Chapter 6

Francisco TomasinoAndy Lachler

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:

Statements• AC>BC; CE>CD• AC+CE > BC+CD• AC+CE=AE; BC+CD=BD• AE>BD

Reasons• Given• Prop. Of Ineq.• SAP• Substitution

Given: AC>BC ; CE >CDProve: AE > BD

B E

A D

C

Exterior Angle Inequality Theorem (EAIT)

• The measure of an exterior angle of a triangle is grater than the measure of either remote interior angle

A

B C

A < C and B < C

Example:

Statements

• 2 > 1• 1 > 3• 2 > 3• 3 = 4• 2 > 4

Reasons

• Given• EAIT• Substitution• VAT• Substitution

3

1

Given: 2 > 1Prove: 2 > 4

2

4

Indirect Proof

How to Write an Indirect Proof• Assume temporarily that the conclusion is not

true• Reason logically until you reach a

contradiction of a known fact• Point out that the temporary assumption must

be false, and that the conclusion must then be true

Example:

Statements• 1. X = 100°• 2. Assume Y is a right angle• 3. X + Y + Z = 180°• 4. 100+90+Z= 180• 5. Y is not a right angle

Reasons• 1. Given• 2. Assumption• 3. Sum of angles of a

triangle is 180° • 4. Substitution • 5. Contradiction

X

Y Z

Given: X = 100°Prove: Y is not a right angle

Practice Problems V

X

Y Z

Given: VY is perpendicular to YZProve: angle VXZ is an obtuse angle

1

32

t

b

aGiven: transversal “t” cuts lines “a” and “b”; angle 1 ≠ angle 2

Prove: “a” is not parallel to “b”

Inequalities for One Triangle

R

ST

Angle RST > Angle TBecause side RT is longer than RS or ST 5 3

4

•In any triangle, the largest interior angle is opposite the largest side. •The smallest interior angle is opposite the smallest side •The middle-sized interior angle is opposite the middle-sized side

Theorems and Corollaries

S

R

T

Z

1

3

2

Given: Triangle RST; RT>RSProve: m of angle RST > m of angle T

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

•By the ruler postulate there is a point Z on segment RT such that RZ = RS. Draw segment SZ•In isosceles triangle RZS, the measure of angle 3 = the measure of angle 2•Because the measure of angle RST = the measure of angle 1 + the measure of angle 2, you have measure of angle RST > measure of angle 2•Substitution of measure of angle 3 for the measure of angle 2 yields the measure of RST > measure of angle 3•Because angle 3 is and exterior angle of triangle ZSR, you have measure of angle 3 > measure of angle T•From the m of RST > m of 3 and m of 3 > m of T, you get m RST > m T

Theorems and CorollariesTheorem 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

Given: Triangle RST; m of S > m of TProve: RT > RS

R

T S

•Assume temporarily that RT is NOT > RS. The either RT = Rs or RT < RS• Case 1: If RT = RS, then m of S = m of T• Case 2: If RT< RS, then m of S < m of T

•In either case there is a contradiction of the given fact that m of S > m of T•The assumption that RT is NOT > RS must be false. It follows that RT > RS

Theorems and CorollariesCorollary 1:

The perpendicular segment from a point to a line is the shortest segment from the point to the line

http://www.mathopenref.com/coordpointdist.html

Corollary 2:The perpendicular segment from a point to a plane is the shortest

segment from the point to the plane

Theorems and CorollariesTheorem 6-4:

The sum of the length of any two sides of a triangle is greater than the length of the third side

C

A BZ

Given: Triangle ABCProve: (1) AB + BC > AC (2) AB + AC > BC (3) AC + BC > AB

• One of the sides. Say segment AB, is the longest side. Then (1) and (2) are true. To prove (3), draw a line, line CZ, through C and perpendicular to line AB.

• By Corollary 1 of Theorem 6-3, segment AZ is the shortest segment from A to segment CZ. Also, Segment BZ is the shortest segment from B to segment CZ.

Theorems and CorollariesInequalities for Two Triangles

SAS Inequality Theorem:If two sides of a triangle are congruent to two sides of another

triangle and the included angle in one triangle has a greater measure than the included angle in the other, then the third side of the first triangle is longer than the third side of the second triangle.

http://www.geogebra.org/en/upload/files/english/nebsary/SasInequality/SAS.html

Theorems and Corollaries

SSS Inequality Theorem:If two sides of a triangle are congruent to two sides of

another triangle and the third side in one triangle is longer than the third side in the other, then the angle between the pair of congruent sides in the first triangle is greater than the corresponding angle in the second triangle

http://www.geogebra.org/en/upload/files/english/nebsary/SssInequality/SssFinal.html

Practice Problems

A

B C

D

FE

Given: Segment BA is congruent to segment ED; Segment BC is congruent to segment EF; m of B > m of EProve: AC > DF

Given: Segment BA is congruent to segment ED; Segment BC is congruent to segment EF; AC > DFProve: m of B > m of E