Download 4.1 Notes

Geometry
Unit III
Section 4.1: Triangle Sum Theorem
The figure formed by 3 segments joining three noncollinear points
A triangle is ___________________________________________________________________________
___________________________________________________.
C
Triangles are classified according to their angles and sides..
Angle classification:
3 equal angles
equiangular _____________________
3 acute angles
acute _____________________
1 right angle
right _____________________
1 obtuse angle
obtuse _____________________
A
B
Side classification:
3 equal sides
equilateral _____________________
2 equal sides
isosceles _____________________
NO equal sides
scalene _____________________
Examples: Classify each triangle according to its angles and sides.
a) In  ABC, m A  90 and AB  AC .
b) In  DEF , m D  m E  m F  90 with no
congruent sides.
Right isosceles
Acute scalene
Theorem 4.1 Triangle Sum Theorem
degrees (proof on p.196 of the text)
The sum of the measures of the three angles of a triangle is 180
__________.
Examples: Find the value of x.
a)
35°
118°
x°
b)
71°
𝑥 + 35 + 118 = 180
𝑥 + 153 = 180
𝑥 = 27
c)
65°
x°
180 − 71 − 65 = 𝑚𝑖𝑠𝑠𝑖𝑛𝑔 𝑎𝑛𝑔𝑙𝑒
44 = 𝑚𝑖𝑠𝑠𝑖𝑛𝑔 𝑎𝑛𝑔𝑙𝑒
180 − 44 = 𝑥
𝑥 = 136
(x–16)°
x°
90 + 𝑥 + 𝑥 − 16 = 180
2𝑥 + 74 = 180
2𝑥 = 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 ___________________________________________________.
Note that the exterior angle will
Extending a side of the triangle
Linear pair
form a ________________________
with an interior angle of the triangle.
136 Note that __________________.
65 + 71 = 136 This work leads us to the
In example b) we found x to equal _____.
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 measures of the 2 remote interior angles
_____________________________________________________________.
2
1  m _____  m _____
3
For the triangle to the right, m _____
2
1
3
2𝑥 + 10 = 𝑥 + 65
Example: Find the value of x.
𝑥 + 10 = 65
x°
𝑥 = 55
65°
(2x + 10)°
A statement easily proven using a particular theorem
A corollary is __________________________________________________________________________.
Example c) on the previous page illustrates the following corollary:
Corollary to the Triangle Sum Theorem
Complementary or sum = 90 degrees
The acute angles of a right triangle are __________________________.