Transcript 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
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.
Exterior
Angle
Theorem
Corollaries
X
mW + mX
+mY = 180
If A F and
B D, then
C E, then
The measure of an exterior
mYZP =
angle of a triangle is equal to
mX +mY
the sum of the measures of
the two remote interior angles.
The acute angles of a right
triangle are complementary
There can be at most one
right or obtuse angel in a
triangle
Y
W
F
A
B
D
C
E
Y
P
X
mG + mJ =
90
Acute
S
G
H
J
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
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.
Exterior
Angle
Theorem
Corollaries
X
mW + mX
+mY = 180
If A F and
B D, then
C E, then
The measure of an exterior
mYZP =
angle of a triangle is equal to
mX +mY
the sum of the measures of
the two remote interior angles.
The acute angles of a right
triangle are complementary
There can be at most one
right or obtuse angel in a
triangle
Y
W
F
A
B
D
C
E
Y
P
X
mG + mJ =
90
Acute
S
G
H
J
Angle Sum Theorem
X
Given ABC
Prove: mA+mB+mC = 180
Statement
ABC
Line XY through A || CB
1 and CAY form a linear pair
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
1.
2.
3.
4.
A
1 2
C
Y
3
B
Reasons
1.
2.
3.
4.
Given
Parallel Postulate
Def of linear pair
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
m1 + 28 + 82 = 180
Find the missing Angles
m1 + 110 = 180
m1 = 70
82
m1 = m2 vertical angles
28
1
m2 + m3 + 68 = 180
70+ m3 + 68 = 180
m3 + 138 = 180
m3 = 42
2
68
3
m1 + 74 + 43 = 180
Find the missing Angles
m1 + 117 = 180
m1 = 63
79
43
1
m1 = m2 vertical angles
2
74
3
m2 + m3 + 79 = 180
63 + m3 + 79 = 180
m3 + 142 = 180
m3 = 38
Find the angle measures
3
2
50
78
1
120 4
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
56
Find the angle measures
5
4 3
32
41 64
38
2
1
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
29
Right Triangle
m1 =
90 – 27
m1 = 63
27
1
Practice Assignment
• Standard - page 248, 12 -32 Even
• Honors - Page 189 24 – 44 Even