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Section 1 – 200pts

Define convex.

Section 1 – 200pts

A set in which every segment that connects two points in the set lies entirely in the set.

Or

A set with no “dents”.

Section 1 – 400pts

Define non-convex.

Give another term for non-convex.

Section 1 – 400pts

A set where there exists a segment that connects two points in the set that does not lies entirely in the set.

Or

A set that contains at least one “dent”.

Alternate term: Concave.

Section 1 – 600pts

Categorize the following shape as convex or non-convex. If non-convex, prove it.

Section 1 – 600pts

Convex

Section 1 – 800pts

Categorize the following shape as convex or non-convex. If non-convex, prove it.

Section 1 – 800pts

Non-convex

Section 1 – 1000pts

Categorize the following shape as convex or non-convex. If non-convex, prove it.

A

B

C

Section 1 – 1000pts

Non-convex. An angle is the union of two rays and does NOT include the space between them.

A

B

C

Section 2 – 200pts

Another term for an “if-then” statement is . . .

Section 2 – 200pts

A conditional

a. The statement that follows the “then” in a conditional is called . . .

b. The statement that follows the “if” in a conditional is called . . .

Section 2 – 400pts

Section 2 – 400pts

a. The consequent.

b. The antecedent.

Section 2 – 600pts

Identify the antecedent and the consequent in the following statement.

If a figure is a square, then it is a polygon.

Section 2 – 600pts

If a figure is a square, then it is a polygon.

antecedent consequent

Notice that the “if’ and the “then” are NOT included in the antecedent or consequent.

Section 2 – 800pts

Consider the statement:

Every bird can fly.

a. Rewrite it as a conditional.

b. Is it true? If so give an example, if not give a counter example.

Section 2 – 800pts

Consider the statement:

Every bird can fly.

a. If an animal is a bird, then it can fly.

b. False. A counterexample would be an ostrich, dodo, penguin, kiwi, . . .

Section 2 – 1000ptsp = a figure is a squareq = If, then it is a polygon.

a. Write p q.

b. Is it true? If so give an example, if not give a counter example.

c. Write q p. What is this called?

d. Is it true? If so give an example, if not give a counter example.

Section 2 – 1000ptsp = A figure is a squareq = It is a polygon.

a. “If a figure is a square, then it is a polygon.”

b. True. An example would be a floor tile in the lab.

c. “If a figure is a polygon, then it is a square.” This is called the converse.

d. False. A counterexample would be a pentagon.

Section 3 – 200pts

What do the following symbols mean?

a.

b.

Section 3 – 200pts

What do the following symbols mean?

a. If p , then q.p implies q.the conditional symbol

b. p if and only if q.the bi-conditional sign

Section 3 – 400ptsConsider the statement:

A polygon has four sides, if it is a quadrilateral.

Write the bi-conditional.

Section 3 – 400ptsConsider the statement:A polygon has four sides, if it is a quadrilateral.

The figure is a quadrilateral if and only if it is a polygon with four sides.

Section 3 – 600ptsp = You can go to the movies.q = Your homework is done.

a. Write p q.

b. Write q p. What is this called?

c. Write p q. What is this called?

Section 3 – 600ptsp = You can go to the movies.q = Your homework is done.

a. If you can go to the movies, then your homework is done.

b. If your homework is done, then you can go to the movies. This is the converse.

c. You can go to the movies if and only if your homework is done. This is the bi-conditional.

Section 3 – 800ptsp = x 9.q = x > 8.

a. Write p q.

b. Is it true? If so give an example, if not give a counter example.

c. Write q p. What is this called?

d. Is it true? If so give an example, if not give a counter example.

Section 3 – 800ptsp = x 9.q = x > 8.

a. If x 9, then x > 8.

b. Yes, it is true. Some examples would be 9.2, 11, 15½

c. If x > 8, then x 9. This is the converse.

d. This is false. Some counterexamples are 8.02, 8.5, 8¾. In fact all examples could be written as 8 < x < 9.

Section 3 – 1000ptsp = x is positive.q = x > 0.

a. Write p q.b. Is it true? If so give an example, if not give a counter

example.c. Write q p. What is this called?d. Is it true? If so give an example, if not give a counter

example.e. Write p q.

Section 3 – 1000ptsp = x is positive.q = x > 0.

a. If x is positive, then x > 0.

b. Yes it is true. Some examples are 1, 3, pi, 167.2, . . . .

c. If x > 0, then x is positive.

d. Yes it is true. Some examples are 1, 3, pi, 167.2, . . . .

e. X is positive if and only if x > 0.

Section 4 – 200ptsA good definition should be able to be written as a ______________ that is true.

Section 4 – 200ptsBi-conditional

Section 4 – 400ptsThe symbol is read.

Section 4 – 400pts“if and only if”

Section 4 – 600ptsQ is the midpoint of and QR = 8.

a. Draw a sketch.b. RT =c. QT =

RT

Section 4 – 600ptsQ is the midpoint of and QR = 8.

a.b. RT = 16c. QT = 8

RT

R TQ

Section 4 – 800ptsA is the midpoint of .LA = 7x - 4 and AB = 4x + 5.

a. Find x.b. Find LA & AB.c. Find LB.

LB

LB

Section 4 – 800ptsA is the midpoint of .LA = 7x - 4 and AB = 4x + 5.a. 7x - 4 = 4x + 5

+4 + 47x = 4x + 9-4x -4x3x = 93 3x = 3

b. LA = 7x - 4 = 7(3) - 4 LA = 17AB = 4x + 5 = 4(3) + 5AB = 17

c. LB = 2LA = 2(17)LB = 34

Section 4 – 1000ptsI is the midpoint of .ZI = 2x + 5 and ZP = 9x - 12.

a. Find x.b. Find ZI.c. Find ZP.

