Download 6.1 Inequalities

Geometry
6.1 Inequalities
Property Of Inequality (P.O.I.)
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If a > b and c ≥ d, then a + c > b + d
If big > small and big/equal ≥ small/equal,
then big + big/equal > small + small/equal
If x > 7 and 2 ≥ 1, then…
x+2>7+1
If AC > BC
B
and CE ≥ CD, then
AE > BD by POI
E
C
A
D
Property Of Inequality (P.O.I.)
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If a > b and b > c, then…
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a>c
If big > medium and medium > small, then…
big > small
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If x > y and y > 7, then…
x>7
Property Of Inequality (P.O.I.)
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If a > b and c > 0, then ac > bc and a > b
c c
If big > small and a positive > 0,
then big × positive > small × positive
and big small
pos.> pos.
If x > 7 and 2 > 0, then…
2x > 14 and…
x/2 > 7/2
Property Of Inequality (P.O.I.)
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7
D
7
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If a = b + c and c is positive, then
a>b
If big = number + small positive, then…
big > number.
If 10 = 6 + 4, then…
10 > 6 and 10 > 4
m 1= m D+m
7
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E
Why?
Ext angle in a triangle = to sum of 2 remote int angles.
Therefore,
7
7
7
7
E
1
F
m 1 > m D and m 1 > m E by POI
Exterior Angle Inequality Theorem
The measure of the exterior angle of a triangle
is greater than the measure of either remote
interior angle.
C
1
B
7
7
7
m 1 > m A and m 1 > m C
by
Ext. Ineq. Thm
7
A
7
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Using the properties of inequality, classify each conditional as true or false.
If the variables are confusing for you, try to replace them with numbers.
1. If 8 = 3x + 2, then 8 > 3x.
2. If x + y = z and a > y, then x + a > z.
3. If a < b and c < b, then a < c.
4. If p > q and r > q, then p > r.
5. If 4y < 16, then y < 4.
6. If d > e and f > g, then d + e > f + g.
7. If 5a > 25, then a > 125.
8. If 3x > 48 and 4 > 2, then 3x + 4 > 50.
9. If x > 10, then x + 3 > 13.
10. If c<d and –9 = -9, then c – 9 > d – 9.
11. If (1/2)y > 7, then y > 14.
12. If d > e and f > g, then d + f > e + g.
13. If p > 10, then p > 12.
14. If p > 12, then p > 10.
Some information about the diagram is given.
Tell whether the other statements can be deduced from what is given.
W
Given: XW = YW; Z is the midpoint of XW
Z
1
X
Y
a. m< X = m< XYW
b. m<1 > m<W
c. m<W > m<X
d. XW > ZW
e. XY < XW
f. 2XZ = XW
And finally what we have all been waiting for…proofs. Yeah!
4.
Given: AB > CD
Prove: AC > BD
A
5.
B
C
D
Given: mPQR  mSRQ
m2  m3
m1  m4
Prove:
S
P
4
1
Q
2
3
R
HW
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P. 205-207 CE(1-20) WE(1-9, 11)
A Few Together from the HW
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P. 205 #5, #10
P. 206 #4
P. 207 #9