Inequalities in o

1. SECTION 5-3
Inequalities in One Triangle
Tuesday, March 6, 2012

2. ESSENTIAL QUESTIONS
How do you recognize and apply properties of
inequalities to the measures of the angles of a
triangle?
How do you recognize and apply properties of
inequalities to the relationships between the angles
and sides of a triangle?
Tuesday, March 6, 2012

3. VOCABULARY
1. Inequality:
Tuesday, March 6, 2012

4. VOCABULARY
1. Inequality: For any real numbers a and b, a > b IFF
there is a positive number c such that a = b + c
Tuesday, March 6, 2012

5. PROPERTIES OF INEQUALITY
FOR REAL NUMBERS
1. Comparison Property of Inequality:
2. Transitive Property of Inequality:
3. Addition Property of Inequality:
4. Subtraction Property of Inequality:
Tuesday, March 6, 2012

6. PROPERTIES OF INEQUALITY
FOR REAL NUMBERS
1. Comparison Property of Inequality: a < b, a = b, a > b
2. Transitive Property of Inequality:
3. Addition Property of Inequality:
4. Subtraction Property of Inequality:
Tuesday, March 6, 2012

7. PROPERTIES OF INEQUALITY
FOR REAL NUMBERS
1. Comparison Property of Inequality: a < b, a = b, a > b
2. Transitive Property of Inequality: If a < b and b < c,
then a < c; If a > b and b > c, then a > c
3. Addition Property of Inequality:
4. Subtraction Property of Inequality:
Tuesday, March 6, 2012

8. PROPERTIES OF INEQUALITY
FOR REAL NUMBERS
1. Comparison Property of Inequality: a < b, a = b, a > b
2. Transitive Property of Inequality: If a < b and b < c,
then a < c; If a > b and b > c, then a > c
3. Addition Property of Inequality: If a < b, then a + c
< b + c; If a > b, then a + c > b + c
4. Subtraction Property of Inequality:
Tuesday, March 6, 2012

9. PROPERTIES OF INEQUALITY
FOR REAL NUMBERS
1. Comparison Property of Inequality: a < b, a = b, a > b
2. Transitive Property of Inequality: If a < b and b < c,
then a < c; If a > b and b > c, then a > c
3. Addition Property of Inequality: If a < b, then a + c
< b + c; If a > b, then a + c > b + c
4. Subtraction Property of Inequality: If a < b, then a −
c < b − c; If a > b, then a − c > b − c
Tuesday, March 6, 2012

10. THEOREMS
5.8 - Exterior Angle Inequality: The measure of an
exterior angle of a triangle is greater than the
measure of either of its corresponding remote
interior angles
Tuesday, March 6, 2012

11. ANGLE-SIDE RELATIONSHIPS
IN TRIANGLES
5.9: If one side of a triangle is longer than another
side, then the angle oppostie the longer side has a
greater measure than the angle opposite the
shorter side
5.10: If one angle of a triangle has a greater measure
than another angle, then the side oppostie the
larger angle has a greater measure than the side
opposite the smaller angle
Tuesday, March 6, 2012

12. EXAMPLE 1
Use the Exterior Angles Inequality to list all of the
angles that satisfy the stated condition.
a. Measures less than m∠14
b. Measures greater than m∠5
Tuesday, March 6, 2012

13. EXAMPLE 1
Use the Exterior Angles Inequality to list all of the
angles that satisfy the stated condition.
a. Measures less than m∠14
∠7, ∠12, ∠1, ∠8, ∠10,
∠4, ∠5
b. Measures greater than m∠5
Tuesday, March 6, 2012

14. EXAMPLE 1
Use the Exterior Angles Inequality to list all of the
angles that satisfy the stated condition.
a. Measures less than m∠14
∠7, ∠12, ∠1, ∠8, ∠10,
∠4, ∠5
b. Measures greater than m∠5
∠10, ∠16, ∠12, ∠14, ∠15
Tuesday, March 6, 2012

15. EXAMPLE 2
List the angles of ∆ABC in order from smallest to largest.
Tuesday, March 6, 2012

16. EXAMPLE 2
List the angles of ∆ABC in order from smallest to largest.
∠C, ∠A, ∠B
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17. EXAMPLE 3
List the sides of ∆ABC in order from shortest to longest.
Tuesday, March 6, 2012

18. EXAMPLE 3
List the sides of ∆ABC in order from shortest to longest.
AC, AB, BC
Tuesday, March 6, 2012

19. EXAMPLE 4
Ebony is following directions for folding a
handkerchief to make a bandana for her hair. After
she folds the handkerchief in half, the directions
tell her to tie the two smaller angles of the
triangle under her hair. If she folds the
handkerchief with the dimensions shown, which
two ends should she tie?
Tuesday, March 6, 2012

20. EXAMPLE 4
Ebony is following directions for folding a
handkerchief to make a bandana for her hair. After
she folds the handkerchief in half, the directions
tell her to tie the two smaller angles of the
triangle under her hair. If she folds the
handkerchief with the dimensions shown, which
two ends should she tie?
∠Y and ∠Z should be tied, as they are
the smallest angles (opposite
shortest sides)
Tuesday, March 6, 2012

21. CHECK YOUR
UNDERSTANDING
Review problems #1-7 on p. 346
Tuesday, March 6, 2012

22. PROBLEM SET
Tuesday, March 6, 2012

23. PROBLEM SET
p. 346 #9-37 odd, 45, 53, 57
“Health is not simply the absence of sickness.” - Hannah
Green
Tuesday, March 6, 2012