Inequalities In Geometry

Chapter 6Ms. Cuervo

Inequalities and Indirect Proof

Chapter 6 Lesson 1

Example 1 Given: AC>AB; AB>BC

Conclusion: AC BC

A

B

C

Example 2Given: m<BAC+ m<CAD= m<BAD

Conclusion: m<BAD m<BAC;

m<BAD m<CAD

DA

CB

Properties of Inequality

If a>b and c≥ d, then a+c>b+d

If a>b and c>0, then ac>bc and a/c>b/c

If a>b and c<0, then ac<bc and a/c<b/c

If a>b and b>c, then a>c

If a=b+c and c>0, then a>b.

Example 3 Given: AC>BC; CE>CD

Prove: AE>BD

A

E

D

C

B

Statements

Reasons

1. AC>BC; CE>CD

2. AC+CE>BC+CD

3. AC+CE=AE BC+CD=BD

4. AE>BD

1. Given

2. A Prop. Of Inequality

3. Segment Add Post.

4. Substitution Prop

Example 4 Given: <1 is an exterior angle of DEF.

Prove: m<1>m<D;

m<1>m<EF

E

D

1

Statements Reasons

1. m<1=m<D+m<E

2. m<1>m<D; m<1>m<E

1. The measure of an ext. < of a triangle equals the sum of the measures of the two remote int <‘s.

2. A Prop. Of Ineq.

Theorem 6-1

The Exterior Angle Inequality Theorem-

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

Inverses and Contrapositives

Chapter 6 Lesson 2

Vocabulary Statement: If p, then q

Inverse: If not q, then not p.

Contrapositive: If not q, then not p.

Example 1Write (a) the inverse and (b) the contrapositive of the true conditional: If two lines are not coplanar, then they do not intersect.

Solution:

Inverse: If two lines are coplanar, then they intersect (False)

Contrapositive: If two lines intersect, then they are coplanar. (True)

Venn Diagram

If P If Q

If P and Q

Logically Equivalent When both the conditional and contrapositive

are true.

When both the conditional and contrapositive are false.

THEN YOU CAN SAY THAT THE STATEMENTS ARE LOGICALLY EQUIVALENT

Summary of Related If-Then Statements

Given statement: If p, then q.

Contrapositive: If not q, then not p.

Converse: If q, then p.

Inverse: If not p, then not q.

A STATEMENT AND ITS CONTRAPOSITIVE ARE LOGICALLY EQUIVALENT.

A STATEMENT IS NOT LOGICALLY EQUIVALENT TO ITS CONVERSE OR TO ITS INVERSE.

Example 1 Suppose this conditional is true:

ALL RUNNERS ARE ATHLETES

What can you conclude from each additional statement?

All Runners are Athletes: Leroy is a

RunnerGiven: If p, then q; All runners are athletes

p Leroy is a runner

Conclusion: q Leroy is an athlete

LeroyAthletes

Runners

All Runners Are Athletes: Lucia Is Not

An Athlete Given: If p, then q; All runners are

athletes.

not q Lucia is not an athlete

Conclusion: not p Lucia is not a runner

Runners

AthletesLucia

All Runners are Athletes: Linda is an

Athlete Given: If p, then q; All runners are

athletes

q Linda is an athlete

No Conclusion Follows Linda might be a runner or she might not be

Athletes Runners

Linda

All Runners are Athletes: Larry is Not

a Runner Given: If p, then q; All Runners are

Athletes

not p Larry is not a runner.

No Conclusion Follows Larry might be an athlete or he might not be.

Athletes

Runners

Larry

Indirect ProofsChapter 6 Lesson 3

What Is An Indirect Proof?

A proof where you assume temporarily that the desired conclusion is not true. Then you reason logically until you reach a contradiction of the hypothesis or some other known fact.

How Do You Write an Indirect Proof?

1. Assume temporarily that the conclusion is not true.

2. Reason logically until you reach a contradiction of a known fact.

3. Point out that the temporary assumption must be false, and that the conclusion must then be true.

Example 1Given: n is an integer and n^2 is even

Prove: n is even

Proof

Assume temporarily that n is not even. Then n is odd and

n^2 =n x n

= odd x odd x= odd

But this contradicts the given information that n^2 is even. Therefore the temporary assumption that n is not even must be false. It follows that n is even.

Example 2 Prove that the bases of a trapezoid have unequal lengths.

Given: Trap. PQRS with bases PQ and SR

Prove: PQ ≠ SR

Proof:

Assume temporarily the PQ=SR. We know that PQ ||SR by the definition of a trapezoid. Since quadrilateral PQRS has two sides that are both congruent and parallel, it must be a parallelogram, and PS must be parallel to QR. But this contradicts the fact that, by definition, trapezoid PQRS can have only one pair of parallel sides. The temporary assumption that PQ=SR must be false. It follows that PQ ≠ SR.

InequalitiesChapter 6 Lesson 4

Theorem 6-2 If one side of a triangle is longer than a

second side, then the angle opposite the first side is larger than the angle opposite the second side.

Theorem 6-3 If one angle of a triangle is larger than a

second angle, then the side opposite the first angle is longer than the side opposite the second angle.

Corollary 1 The perpendicular segment from a point to a

line is the shortest segment from the point to the line.

Corollary 2 The perpendicular segment from a point to a

plane is the shortest segment from the point to the plane.

The Triangle Inequality

(Theorem 6-4) The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Inequalities For Two Triangles

Chapter 6 Lesson 5

SAS Inequality Theorem

If two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle.

SSS Inequality Theorem

If two sides of one triangle are congruent to two sides of another triangle, but the third side of the first triangle is longer than the third side of the second, hen the included angle of the first triangle is larger than the included angle of the second.