1. 7-2: Inequalities for Numbers, Segments, and Angles.
PROOF Geometry

2. Introduction 1. Is AD=CD? Why? 2. Now let’s take point D and move it:
What can you say about AD and CD now?

3. Inequalities for segments/Angles
Until now we have only talked about segments and angles in terms of congruence or equality. We need a new relation.
A < B if mA < mB

4. Properties of Inequalities
Transitive Property of Inequality:
If x < y and y < z, then x < z.
Addition Property of Inequality:
If a < b and x  y, then a + x < b + y.
Multiplication Property of Inequality:
If x < y and a > 0, then ax < ay.
Trichotomy Property:
For every x and y, one and only one of the following conditions holds: x < y; x = y; x > y.

5. Examples
𝐸𝐹 > 𝐶𝐷 , and 𝐴𝐵 < 𝐶𝐷 . What can you conclude? What property supports your statement?
If x<y and w<z, what can you conclude about x+w?
If AB<CD and EF<GH, what can you conclude about AB+GH?

6. Inequality Theorem If a = b + c and c > 0 then a > b Proof:
a = b + c Given
a – b = c Subtraction prop of eq.
c > 0 Given
a – b > 0 Substitution
a > b Addition prop of inequalities

7. The Parts Theorem (Segments)
If D is a point on 𝐴𝐵 between A and B, then 𝐴𝐵 > 𝐴𝐷 and 𝐴𝐵 > 𝐷𝐵 .
Picture:
Proof:
The whole is greater than any one of its parts

8. The Parts Theorem (Segments)
If D is a point on 𝐴𝐵 between A and B, then 𝐴𝐵 > 𝐴𝐷 and 𝐴𝐵 > 𝐷𝐵 .
Picture:
Proof:
1. AB = AD + DB Given A – D – B
2. DB > 0, AD > 0 Length of segment positive
3. AB > AD, AB > DB Inequality Theorem
Def. of segment inequality

9. The parts Theorem (angles)
If D is a point in the interior of ∠𝐴𝐵𝐶, then ∠𝐴𝐵𝐶>∠𝐴𝐵𝐷 and ∠𝐴𝐵𝐶>∠𝐷𝐵𝐶.
Picture:
Proof:
The whole is greater than any one of its parts

10. The parts Theorem (angles)
If D is a point in the interior of ∠𝐴𝐵𝐶, then ∠𝐴𝐵𝐶>∠𝐴𝐵𝐷 and ∠𝐴𝐵𝐶>∠𝐷𝐵𝐶.
Picture:
Proof:
1. m ABC =m  ABD + m  DBC AAP
2. m  ABD > 0, m  DBC > Measure of angle positive
3. m ABC > m  ABD m ABC > m  DBC Inequality Theorem
4.  ABC >  ABD  ABC >  DBC Def. of angle inequality

11. Example Given: AC > BC, CE > CD Prove: AE > BD
1. AC > BC, CE > CD Given
2. AC + CE > BC + CD Add. Prop of eq.
3. AE > BD Def. of between

12. Given: BC=DC, AC=EC
Prove: ∠𝐸<∠𝐵𝐴𝐷
1. BC=DC, AC=EC Given
2. m ACB =m  DCE VAT
3. ACB  ECD SAS
4. m BAC = m  E CPCTC
5. m  BAD > m BAC Parts Theorem
6. m  BAD > m  E Transitive prop. of ineq.
7.  E <  BAD Def. of angle inequality

13. Homework
pg : #1-5, 7, 9-11, 14