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3-4 Parallel Lines and the Triangle Angle Sum
Theorem
In this study guide you will learn about the interior and exterior angles of a triangle,
different types of triangles and two new theorems.
** Theorem 3-12 Triangle Angle Sum Theorem**
The sum of the measures of the angles of a triangle is 180˚
To prove Theorem 3-12
Draw line CP parallel to line AB----------------------By Protractor Postulate
m1 + m2 + m3=180------------------------------Definition of a Straight Line
mA  m1----------------------------------------------Alternate Interior Angles
mB m2-----------------------------------------------Alternate Interior Angles
mA + mB + m 3=180-----------------------------Substitution
Easy Practice Problems Using Theorem 3-12
Classifying Triangles by Angles and Sides
Exterior Angle of a polygon: the angle formed by a side and the extension of an
adjacent side
Remote Interior Angles: The two angles that are not adjacent to the exterior angle
**Theorem 3-13 Triangle Exterior Angle Theorem**
The measure of each exterior angle of a triangle is the sum of the measures of the its two
remote interior angles
Proof for Theorem 3-12
m2 + m3 + m4=180------------------------------Triangle Angle Sum Theorem
m1 + m4 =180---------------------------------------Definition of a Straight Line
m1 + m2 + m3= m1 + m4 ------------------Substitution
m1 = m2 + m3-------------------------------------Subtraction Property of Equality
Easy Practice Problems Using Theorem 3-13 (#20 may use another theorem as
well. HINT.)
Some Slightly Complicated Problems
27.
And Just One More Proof
ANSWERS
Practice Problems for Theorem 3-12
1. 117+33+ m1=180
150+ m1=180
m1=30
2. 44.7+52.2+ m1=180
96.9+ m1=180
m1=83.1
3. 33+57+ m1=180
90+ m1=180
m1=90
Slightly Complicated Problems
23. 90+(8x-1)+(4x+7)=180
90+12x+6=180
96+12x=180
12x=84
x=7
24. (2x+4)+(2x-9)+x=180
Practice Problems for Theorem 3-13
18. 60+63= m1
m1=123
19. 128.5= m2 +13
115.5= m2
20. m4+ 45+47=180
m4+92=180
m4=88
-Angles are 90, 55, and 35
-This is a Right Triangle
-Angles are 78, 65, and 37
45+47=m3
92= m3
5x-5=180
5x=185
x=37
-This is an Obtuse Triangle
25. y+54+90=180
y+144=180
y=36
26. 55+c=180
c=125
z=90
c+d+32=180
125+d+32=180
157+d=180
d=23
x+z+52=180
x+90+52=180
x+142=180
x=38
32+b=90
b=58
-Triangle ABD and
CBD are Right
Triangles
-Triangle ABC is an
Acute Triangle
a+b+55=180
a+58+55=180
a+113=180
a=67
d+a=e
23+67=e
90=e
-Triangle EFH is an Obtuse Triangle
-Triangle GFH is an Acute Triangle
-Triangle EFG is a Right Triangle
27. (2x-23)+38+x=180
3x-23+38=180
3x+15=180
3x=165
x=55
x=y
55=y
y+ (4w-5)+w=180
55+5w-5=180
50+5w=180
5w=130
w=26
-Angles for Triangle on the
Right: 87, 38 and 55. This
is an Acute Triangle
-Angles for Triangle on the
Left: 55, 26, and 99
34.
mC=90-------------------------------------------------Given
mA + mB + mC=180----------------------------Triangle Angle Sum Theorem
mA+ mB+90=180----------------------------------Substitution
mA+ mB=90-----------------------------------------Subtraction Property of Equality