Chapter_4.2_Angles_of_Triangles_web
"It's okay to make mistakes. Mistakes are our teachers -- they help us to learn." John Bradshaw
Copy these into your Glossary
Theorem Example
Angle Sum
Theorem
The sum of the measures of the angles of a triangle is 180 m
W + m
X
+m
Y = 180
X
W
Third Angle
Theorem
If two angles of one triangle are congruent to two angles of a second triangle, the third angles of the triangles are congruent.
If
B
C
A
F and
D, then
E, then
Exterior
Angle
Theorem
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles.
Corollaries The acute angles of a right triangle are complementary m
YZP = m
X +m
Y m
G + m
J =
90
X
A
B
Y
G
C
H J
F
D
S
There can be at most one right or obtuse angel in a triangle
Acute
E
Y
P
Chapter 4.2 Angles of Triangles:
Objective: Understand and apply the angle sum and exterior angle theorems
.
Check.4.11 Use the triangle inequality theorems (e.g.,
Exterior Angle Inequality Theorem, Hinge Theorem, SSS
Inequality Theorem, Triangle Inequality Theorem) to solve problems.
Check.4.12 Apply the Angle Sum Theorem for polygons to find interior and exterior angle measures given the number of sides, to find the number of sides given angle measures, and to solve contextual problems.
Spi.4.11 Use basic theorems about similar and congruent triangles to solve problems.
Angles of Triangle
Cut out a triangle (1/2 size of a piece of paper)
Label vertices A, B, and C (on front and back)
Fold vertex B so it touches AC the fold line is parallel AC
Fold A and C so they meet vertex B
What do you notice about the sum of angles A, B and C?
Tear of vertex A, and B
Arrange
A and
B so they fill in the angle adjacent and supplementary to
C.
What do you notice about the relationship
A and
B and the angle outside
C?
"It's okay to make mistakes. Mistakes are our teachers -- they help us to learn." John Bradshaw
Demonstrated 2 Theorems
Theorem Example
Angle Sum
Theorem
The sum of the measures of the angles of a triangle is 180 m
W + m
X
+m
Y = 180
X
W
Third Angle
Theorem
If two angles of one triangle are congruent to two angles of a second triangle, the third angles of the triangles are congruent.
If
B
C
A
F and
D, then
E, then
Exterior
Angle
Theorem
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles.
Corollaries The acute angles of a right triangle are complementary m
YZP = m
X +m
Y m
G + m
J =
90
X
A
B
Y
G
C
H J
F
D
S
There can be at most one right or obtuse angel in a triangle
Acute
E
Y
P
Angle Sum Theorem
Given
ABC
Prove: m
A+m
B+m
C = 180
X
C
A
1 2 3
Statement
1.
ABC
2.
Line XY through A || CB
3.
1 and
CAY form a linear pair
4.
1 and
CAY are supplementary
5.
m
1+m
CAY=180
6.
m
CAY= m
2+m
3
7.
m
1+m
2+m
3=180
8.
1
C,
3
B
9.
m
1=m
C, m
3=m
B
10. m
C+m
2+m
B=180
B
Y
Reasons
1.
Given
2.
Parallel Postulate
3.
Def of linear pair
4.
If 2
’s form a linear pair, they are supplementary
5.
Def of supplementary
’s
6.
Angle Addition Postulate
7.
Substitution
8.
Alternate Interior Angle Theorem
9.
Def of congruent angles
10. Substitution
Find the missing Angles
82
28
1
2 m
2 + m
3 + 68 = 180
70+ m
3 + 68 = 180 m
3 + 138 = 180 m
3 = 42 3 m
1 + 28 + 82 = 180 m
1 + 110 = 180 m
1 = 70 m
1 = m
2 vertical angles
68
Find the missing Angles
79
43
1
74
m
2 + m
3 + 79 = 180
63 + m
3 + 79 = 180 m
3 + 142 = 180 m
3 = 38
2
3 m
1 + 74 + 43 = 180 m
1 + 117 = 180 m
1 = 63 m
1 = m
2 vertical angles
Find the angle measures
3
50
78
1
2
120
4
56
m
1 = 50 + 78, exterior angle theorem m
1 = 128 m
1 + m
2 = 180, linear pair are supplemental
128 + m
2 = 180 m
2 = 52 m
2 + m
3 = 120 exterior angle theorem
52+ m
3 = 120 m
3 = 68
120 + m
4 = 180, linear pair are supplemental m
4 = 60 m
4 + 56 = m
5 exterior angle theorem
60+ 56 = m
5
116= m
5
5
Find the angle measures
5
4 3
32
41
64
38
2
1
29
m
1 = 32 + 38 m
1 = 70 m
1 + m
2 = 180, linear pair are supplemental m
2 = 110 m
2 = m
3 +64 exterior angle theorem m
3= 110 – 64 = 46 m
3 + m
4 +32 = 180
46 + m
4 + 32 = 180 m
4 = 102 m
4 + m
5 +41 = 180
102 + m
5 +41 = 100
37= m
5
Right Triangle
27
m
1 =
90 – 27 m
1 = 63
1
Practice Assignment
• Standard - page 248, 12 -32 Even
• Honors - Page 189 24 – 44 Even
