Download Properties of Inequalities

Geo 9 Ch 6
Inequalities
Properties of Inequalities
1
If a>b and c>d,
then
a+c > b+d
If a>b and c>0
then
ac>bc or
If a>b and c<0
then
If a>b and b>c
then
______________________
a b

c c
a b

ac<bc or
c c
a>c
If a = b + c and c>0, then a>b
Exterior Angle Inequality:
______________________
______________________
______________________
EAT >
<3 is equal to <1 + <2
therefore <3 is greater than <1 or <2
2
1
3
1. Using the figure to the right, fill in the blanks.
a)
b)
c)
d)
e)
f)
If x = 40 and y = 30, then w > _______
If y = 54 and z = 68, then w _________
If w = 112, then x __________
If w = 150, then z __________
If x = 25 and z = 90, then w ___________
If z = 90, then x _______ and y __________
R
y
S
w z
Q
B
Prove: <B > <C
D
1
2
A
C
1.
B 1
1. Given
x
P
Geo 9 Ch 6
Inequalities
2
1)
C
D
2
1
3
O
A
B
3 1
Prove:
1. CO  AB,
D in interior of <COB
Prove:
2)
ROT  SOV
S
R
1.
T
2
1
3
O
V
1. Given
ROS  TOV
Geo 9 Ch 6
Inequalities
3
X
3.
Prove: <1 > <X
R
1. Figure as shown
U
1. Given
1
T
S
Prove:
4.
1 U
1. Given diagram
X
1
S
R
T
U
5.
Prove:
R
S
1. PS bisects <P
V
1
C
2
M
P
1
2
1. Given
Geo 9 Ch 6
Inequalities
4
Geo 9 Ch 6
Inequalities
5
Geo 9 Ch 6
Inequalities
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Geo 9 Ch 6
Inequalities
Ch. 6.4 Inequalities for one triangle
7
B
A
C
Th 6-2: If one side of a triangle is longer than a second side, then ____________________________
________________________________________________________________________________
AB = 6, BC = 8, AC = 10
Which is largest angle?
Th 6-3: If one angle of a triangle is larger than a second triangle, then ________________________
________________________________________________________________________________
<A = 50  , <B = 100  , <C = 30  Which is the largest side?
Corr 1: The perpendicular segment from a point to a line is ______________________________
______________________________________________________________________________
Corr 2: The perpendicular segment form a point to a plane is ____________________________
_____________________________________________________________________________
Th 6-4: The sum of the lengths any two sides of a triangle is greater than __________________
_____________________________________________________________________________
Possible triangles
a)
b)
c)
d)
6,8,20
3,4,8
2.5, 4.1, 5.0
6, 5, x what does x have to be?
1. In triangle ABC, AB = 12, BC = 7, AC = 9. Name the largest angle, the smallest angle.
2. In triangle PQR, <P = 72, <Q = 37 and <R = 71. Name the largest side, the shortest side.
S
3. For triangle MPS, name the angles in order of
increasing length.
20
M
M
15
M
P
4
Geo 9 Ch 6
Inequalities
8
EXAMPLES
4. Given triangle GKH as marked. Arrange the angles <k, <1, <G, and <H
in order of increasing size.
K
1
15
13
12
G
9
M
H
5
5. Which segment is the longest?
R
60
P
46
65
74
25
M
K
6. Which segment is the shortest?
S
70
P
40
51
52
70
77
Q
R
Geo 9 Ch 6
Inequalities
9
Prove: < C is the smallest angle of the triangle
7.
B
1. AC > BC
BC > AB
A
C
8.
Prove: AB > CD
1. <C > < A,
B
D
<D > <B
E
C
A
Geo 9 Ch 6
Inequalities
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Geo 9 Ch 6
Inequalities
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Geo 9 Ch 6
Inequalities
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Geo 9 Ch 6
Inequalities
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Geo 9 Ch 6
Inequalities
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Ch 6 Geometry Review Worksheet
A
(1)
(2)
5
C
4
4
3
E
5
6
D
1
3
6
2
2
1
C
B
8
7
A
D
B
Given: AB = AC , m5 > m4
Given: AD = CE , AE > CD
Prove: m2 > m1
Prove: AB > BC
D
(3)
A
(4)
6
A
5
4
3
2
1
B
B
C
D
C
Given: AC > AB
Prove: CD > BD
Prove: AB + AD + BD > 2 AC
(5)
(6)
A
B
A
B
6
5
1
D
C
Given: ABCD is a parallelogram
mBCD > mADC
4
2
C
Given: AC = BC
Prove: AD > BD
3
D
Geo 9 Ch 6
Inequalities
15
Prove: BD > AC
A
(7)
(8)
B
3
C
C
5
A
1
D
6
B
2
D
E
Given: mBCD = mBDC
mACD > mADC
Given: AB = AE = CD = DE
Prove: mABD > mABC
Prove: m5 > m3
A
(9)
1
4
(10)
A
2
4
6
5
4
1
C
C
B
D
3
3
2
B
Given: BD = CD , AD  BC
Given: AB  BC
Prove: AB  AC
Prove: AC > BC
(11) Find the longest and shortest segments
(12) If two sides of a triangle are 17 and 26,
in the figure below.
then the third side must be between
and
B
61 60
A
C
60
D
59
.
Geo 9 Ch 6
Inequalities
Additional Review A
13)
16
Given: AB = AC
BD = BC
Prove: BC > CD
6
D
5
4
1
B
3
2
C
A
14)
8
7
B
1
2
Given: AB = AD
Prove: AE > AC
9
5
4
3
D
C
6
E
A
15)
Given: AB > AC
BD = EC
Prove: BE > CD
E
D
1
B
4
2
3
C
Geo 9 Ch 6
Inequalities
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SUPPLEMENTARY PROBLEMS
1. Graph triangle ABC with A = (-8,7) B = (-4, 1) and C = ( 5,7).
List the sides of the triangle in increasing order.
What do you think will happen to the order if I add 2 to AB, 3 to BC and 4 to AC?
2. Which is the biggest angle, <1, <2 or <3.
3
1
2
3. What are the key steps to writing an indirect proof?
4. Cut out two strips of paper the length of one side of a piece of paper.
Take one of them and break it into 2 pieces, not the same length.
Try to make a triangle out of the 3 pieces.
5. Go back to triangle ABC from the first problem. Measure the angles.
List the angles in increasing order.
6. Graph triangle ABC with A=(-10,8) B = (-10,2) C = (0,2).
Find it’s sides.
Graph triangle SNO with S = (3,0) N = (5,6) and O = (11,6)
Find its sides.
Which angle is bigger, < ABC or < NOS. Why do you think what you do?
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1
3