Lauretta J. Fox
The triangle is the simplest and one of the most familiar of all polygons. It is used in construction and design of every description. We see it in the framework of buildings and bridges. Because it is a rigid figure, the shape of a triangle cannot be changed when pressure is applied to it. For this reason the triangle provides an excellent support for many structures.
A
triangle
is a polygon that has three sides. The symbol used to denote a triangle is ’ . An
altitude
of a triangle is a segment drawn from a vertex perpendicular to the side opposite that vertex, or perpendicular to that side extended. A
median
of a triangle is a segment drawn from a vertex to the midpoint of the side opposite that vertex. Every triangle has three altitudes and three medians.
Example:
(figure available in print form)
Triangle ABC is shown.
CD is an altitude.
CE is a median.
As its name implies, a triangle has three angles. The sum of the three angles of a triangle is 180 degrees.
Example:
(figure available in print form)
Triangles may be classified by their sides. A
scalene triangle
has no equal sides. An
isosceles triangle
has two equal sides. The equal sides of the isosceles triangle are the legs and the third side is the base of the triangle. If three sides of a triangle are equal, the triangle is
equilateral
. In every triangle, the sum of any two sides is greater than the third side.
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Scalene Triangle
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Isosceles Triangle
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Equilateral Triangle
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(figure available in print form)
Triangles also may be classified by their angles. An
acute triangle
is a triangle in which each angle is less than 90¼. A
right triangle
contains one right angle. The sides that form the right angle are called legs, and the side opposite the right angle is the hypotenuse of the triangle. If a triangle contains one obtuse angle, it is an
obtuse triangle
. An
equiangular triangle
has three equal angles.
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Acute Triangle
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Right Triangle
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Obtuse Triangle
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(figure available in print form)
There is a relationship between the number of equal sides and the number of equal angles in a triangle. If all sides of a triangle are unequal, the angles opposite these sides are unequal in the same order, that is, the largest angle is opposite the largest side, the middle angle is opposite the middle side, and the smallest angle is opposite the smallest side.
In an isosceles triangle, the angles opposite the equal sides are equal. They are called
base angles
, and the third angle of the isosceles triangle is the
vertex angle
. An equilateral triangle is always equiangular.
Suggested Assignment: In your home or neighborhood, identify as many types of triangles as you can. Name the places where triangles are used most often. For what purpose are triangles used in architecture?
Exercises
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1.) The sides of ’MNP are ___, ___, and ___.
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(figure available in print form)
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2.) The vertices of ’ MNP are ___, ___, and ___.
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3.) If MR = RN, PR is a ___ of ’ MNP.
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4.) If MT is perpendicular to PN, MT is an ___ of ’ MNP.
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5.) In ’ ABC, A = 67° and B = 36¼ , C = ___¼.
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6.) Can the sides of a triangle be (a) 2”, 3”, 7”? (b) 4”, 5”, 6” ?
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7.) In right triangle RST, S is the right angle. (a) The legs of ’ RST are ___ and ___. (b) The hypotenuse of ’RST is ___.
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8.) In isosceles ’ XYZ, XY = XZ.
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(a) The legs of ’ XYZ are ___ and ___.
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(b) The base of ’ XYZ is ___.
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(c) The base angles of ’ XYZ are ___ and ___.
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(d) The vertex angle Of ’ xYz is ___.
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9.) In ’ EFG, E=100°, F = 50°, and G = 30°
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(a) The largest side of ’ EFG is ___.
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(b) The smallest side of ’ EFG is ___.
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lO.) Classify each triangle shown as scalene, isosceles, or equilateral:
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(figure available in print form)
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11.) Classify each triangle shown as right, obtuse, or equiangular
(figure available in print form)
