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In this lesson, we will study special triangles and their properties . We will also use special triangles to derive the formulas for the areas of special quadrilaterals.
A scalene triangle is a triangle in which no two sides are congruent.
An isosceles triangle in which at least two sides are congruent.
In an isosceles triangle, the two congruent sides are called the legs and the third side is called the base.
An equilateral triangle in which all three sides are congruent.
How do we classify triangles according to their angles?
An acute triangle is a triangle in which all three angles are acute angles.
A right triangle is a triangle in which there is only one right angle.
In a right triangle, the side opposite the right angle is the hypotenuse, and the other two sides are the legs.
An obtuse triangle is a triangle in which there is only one obtuse angle.
For any triangle, we also have the following parts:
A median of a triangle is the segment from any vertex to the midpoint of the opposite side.
An altitude of a triangle is the segment from any vertex that is perpendicular to the line containing the opposite side.
You may recall the notion of an exterior angle from the previous section on convex polygons . Every triangle has six exterior angles, as shown below.
In the figure above, the angles angle B and angle C of the triangle ABC are referred to as the remote interior angles of the exterior angles angle 1 and angle 2.
An exterior angle of a triangle is equal to the sum of its remote interior angles.
The Exterior Angle Theorem follows from the fact that a fixed angle and one of its exterior angles are supplementary, and that the sum of the angles of a triangle is 180°.
What is the measure of the angle marked y in the triangle shown below?
The correct answer is B. The measure of x is 20 ° because it is the supplement of the angle indicated.
Since the sum of the interior angles is 180 °, we have 20 + 20 + y = 180 . Thus, y = 140 °.
Two sides of a triangle are congruent if and only if the angles opposite those sides are congruent.
This means that an isosceles triangle has at least two congruent angles. Conversely, if a triangle has at least two congruent angles it is an isosceles triangle.
A triangle has three congruent sides if and only if it has three congruent angles.
Any equilateral triangle is equiangular and vice versa.
If angle ABC is an isosceles triangle with congruent sides
and
, then the median from B to
is the altitude from B to
What is the formula for the altitude of an equilateral triangle if s = length of a side?
The correct answer is D. By the preceding theorem the altitude is the median in an equilateral triangle, so the shorter side has length 
. In fact, the altitude divides the triangle into two congruent 30-60-90 triangles. Therefore, the longer leg has length 
= 
.
In an equilateral triangle, the altitudes (which are also the medians) and angle bisectors all intersect at one point in the interior called the center of gravity.
If the hypotenuse and one leg of a right triangle are congruent to corresponding sides of another right triangle, then the triangles are congruent.
In a 30-60-90 triangle, the length of the hypotenuse is twice the length of the shorter leg and the length of the longer leg is 
times the shorter leg.
In a 45-45-90 triangle, the length of the hypotenuse is 
times the shorter leg.
On a baseball diamond, the distance between consecutive bases is 90 feet and the diamond is actually a square. What is the distance from second base to home plate?
feet
feet
feetThe correct answer is D. This distance is in fact a diagonal, and since a diagonal divides a square into two 45-45-90 triangles . A leg has length 90, so the hypotenuse has length 
.