ZP

Section 4 – 1000pts

a. 2x + 5 + 2x + 5 = 9x - 124x + 10 = 9x - 12

+ 12 +124x + 22 = 9x-4x -4x

22 = 5x4.4 = x

I is the midpoint of .ZI = 2x + 5 and ZP = 9x - 12.

b. ZI = 2x + 5ZI = 2(4.4) + 5ZI = 13.8

c. ZP = 9x - 12ZP = 9(4.4) - 12ZP = 27.6

ZP

Section 5 – 200ptsa.What is the definition of the intersection?

b.What is the symbol used?

Section 5 – 200ptsa. The set of elements that are in both sets.

b.

Section 5 – 400ptsa.What is the definition of the union?

b.What is the symbol used?

Section 5 – 400ptsa. The set of elements that are in either set

or both sets.

b.

Section 5 – 600ptsLet A = the set of numbers y with y -1.Let B = the set of numbers with y < 7.

In both cases, state answer as an inequality and on a number line.a. Find A B.b. Find A B.

Section 5 – 600ptsLet A = the set of numbers y with y -1.Let B = the set of numbers with y < 7.

a. A B : -1 y < 7

b. A B : all reals

0

0

Section 5 – 800ptsLet STIX be the quadrilateral and SIX be the triangle, as shown.

a. Find STIX SIX.b. Find STIX SIX.

S

T

I

X

Section 5 – 800ptsLet STIX be the quadrilateral and SIX be the triangle, as shown.

a.b.

S

T

I

X

.SI and STIX quad :as written be could alsoSI,XS,IX,TI,ST

XI,SX

Let A = the set of numbers {1, 3, 4, 6, 7, 12}Let B = the set of numbers {1, 2, 5, 6, 8, 11, 13}Let C = the set of numbers {-2, 0, 1, 7, 12, 13, 17}

a. Find A B.b. Find B C.c. Find A C.d. Draw a venn diagram showing all the

intersections and unions.

Section 5 – 1000pts

Section 5 – 1000ptsa. A B = {1, 6}b. A B = {1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 13}c. A C = {1, 7, 12}d. venn diagram.

1

AB

C

2

3

4

56

7

8

11

12

13

17-20

Section 6 – 200ptsDefine Polygon.

Section 6 – 200ptsA polygon is the union of segments in the same plane such that each segment intersects exactly two others only at the endpoints.

Section 6 – 400ptsDetermine whether each is a polygon or not.

Section 6 – 400pts

yesno

yes yes

yes yes

no no

no

no

no

no

Section 6 – 600ptsSketch a convex octagon.

Section 6 – 600pts

Section 6 – 800ptsIdentify each figure.

Section 6 – 800pts

triangle

Quadrilateralregion

Pentagon

Hexagon12-gon

Heptagonalregion

Section 6 – 1000ptsSketch the triangle hierarchy. Incorporate the following terms: scalene, figure, equilateral, polygon, triangle, isosceles.

Draw the triangles with the appropriate markings to distinguish one type from another.

Section 6 – 1000pts

Polygon

Figure

Triangle

ScaleneIsosceles

Equilateral

Section 7 – 200ptsState the triangle inequality postulate.

Section 7 – 200ptsThe sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Section 7 – 400ptsTell whether the numbers given can be lengths of the three sides of a triangle.

a. 5, 5, 5b. 13, 16, 10c. 17, 4, 12d. 3, 4, 7

Section 7 – 400ptsTell whether the numbers given can be lengths of the three sides of a triangle.

a. yesb. yesc. No since 4 + 12 = 16 which is less than 17d. No since 3 + 4 = 7 which is equal to 7

Section 7 – 600ptsSuppose two sides of a triangle have lengths 16inches and 11inches. What are the possible lengths of the remaining side?

1116

Section 7 – 600pts5 < x < 27

511

16

11 16

27

Section 7 – 800ptsA hotel in Houston is 40 miles from the airport and 25 miles from the convention center. With that information, determine the possible range of distances between the airport and convention center.

Section 7 – 800pts15 < distance < 65

Section 7 – 1000ptsWhen you move point B off of line AC, a triangle is formed. Using the endpoints, state the triangle inequality postulate.

AB

C

Section 7 – 1000ptsAB + BC > ACAC + BC > ABAC + AB > BC

A

B

C

Section 8 – 200ptsWhat is a conjecture?

Section 8 – 200ptsA conjecture is an educated guess, a hypothesis, or an opinion.

Section 8 – 400ptsWhat is needed to show that a conjecture is false?

Section 8 – 400ptsJust one counterexample.

Section 8 – 600ptsIn mathematics, for a conjecture to be true, it must be ________.

Section 8 – 600ptsPROVED!!!

Section 8 – 800ptsDraw several instances of the following conjecture and conclude whether or not it is true. Use a straight edge.

“If the midpoints of two sides of a triangle are joined the segment is parallel to the third side.”

Section 8 – 800ptsThis conjecture is true.

Example.

Section 8 – 1000ptsDraw several instances of the following conjecture and conclude whether or not it is true. Use a straight edge.

“If the midpoints of the four sides of a rectangle are connected, the resulting figure is a rectangle.”

Section 8 – 1000ptsThis conjecture is not true.

Example.