FINAL EXAM REVIEW

Chapter 6-7Key Concepts

VocabularyVocabulary

Chapter 6Chapter 6

inequalityinequality

inverseinverse

contrapositivecontrapositive

logically logically equivalentequivalent

indirect proofindirect proof

Chapter 7Chapter 7

ratioratiomeans/extremesmeans/extremesproportionproportionscale factorscale factorPOP’sPOP’ssimilar figuressimilar figuresproportional segmentsproportional segments

Exterior Angle Inequality TheoremExterior Angle Inequality Theorem

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

A

B

C

1

m 1 > m A and m 1 > m C by Ext. Ineq. Thm

7 7 7 7

7

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.Inverse negates given statement.

Contrapositive negates converse.

An Indirect Proof An Indirect Proof TemplateTemplate

1) Assume temporarily1) Assume temporarily that the that the conclusion is not true.conclusion is not true.

2) Then2) Then…Reason logically until you reach …Reason logically until you reach a contradiction of a known fact or a a contradiction of a known fact or a given.given.

3) But this contradicts that3) But this contradicts that…state the …state the contradiction.contradiction.

4) Thus our assumption is false, 4) Thus our assumption is false, therefore . .therefore . . , the conclusion is true. , the conclusion is true.

.

Theorem

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.

A

C B

If BC > AB, then

big

smallm<A > m<C.

                                                                                                           

   

                                                        

    

                                                                                                                               

Crikey,that seems easy! Longest side opposite biggest angle, shortest side opposite smallest angle, medium side opposite medium angle.

Theorem

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

A

C B11

914

9 + 11 = 20 20 > 14

11 + 14 = 25 25 > 9

14 + 9 = 23 23 > 11

Corollaries

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

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

●

●

SAS Inequality TheoremSAS Inequality Theorem If two sides of one triangle are congruent to two sides of anotherIf two sides of one triangle are congruent to two sides of another

triangle, but the included angle of the first triangle is larger than thetriangle, 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 included angle of the second, then the third side of the first triangle

is longer than the third side of the second triangle.is longer than the third side of the second triangle.

small

small

BIG

BIG

SSS Inequality TheoremSSS Inequality Theorem

If two sides of one triangle are congruent to two sides of another triangle, If two sides of one triangle are congruent to two sides of another triangle,

but the third side of the first triangle is larger than the third side of the second, but the third side of the first triangle is larger than the third side of the second,

then the included angle of the first triangle is larger than the included angle of then the included angle of the first triangle is larger than the included angle of

the second triangle.the second triangle.

BIG

BIG small

small

P.O.P.’sP.O.P.’s

3

15

2

10

15

3

10

2

(10)3 = 2(15)Given: Is equivalent to:

3

2

15

10 &

Is equivalent to:

10

15

2

3 &

1

5

3

15

2

10

6

30

132

51510

3

315

2

210

Also

Also

Similar Polygons

Their vertices can be paired so that:•Corresponding angles are congruent•Corresponding sides are in proportion

(their lengths have the same ratio)

4 8

6

12

84

3

6

Similarity Chart

•Definition:•All angles congruent•All sides proportional

•SAS Similarity Theorem•SSS Similarity Theorem

All Polygons

•AA Postulate (2 <‘s = 2 <‘s)

Triangles

AA Similarity Postulate

If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

~

SAS Similarity Theorem

If an angle of one triangle is congruent to an angle of another triangle and the sides including those angles are in proportion, then the triangles are similar.

A

B C

D

E F

36

4 8

∆ABC ~ ∆DEF

m<A = m<D

36

= 48

smallbig

SSS Similarity Theorem

If the sides of two triangles are in proportion, then the triangles are similar.

A

B C

D

E F

46

6 9

∆ABC ~ ∆DEF

8 12

46

= 69

= 812

smallbig

Triangle Proportionality Thm. If a line parallel to one side of a triangle intersects

the other two sides, then it divides those sides

proportionally. big A

small A

whole Abig B

small B

whole B=

side C1

Side C2

=

=

= = = =

whole A

small A

whole B

small B

side C1

side C2

big A

small A

big B

small B

big A

whole A

big B

whole B

whole Bwhole A

big A big B

small A

small B

big A

big B

whole A

whole BThink of it as two separate similar triangles.

Corollary

If three // lines intersect two transversals,…then they divide the transversals proportionally.

a

b

c

d =ab

cd

Triangle Angle-Bisector Thm. If a ray bisects an angle of a triangle,… then it divides the opposite side into

segments proportional to the other two sides.a

bc

d

=ab

cd

HomeworkHomework

►Study for final examStudy for final exam►Suggestion: Work problems on pg. Suggestion: Work problems on pg.

281-283281-283 (answer key will be handed out)(answer key will be handed out)

Review Constructions #1-7, 10-13 in Ch. 10Review Constructions #1-7, 10-13 in Ch. 10

Ch. 7 Multiple Choice Refresher on page 632Ch. 7 Multiple Choice Refresher on page 632