Transcript Honors Geometry Section 3.5 Triangle Sum Theorem

Honors Geometry Section 3.5
Triangle Sum Theorem
A triangle is the figure formed by
three line segments joining three
noncollinear points.
A
ABC
B
C
Triangles are classified according to their angles and sides.
Angle classification:
equiangular
3 congruent angles
acute
3 acute angles
right
1 right angle
obtuse
1 obtuse angle
Side classification:
equilateral
3 congruent sides
isosceles
2 congruent sides
scalene
0 congruent sides
Examples: Classify each triangle
according to its angles and sides.
right isosceles
acute scalene
In Unit III, we discussed parallel lines in rather great
detail. We did, however, fail to discuss Euclid’s Parallel
Postulate. Let’s remedy that now.
Theorem 3.5.1 The Parallel Postulate
Given a line and a point not on the
line, there is exactly one line
through the point parallel to the
given line.
As with many of the postulates that we have
discussed thus far, this may seem obvious but it plays
a very important role in Euclidean geometry. In
Euclidean geometry, planes are flat, but there are
other ways of thinking of a plane. In spherical
geometry planes are the surface of a sphere (i.e. a
globe) and lines are great circles (i.e. the equator or
any of the lines of longitude). In spherical geometry
the Parallel Postulate would read
“given a line and a point not on
the line there are no lines through
the given point parallel to the
given line.
Theorem 3.5.2: Triangle Sum Theorem
The sum of the measures of the

three angles of a triangle is 180
____.
(You will be asked to complete the proof of
the Triangle Sum Theorem in the
homework.)
Examples: Find the value of x.
35  118  x  180
153  x  180
x  27
Examples: Find the value of x.
180  65  71  44
180  44  136
Examples: Find the value of x.
90  x  16  x  180
x  16  x  90
2 x  106
x  53
The angle of x° in example b) is called
an exterior angle of the triangle. An
exterior angle of a triangle is formed
by extending a side
of the triangle.
Note that the exterior angle will form a
_________with
an interior angle of
linear pair
the triangle.
In example b) we found x to equal
136. Note that ____________.
65  71  136
This work leads us to the following
theorem.
Theorem 3.5.3 Exterior Angle Theorem
The measure of an exterior angle
of a triangle is equal to the sum of
the two remote interior angles.
For the triangle to the right,
m1  m2  m3
Example: Find the value of x.
2 x  10  x  65
x  55
A corollary is a statement easily
proven using a particular theorem.
Example c) on the previous page
illustrates the following corollary:
90  x  16  x  180
x  16  x  90
Corollary to the Triangle Sum
Theorem
The acute angles of a right triangle
are complementary
_____________.