Types of Triangles Study Guide: GED Math (page 2)
Practice problems for these concepts can be found at:
Triangles are three-sided polygons. The three interior angles of a triangle add up to 180 degrees. Triangles are named by their vertices. The triangle pictured is named ABC because of the vertices A, B, and C, but it could also be named ACB, BCA, BAC, CBA, or CAB. The vertices must be named in order, but can start from any one of the vertices.
If you know the measure of two angles of a triangle, you can find the measure of the third angle by adding the measures of the first two angles and subtracting that sum from 180. The third angle of triangle ABC at left is equal to 180 – (50 + 60) = 180 – 110 = 70 degrees.
The exterior angles of a triangle are the angles that are formed outside the triangle.
Adjacent interior and exterior angles are supplementary. Angle y and the angle that measures 50 degrees are supplementary. Angle z is also supplementary to the angle that measures 50 degrees, because these angles form a line. The measure of angle y is equal to 180 – 50 = 130. Because angle z is also supplementary to the 50-degree angle, angle z also measures 130 degrees. Notice that angles y and z are vertical angles—another reason why these two angles are equal in measure.
The measure of an exterior angle is equal to the sum of the two interior angles to which the exterior angle is not adjacent. You already know angle y measures 130 degrees, because it and angle BAC are supplementary. However, you could also find the measure of angle y by adding the measures of the other two interior angles. Angle ABC, 70, plus angle ACB, 60, is equal to the measure of the exterior angle of BAC: 70 + 60 = 130, the measure of angles y and z.
If you find the measure of one exterior angle at each vertex, the sum of these three exterior angles is 360 degrees. The measure of angle y is 130 degrees. The measure of angle u is 110 degrees, because it is supplementary to the 70-degree angle (180 – 70 = 110) and because the sum of the other interior angles is 110 degrees (50 + 60 = 110). The measure of angle w is 120 degrees, because it is supplementary to the 60-degree angle (180 – 60 = 120) and because the sum of the other interior angles is 120 degrees (70 + 50 = 120). The sum of angles y, u, and w is 130 + 110 + 120 = 360 degrees.
Types of Triangles
If the measure of the largest angle of a triangle is less than 90 degrees, the triangle is an acute triangle. The largest angle in this triangle measures 70 degrees; therefore, it is an acute triangle.
If the measure of the largest angle of a triangle is equal to 90 degrees, the triangle is a right triangle. The largest angle this triangle measures 90 degrees; therefore, it is a right triangle.
If the measure of the largest angle of a triangle is greater than 90 degrees, the triangle is an obtuse triangle. The largest angle in this triangle measures 150 degrees; therefore, it is an obtuse triangle.
There are three other types of triangles. If no two sides or angles of a triangle are equal, the triangle is scalene. If exactly two sides (and therefore, two angles) of a triangle are equal, the triangle is isosceles. If all three sides (and therefore, all three angles) of a triangle are equal, the triangle is equilateral.
In a triangle, the side opposite the largest angle of the triangle is the longest side, and the side opposite the smallest angle is the shortest side. You can see this in the scalene triangle pictured. In an isosceles triangle, the sides opposite the equal angles are the equal sides. In a right triangle, the angle opposite the right angle is the hypotenuse, which is always the longest side of the triangle.
Triangles are congruent if they are exactly the same size and the same shape. You might look at two triangles and guess whether they are the same—you're probably a pretty good judge as to whether two triangles are the same size and shape. Or, you could cut out one of the triangles and see if it fits exactly on top of the other one. If so, they are congruent. However, geometry is largely about proving things, not guessing about them. So, you need to be able to use basic geometry rules, such as theorems (formulas or statements in mathematics that can be proved true) to show that two triangles are congruent.
Though geometry rules come in different forms, they can be generally understood as statements that all mathematicians have agreed to accept as true or that can be proved true. Postulates are statements that are accepted without proof. Theorems are statements that can be proved true. You will learn more about these principles when you take a geometry course (if you have not already taken one). For now, you just need to know that you must follow certain steps to prove that two triangles are congruent.
In fact, there are three rules for proving that two triangles are congruent to one another. Notice that the symbol ≅ means congruent to. When you see this symbol, say the word congruent.
Triangles are similar if they have the same shape and their sizes are proportional to one another. You can prove that two triangles are similar using one of the following three rules. Notice that the symbol ˜ means similar to.
Parts of a Right Triangle
A triangle that has a right angle is called a right triangle. The sides and angles of right triangles have special relationships.
As you know, a right triangle has three sides. Two of the sides come together to form the right angle. These two sides are called legs. The third side of the triangle is called the hypotenuse. The hypotenuse is always the longest side of a right triangle. It is directly across from the right angle.
The Pythagorean Theorem
Right triangles are special triangles used for measuring. In a right triangle, the base and one side are perpendicular.
In right triangles, there is a special relationship between the hypotenuse and the legs of the triangle. This relationship is always true and it is known as the Pythagorean theorem. The following equation summarizes the Pythagorean theorem.
a2 + b2 = c2
In the equation, a and b are the legs of the right triangle, and c is the hypotenuse.
In words, the Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.
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