Download Section 1.1 and 1.2

Section 1.1: Angles, Degrees, and Triangles
Angles and Degree Measure
First, let’s start with a definition from good ol’ Wikipedia:
“Trigonometry (from Greek trigōnon "triangle" + metron "measure") is a
branch of mathematics that studies triangles and the relationships between
the lengths of their sides and the angles between those sides.”
Basic Terminology:
Line – the straight path connecting two points (generally A and B)
and extending beyond the points in both directions.
Line segment – the portion of the line between those two points.
Ray – a portion of the line that starts at the point A and continues
beyond B to infinity.
Angle – formed when a ray is rotated about its endpoint.
When we measure lengths or distances, we use units like:
When we measure angles, we use units like:
An angle formed by one complete counterclockwise rotation has a measure
of ______________
How many degrees would ½ a rotation be?
What about of a rotation?
What about
of a rotation?
Heads up: Greek letters are commonly used to denote angles in trig.
Types of angles that could be good to know about:
Example 1: Find the measure of each angle:
a. Find the compliment of 25°.
b. Find the supplement of 135°.
c. Represent the compliment of in terms of .
d. Find two supplementary angles such that the first angle is three times
as large as the second angle.
Triangles
The sum of the measures of the angles of any triangle is ___________________.
Example 2: If two angles of a triangle have measures 16° and 96°, what is
the measure of the third angle?
In any right triangle, the square of the length of the longest side
(hypotenuse) is equal to the sum of the squares of the lengths of the other
two sides (legs).
Example 3: Using the Pythagorean Theorem to Find the Side of a Right
Triangle
Suppose you have a 13-foot ladder and want to reach a height of 12 feet to
clean out the gutters on your house. How far from the base of the house
should the base of the ladder be?
Special Right Triangles
If a triangle has angles measuring 45°-45°-90°, what are the lengths of each
of the sides?
Example 5: Solving a 45°-45°-90° Triangle
A house has a roof with a 45° pitch. If the house is 48 feet wide, what are
the lengths of the sides of the roof that form the attic? Round to the nearest
foot.
What about a 30°-60°-90° Triangle? How do each of the sides in this
triangle relate?
Example 6: Solving a 30°-60°-90° Triangle
Before a hurricane strikes it is wise to stake down trees for additional
support during the storm. If the branches allow for the rope to be tied 20
feet up the tree and a desired angle between the rope and the ground is
60°, how much total rope is needed? How far from the base of the tree
should each of the two stakes be hammered?
Section 1.2: Similar Triangles
Finding Angle Measures Using Geometry
Definitions:
Vertical Angles – angles of equal
measure that are opposite one
another and share the same vertex.
Transversal – a line that intersects
two other lines.
Heads up: There are two ways to indicate the two or more angles have the
same measure:
Properties of Angles
Example 1: Finding Angle Measures
Given that angle 1= 110° and ||, find the measure of angle 7.
Classification of Triangles
What are the three different types of triangles (based on the number of
sides of equal length)?
Similar Triangles – triangles with equal corresponding angle measures
(equal angles).
Congruent Triangles – triangles with equal corresponding angle measures
(equal angles) and corresponding equal side lengths.
A sweet property of Similar Triangles:
1) Corresponding angles must have the same measure
2) Corresponding sides must be proportional (ratios must be equal)
= =
′ ′ ′
Example 2 : Finding Lengths of Sides in Similar Triangles
Given that the following two triangles are similar, find the length of each of
the unknown sides ( and ).
Applications Involving Similar Triangles
Example 3: Calculating the Height of a Pole
Rex is trying to figure out the height of a utility pole. He has measured its
shadow to be 8 feet long while his 3-foot-tall mailbox has a shadow of 1.3
feet. How tall is the pole?