We have therefore shown that the area of a triangle is one-half the product of the lengths of the base and altitude (or height). Thus, the area of one of the triangles is half the area of the rectangle. In this case, we combined two equivalent triangles to form the rectangle. But we know that the area of a rectangle is simply the length times the width. Notice that now we have formed a rectangle with a length of b + c (which is the length of the base of the triangle) and a width of a (which is the altitude, or height, of the triangle). An altitude is shown as a dashed line in the triangle below. This opposite side, in this case, is called the base of the triangle. The altitude is a line segment that passes through a vertex and that is perpendicular to the side opposite that vertex.
When analyzing triangles, it is sometimes helpful to identify the altitude of a triangle. Note that we use the positive square root in the calculation above a negative result would not have any meaning in this context. Thus, the unknown side has a length of 4 (technically, the length is "4 units," but the units are unspecified and therefore are occasionally unmentioned). The unknown side is a leg, which we can call x, and the side of length 5 is the hypotenuse. Solution: Because this triangle is a right triangle, we can use the Pythagorean theorem. Practice Problem: Find the length of the unknown side in the triangle below. Thus, each angle in an equilateral triangle has a measure of 60°. Combine this with the fact that the sum of all the angles in a triangle is 180° we can then write Solution:Recall from the definition that the angles of an equilateral triangle all have equal measures. Practice Problem:How many degrees are in each angle of an equilateral triangle? If we know the lengths of two of the sides for a right triangle, we can calculate the length of the third side.
This theorem only applies to right triangles. We can relate the lengths of the sides by way of the following formula (the Pythagorean theorem): The side c, which is opposite the right angle, is called the hypotenuse. (Although we learned about supplementary angles in the context of only two angles, the same principle also applies to three or more angles as well.) Thus, By the rules of corresponding angles, we can infer the following:īut notice that the three angles A, B, and C formed with the line q are supplementary angles.
We can therefore use the properties concerning parallel lines cut by transversals to make inferences about the angles shown in the diagram. Note that the two line segments are simply transversals: they intersect two parallel lines. We now use the vertex labels as angle labels (in italics) to avoid cluttering the diagram. Now, extend line segment AB (which we can write as ), forming line p, and add a line q that is parallel to p and that intersects point C. Note that the figure is not necessarily drawn to scale. Let's start with an arbitrary triangle ABC, which is shown below. Using what we have learned about geometry so far, we can prove a well-known property of triangles: the sum of the measures of the angles in a triangle is 180°. These intersection points are called vertices (which is the plural form of vertex). Of course, three intersecting line segments are sufficient to form a triangle as well in either case, the intersection points may or may not be indicated with labeled (or unlabeled) points.
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