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SECTION 5-3Inequalities in One Triangle
Tuesday, March 6, 2012
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
VOCABULARY
1. Inequality:
Tuesday, March 6, 2012
VOCABULARY
1. Inequality: For any real numbers a and b, a > b IFF there is a positive number c such that a = b + c
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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
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
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
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
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
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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
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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
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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
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
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
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EXAMPLE 2
List the angles of ∆ABC in order from smallest to largest.
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EXAMPLE 2
List the angles of ∆ABC in order from smallest to largest.
∠C, ∠A, ∠B
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EXAMPLE 3
List the sides of ∆ABC in order from shortest to longest.
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EXAMPLE 3
List the sides of ∆ABC in order from shortest to longest.
AC, AB, BC
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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
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
CHECK YOUR UNDERSTANDING
Review problems #1-7 on p. 346
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PROBLEM SET
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PROBLEM SET
p. 346 #9-37 odd, 45, 53, 57
“Health is not simply the absence of sickness.” - Hannah Green
Tuesday, March 6, 2012