angle is opposite the longest side, while the smallest angle is opposite the shortest side. Additionally, if two sides are equal, their opposite angles are also equal.

Common Right Triangles

Common Combinations: The most popular 3-4-5 3^2 + 4^2 = 5^2 ==> 9 + 16 = 25 Also quite popular 5-12-13 5^2 + 12^2 = 13^2 ==> 25 + 144 = 169 Not as popular 8-15-17 8^2 + 15^2 = 17^2 ==> 64 + 225 + = 289 Watch out for impostor triangles. A non-right triangle with one side equal to 3 and another side equal to 4 does not have a third side of length 5.

Equilateral Triangles and the 30-60-90 Triangle

An equilateral triangle is one in which all three sides (and all three angles) are equal. Each angle of an equilateral triangle is 60 degrees. A close relative of the equilateral triangle is the 30-60-90 triangle. Notice that two of these triangles, when put together, form an equilateral triangle. The lengths of the legs of every 30-60-90 triangle have a set ratio: (see screenshot, easier to understand) x : x(3)^1/2 : 2x Leg (short): Angle = 30 degrees Length = x Leg (long): Angle = 60 degrees Length = x(3)^1/2 Hypotenuse: Angle = 90 degrees Length = 2x

Exterior Angles of a Triangle

An exterior angle of a triangle is equal in measure to the sum of the non-adjacent (opposite) interior angles of the triangle. For example (see screenshot, easier to understand) (two equations): a + b + c = 180 b + x = 180 Since a + b + c = 180 and b + x = 180 (and b = 180 - x) Then a + (180 - x) + c = 180 Therefore, a + c = x

Isosceles Triangles and the 45-45-90 Triangle

An isosceles triangle is one in which two of the three sides are equal. The two angles opposite those two sides will also be equal. An isosceles right triangle has one 90 degrees angle and two 45 degrees angles. This triangle is called the 45-45-90 triangle. The lengths of the legs of every 45-45-90 triangle have a set ratio: (see screenshot, easier to understand) x : x : x(2)^1/2 Leg: Angle = 45 degrees Length = x Leg: Angle = 45 degrees Length = x Hypotenuse: Angle = 90 degrees Length = x(2)^1/2 Interestingly, the 45-45-90 triangle is exactly half of a square. That is, two 45-45-90 triangles put together make up a square. Thus, if you are given the diagonal of a square, you can use the 45-45-90 ratio to find the length of a side of a square.

Triangles and Area

Area of a Triangle = (1/2) * Base * Height The height always refers to the a line drawn from the opposite vertex to the base, creating a 90 degrees angle. Although you may commonly think of "the base" of a triangle as whichever side is drawn horizontally, you can designate any side of a triangle as the base. (see screenshot) Right triangles have three possible bases just as other triangles do, but they are special because their two legs are perpendicular. Therefore, if one of the legs is chosen as the base, then the other leg is the height. Of course, you can also choose the hypotenuse as the base.

Similar Triangles

Often, looking for similar triangles can help you solve complex problems. Triangles are defined as similar if all their corresponding angles are equal and their corresponding sides are in proportion (see screenshot). While the above is the definition of a similar triangle, in order to find similar triangle, find congruent angles. Once you find that two triangles have two pairs of equal (or congruent) angles, you know that the triangles are similar. If two sets of angles are congruent, then the third set of angles must be congruent.

The Sides of a Triangle

Triangle Inequality Law: The sum of any two sides of a triangle must be "greater than" the third side. (See screenshot) Note that the sum of two sides cannot be equal to the third side. The sum of two sides must always be greater than the third side. Likewise, the difference cannot be equal to the third side. The difference between two sides must be less than the third side. If you are given two sides of a triangle, the length of the third side must lie between the difference and the sum of the two given sides. For instance, if you are told that two sides are of length 3 and 4, then the length of the third side must be between 4 - 3 = 1 and 4 + 3 = 7. (see screenshot)