Download Lesson 4_1-4_3 Notes

Lesson 4.1-4.3:
Triangle Sum Conjecture
As you’ve learned, a triangle is the simplest polygon, having 3 sides and three angles.
The sum of the three angles is equal to 180 degrees.
Another application of triangles is a procedure used in surveying called triangulation. This
procedure allows surveyors to locate points or positions on a map by measuring angles and distances
and creating a network of triangles. Triangulation is based on an important property of plane
geometry.
Triangles are classified by sides and angles. Just a reminder*
By Angles:
Less than 90°
Exactly 90°
Greater than 90°
By Sides:
No equal sides
Each angle measures
exactly 60 °
Vertex
Legs
Legs
Base
Side a = Side b
Side a = Side b = Side c
It goes like this…..
An isosceles triangle is a triangle with at least two congruent sides. In an isosceles
triangle, the angle between the two congruent sides is called the vertex angle, and the
other two angles are called the base angles. The side between the two base angles is
called the base of the isosceles triangle. The other two sides are called the legs.
Equilateral triangles have at least two congruent sides, so they fit the definition of
isosceles triangles. That means any properties you discover for isosceles triangles
will also apply to equilateral triangles.
Used as an
example.
Not
accurate.
So far you have studied interior angles of triangles. Triangles also have exterior angles.
If you extend one side of a triangle beyond its vertex, then you have constructed an
exterior angle at that vertex.
Each exterior angle of a triangle has an adjacent interior angle and a pair of remote
interior angles. The remote interior angles are the two angles in the triangle that do
not share a vertex with the exterior angle.
You can also think of it in another way: The shortest path between two points is along
the segment connecting them. In other words, the path from A to C to B can’t be shorter
than the path from A to B.
C
A
B
Side-Angle Inequality:
Angle-Side Relationships
If one side of a triangle is longer than another side, then the angle opposite the longer
side will have a greater degree measure than the angle opposite the shorter side.
Converse also true: If one angle of a triangle has a greater degree measure than
another angle, then the side opposite the greater angle will be longer than the side
opposite the smaller angle.
*In short, we just need to understand that the larger sides of a triangle lie opposite of
larger angles, and that the smaller sides of a triangle lie opposite of smaller angles. Let's
look at the figures below to organize this concept pictorially.
Since segment BC is the
longest side, the angle
opposite of this side,  A, is
the one that has the largest
measure in  ABC.
Our smallest angle, C, tells us
that segment AB is the smallest
side of ABC.
In the figure below, what range of length is possible for the third side, x, to be.
When considering the side lengths of a
triangle, we want to use the Triangle
Inequality Theorem. Recall, that this
theorem requires us to compare the length
of one side of the triangle, with the sum of
the other two sides. The sum of the two
sides should always be greater than the
length of one side in order for the figure to
be a triangle. Let's write our first
inequality.
So, we know that x must be greater than
3. Let's see if our next inequality helps us
narrow down the possible values of x.
This inequality has shown us that the
value of x can be no more than 17. Let's
work out our final inequality.
This final inequality does not help us
narrow down our options because we
were already aware of the fact that x had
to be greater than 3. Moreover, side
lengths of triangles cannot be negative, so
we can disregard this inequality.
Combining our first two inequalities yields:
So, using the Triangle Inequality Theorem shows us that x must have a
length between 3 and 17.