Every triangle has three sides and three interior angles. The measures of the interior angles add up to
. The length of each side must be less than the sum of the lengths of the other two sides. For e.g. the sides of a triangle could not have the lengths
is greater than
The following are special triangles.
A triangle with three congruent sides is called an equilateral triangle
. The measures of the three interior angles of such a triangle are also equal, and each measure is
A triangle with at least two congruent sides is called an isosceles triangle
. If a triangle has two congruent sides, then the angles opposite the two sides are congruent. The converse is also true.
For e.g. in
below, since both
, it follows that
. Also, since
, the measure of
A triangle with an interior right angle is called a right triangle. The side opposite the right angle is called the hypotenuse; the other two sides are called legs.
In right triangle DEF above, EF is the hypotenuse and DE and DF are legs. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. Thus, for triangle DEF above,
This relationship can be used to find the length of one side of a right triangle if the lengths of the other two sides are known. For e.g. if one leg of a right triangle has length 5 and the hypotenuse has length 8 then the length of the other side can be determined as follows.
Since , and x must be positive, it follows that , or approximately .
The Pythagorean Theorem can be used to determine the ratios of the sides of two special right triangles. One special right triangle is an isosceles right triangle. Applying the Pythagorean Theorem to such a triangle shows that the lengths of its sides are in the ratio to to, as indicated below.
The other special right triangle is a right triangle, which is half of an equilateral triangle, as indicated below.
Note that the length of the shortest side, x, is one-half the length of the longest side, . By the Pythagorean theorem, the ratio of to is to because
Hence, the ratio of the lengths of the three sides of such a triangle is to to .
In all three triangles above, the area is , or .
Two triangles that have the same shape and size are called congruent triangles. More precisely, two triangles are congruent if their vertices can be matched up so that the corresponding angles and the corresponding sides are congruent.
The following three propositions can be used to determine whether two triangles are congruent by comparing only some of their sides and angles.
If the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent.
If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.
Two triangles that have the same shape but not necessarily the same size are called similar triangles
. More precisely, two triangles are similar if their vertices can be matched up so that the corresponding angles are congruent or, equivalently, the lengths of corresponding sides have the same ratio, called the scale factor of similarity. For e.g. For example, all
right triangles, discussed above, are similar triangles, though they may differ in size.
When we say that triangles ABC and DEF are similar, it is assumed that angles A and D are congruent, angles B and E are congruent, and angles C and F are congruent, as shown in the figure below. In other words, the order of the letters indicates the correspondences.
Since triangles ABC and DEF are similar, we have . By cross multiplication, we can obtain other proportions, such as .