Chapter 7: Geometric Inequalities

Question 0

Transitive Postulate

Answer 0

If a, b, and c are real numbers such that a>b and b>c, then a>c.

Ex) If 12 > 7 and 7 > 3, then 12 > 3.

Question 1

A whole is equal to the sum of all its parts

Answer 1

Ex) 14 = 9 + 5

	14 > 9 and 14 > 5

Question 2

Substitution Postulate

Answer 2

A quantity may be substituted for its equal in any statement of inequality.

Ex) If 10 > 2 + 5 and 2 + 5 = 7, then 10 > 7.

Question 3

Trichotomy Postulate

Answer 3

Given any two points, a and b, one and only one of the following is true:

	a < b	or	a = b	or	a > b

Question 4

Addition Postulate of Inequality

Answer 4

If equal quantities are added to unequal quantities, then the sums are unequal in the same order.

If unequal quantities are added to unequal quantities in the same order, then the sums are unequal in the same order.

Ex) Since 12 > 5, then 12 + 3 > 5 + 3 or 15 > 8.
Since 12 > 5 and 3 > 2, then 12 + 3 > 5 + 2 or
15 > 7.

Question 5

Subtraction Postulate of Inequality

Answer 5

If equal quantities are subtracted from unequal quantities, then the differences are unequal in the same order.

Ex) 12 > 5, then 12 – 3 > 5 – 3 or 9 > 2.

However:
When unequal quantities are subtracted from unequal quantities, the results may or may not be unequal and the order of the inequality may or may not be the same.

Ex) 5 > 2 and 4 > 1, but it is not true that 5 – 4 > 2 – 1 since 1 = 1.

12 > 10 and 7 > 1, but it is not true that 12 – 7 > 10 – 1.

12 > 10 and 2 > 1, and it is true that 12 – 2 > 10 – 1.

Question 6

Multiplication Postulate of Inequality

Answer 6

If unequal quantities are multiplied by positive equal quantities, then the products are unequal in the same order.

If unequal quantities are multiplied by negative equal quantities, then the products are unequal in the opposite order.

Ex) If 9 > 3, then 9(4) > 3(4).
If 9 > 3, then 9(-4) < 3(-4).

Question 7

Division Postulate of Inequality

Answer 7

If unequal quantities are divided by positive equal quantities, then the quotients are unequal in the same order.

If unequal quantities are divided by negative equal quantities, then the quotients are unequal in the opposite order.

Question 8

What is the shortest distance between two points?

Answer 8

The shortest distance between two points is the length of the line segment joining these two points.

Question 9

Triangle Inequality Theorem

Answer 9

The length of one side of a triangle is less than the sum of the sum of the lengths of the other two sides.

Question 10

Exterior Angles of a Polygon

Answer 10

An angle that forms a linear pair with one of the interior angles of the polygon

Question 11

Exterior Angles of a Triangle

Answer 11

The measure of an exterior angle of a triangle is greater than the measure of either nonadjacent interior angle

Question 12

Theorem

Answer 12

If the lengths of two sides of a triangle are equal, then the measures of the angles opposite those sides are equal

Question 13

Theorem

Answer 13

If the lengths of two sides of a triangle are unequal, then the measures of the angles opposite these sides are unequal and the larger angle lies opposite the longer side

Question 14

Theorem

Answer 14

If the measures of two angles of a triangle are unequal, then the lengths of the sides opposite these angles are unequal and the longer side lies opposite the larger angle

Question 15

Exterior Angle Inequality Theorem

Answer 15

The measure of an exterior angle of a triangle is greater than the measure of either non adjacent angle