Types of Triangles Study Guide: GED Math
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.
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