Download Section 6.3 Triangles

Section 6.3
Triangles
OBJECTIVE 1:
Classifying Triangles
The word “Trigonometry” comes from the Greek words “Trigonon”, meaning triangle, and “metron”, meaning
measure. In the first section of Chapter 6, we introduced angle measure. In this section, we introduce some
fundamentals of triangles.
Every triangle has three angles and three sides. The sum of the measures of the three angles is equal to 180 or 
radians. Two angles of a triangle are congruent if the measure of the two angles is the same. Similarly, two sides
of a triangle are congruent if the lengths of the two sides are equal.
We can classify triangles based on the measure of their angles or the lengths of their sides. We start by classifying
triangles based on their angle measures. We will use the notation, , to represent an angle. The notation, A ,
refers to angle A. A triangle with three acute angles is called an acute triangle. A triangle with one obtuse angle is
called an obtuse triangle. A triangle with one right angle is called a right triangle. The side opposite the right
angle of a right triangle is called the hypotenuse and the two other sides of a right triangle are called legs. Also,
note that the hypotenuse and both legs can be referred to as “sides” but that the hypotenuse is never referred to as a
“leg.”
Draw an Acute Triangle
Draw an Obtuse Triangle
Draw a Right Triangle
We can also classify triangles based on their side lengths. We can represent that two sides of a triangle
are congruent by drawing the same amount of “tick marks” through the congruent sides. A triangle with
no congruent sides is called a scalene triangle. A triangle with two congruent sides is called an isosceles
triangle. The angles opposite the congruent sides of an isosceles triangle are also congruent. A special
triangle with three congruent sides is called an equilateral triangle. Note that all three angles of an

equilateral triangle are congruent with a measure of 60 or radians .
3
Draw a Scalene Triangle. Use tick marks to designate different lengths.
Draw an Isosceles Triangle. Use tick marks to designate different lengths.
Draw an Equilateral Triangle. Use tick marks to designate different lengths.
OBJECTIVE 2:
Using the Pythagorean Theorem
The Pythagorean Theorem
Given any right triangle, the sum of the squares of the lengths
of the legs is equal to the square of the length of the hypotenuse.
In other words, if a and b are the lengths of the two legs and if
c is the length of the hypotenuse, then a 2  b2  c 2 .
6.3.9 An official NCAA women’s softball “diamond” is really a square. The distance between each consecutive
base is 60 feet. What is the distance between home plate and second base?
6.3.10 How far up the side of a building will a ______m ladder reach, if the foot of the ladder is _____m
from the base of the building?
OBJECTIVE 3:
Understanding Similar Triangles
Triangles that have the same shape but not necessarily the same size are called similar triangles.
Properties of Similar Triangles
1. The corresponding angles have the same measure.
2. The ratio of the lengths of any two sides of one triangle is equal
to the ratio of the lengths of the corresponding sides of the other
triangle.
Definition
Proportionality Constant of Similar Triangles
If two triangles are similar, there exists a constant k  1 called the
proportionality constant of similar triangles equal to the ratio
of the lengths of corresponding sides.
a b c
Given the similar triangles in the figure below, k    , where a  x ,
x y z
b  y , and c  z .
6.3.18 Find the missing sides of the given similar right triangles.
Understanding the Special Right Triangles – KNOW THESE!!
OBJECTIVE 4:
The
  
, ,
4 4 2
 45, 45, 90  Right Triangle


4
a
a 2
a
The
  
, ,
6 3 2
1
4
2


4
4
 30, 60, 90 
1
Right Triangle
6.3.22 and 6.3.24 Find the missing sides of each special right triangle.
OBJECTIVE 5:
Solving Applied Problems Using Similar Triangles
6.3.31 A _________ft tall man is standing next to a building. The man’s shadow is ________ ft long.
At the same time, the building casts a shadow that is ______ feet long. How tall is the building?
6.3.34 The angle of elevation is the angle from the horizontal looking up at an object. The angle of
elevation of an airplane is ___________ radians, and the altitude of the plane is _________ ft. How far
away is the plane “as the crow flies” which means through the air rather than on the ground?