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SMCHS
Geometry B
Mr. Ricks
Chapter 6
Section 6-1:
So far we have worked with equal angles and segment lengths. In this chapter, you will
begin working with and comparing unequal segment lengths and angles.
Example:
Given: AC > AB and AB > BC
Fill in the following blank with either <, >, or =
Conclusion: AC _____ BC
Properties of Inequality
If a > b and c > d, then a + c > b + d.
If a > b and c > 0, then ac > bc and a/c > b/c
If a > b and c < 0, then ac < bc and a/c < b/c
If a > b and b > c, then a > c
If a = b + c and c > 0, then a > b
THM 6-1:
The Exterior Angle Inequality Theorem
The measure of an exterior angle of a triangle is greater than the measure of either remote
interior angle.
B
mDCB = 125.26
mCBA = 66.84
mBAC = 58.41
D
A
C
HOMEWORK: p. 205-206: 1-15 odd, 17, 18, 19 (CE) and 1-6 all (WE)
Section 6-2: Inverses and Contrapositives
We already discussed the if-then statement and its’ converse. In this section we will
discuss the inverse and the contrapositive statements of an if-then statement.
Given if-then statement:
Contrapositive:
Converse:
Inverse:
If p, then q
If not q, then not p
If q, then p
If not p, then not q
Note: In the above definitions, p is the hypothesis and q is the conclusion.
A Venn diagram can be used to represent a conditional statement.
Example:
A conditional and its contrapositive are logically equivalent.
Example 1:
Conditional:
Contrapositive:
Converse:
Inverse :
If x  4 , then 6x  4  20 .
If 6x  4  20 , then x  4 .
If 6x  4  20 , then x  4 .
If x  4 , then 6x  4  20 .
Example 2:
Conditional:
Contrapositive:
Converse:
Inverse:
If a parallelogram is a square, then it is a rectangle.
If a parallelogram is not a rectangle, then it is not a square.
If a parallelogram is a rectangle, then it is a square.
If a parallelogram is not a square, then it is not a rectangle.
HOMEWORK:
p. 210-211: 1-8 all (CE) and 2-10 even (WE)
Section 6-3: Indirect Proof
How to write an indirect proof
1. Assume temporarily that the conclusion is not true.
2. Reason logically until you reach a contradiction of a known fact.
3. Point out that the temporary assumption must be false, and that the conclusion
must then be true.
Section 6-4: Inequalities for One Triangle
THM 6-2:
If one side of a triangle is longer than a second side, then the angle opposite the first side
is larger than the angle opposite the second side.
G
m EF = 7.39 cm
mGFE = 32.30
m FG = 6.63 cm mFEG = 63.30
m GE = 3.97 cm mEGF = 84.40
F
E
THM 6-3:
If one angle of a triangle is larger than a second angle, then the side opposite the first
angle is longer than the side opposite the second angle.
H
m HJ = 5.07 cm
m JI = 4.26 cm
J
m IH = 3.28 cm
mIJH = 40.02
mJHI = 56.57
mHIJ = 83.41
I
COROLLARY 1:
The perpendicular segment from a point to a line is the shortest segment from the point to
the line.
m RL = 2.66 cm
R
m RM = 1.99 cm
m NR = 1.94 cm
K
L
M
N
RN  KQ
O
P
Q
m RO = 2.72 cm
m RP = 4.89 cm
COROLLARY 2:
The perpendicular segment from a point to a plane is the shortest segment from the point
to the plane.
THM 6-4:
The Triangle Inequality
The sum of the lengths of any two sides of a triangle is greater than the length of the third
side.
U
S
m US = 5.07 cm
m US+m ST = 9.33 cm
m ST = 4.26 cm
m ST+m TU = 7.54 cm
m TU = 3.28 cm
m US+m TU = 8.35 cm
T
HOMEWORK: p. 221-223: 2-12 even (CE), 1-12 all (WE)
Section 6-5: Inequalities for Two Triangles
THM 6-5:
SAS Inequality Theorem
If two sides of one triangle are congruent to two sides of another triangle, but the
included angle of the first triangle is larger than the included angle of the second, then the
third side of the first triangle is longer than the third side of the second triangle.
THM 6-6:
SSS Inequality Theorem
If two sides of one triangle are congruent to two sides of another triangle, but the third
side of the first triangle is longer than the third side of the second triangle, then the
included angle of the first triangle is larger than the included angle of the second triangle.
HOMEWORK: p. 230-232: 1-8 all (CE) and 1-8 all (WE)
CHAPTER 6 REVIEW: p. 235-236: 1-18 all (Chapter Review)
CHAPTER 6 EXTRA CREDIT: p. 236-237 1-14 all (Chapter Test)