Area and Right Triangles and Fundamentals of Proof

Question 1

definition and area postulates

Postulate

Answer 1

Assumption

• If two closed regions intersect only along a common boundary, then the area of their union is equal to the sum of their individual area
• Every closed reagion has an area
• If two closed figures are congruent, then their areas are equal
• The area of a rectangle is equal to the product of the base and the height of that base

Question 2

area

Square

Answer 2

The area of a square is equal to the length of its side squared.

A_sq = s²

s= side

Question 3

area

Rectangle

Answer 3

Derived from the area of a square:

Arect = ab = bh

b= base

h= height

Question 4

area

Parallelogram

Answer 4

Quadrilateral with opposite sides parallel

A_{parallelogram} = bh

Question 5

area

Triangle

Answer 5

A_tri = 1/2bh

Question 6

area

Trapezoid

Answer 6

Exactly one pair of parallel sides

A_trap = 1/2[(b₁ + b₂)h]

Question 7

area

Rhombus

Answer 7

Quadrilateral with all sides congruent

diagnoals = d₁ and d₂

A_rhom = bh = 1/2(d₁d₂₎

Question 8

area

Kite

Answer 8

diagnoals = d₁ and d₂

A_rhom = bh = 1/2(d₁d₂₎

Question 9

Pythagorean Theorem

Answer 9

The square of the hypotenuse of a right triangle equals the sum of the squares of the two legs

Question 10

45-45-90 Triangle

Answer 10

Ratio of leg to hypotenuse

1:sqrt(2)

Question 11

30-60-90 Triangles

Answer 11

Ratio of short leg to long leg to the hypotenuse

1: sqrt(3): 2

Question 12

Equilateral Triangle Area

Answer 12

A = [s²sqrt(3)]/4

Question 13

Ratio of Areas of Triangles (TART)

Answer 13

Theorem: The ratio of areas of two triangles equals the product of the ratios of the bases and the ratio of the heights

A₁/A₂ = (b₁/b₂)(h₁/h₂)

corolaaries: if two triangles have equal bases, the ratio of their areas is equal to the ratio of the heights

if two triangles have equal heights, the ratio of their areas is equal to the ratio of the bases

Question 14

Segment Addition Postulate

Answer 14

PQ + QR = PR whenever Q is between P and R

Question 15

Midpoint Definition

Answer 15

Point M, in between points A and B, such that AM = MB

Question 16

Bisector

Answer 16

A line, segment, ray or plane that passes through the midpoint

Question 17

Angle Bisector

Answer 17

Line/Segment/Ray BX, in the interior of angle ABC such that angle ABX = angle CBX

Question 18

Angle Addition Postulate

Answer 18

angle ABX + angle CBX = angle ABC where X is in the interior of the angle

Question 19

Linear Pairs

Answer 19

= 180 degrees

Question 20

Midpoint Theorem

Answer 20

If M is the midpoint of AB, then AM = 1/2 AB and MB = 1/2 AB

Question 21

Definition vs. Theorem

Answer 21

the theorem is something deductively proven based on the definition

Question 22

Vertical Angles

Answer 22

are congruent

Question 23

Addition/Subtraction Property of Equality

Answer 23

a = b, c = d

a + c = b + d

Question 24

Multiplication/Division Property of Equality

Answer 24

a = b, c = d

ac = bd

Question 25

Reflexive Property of Equality

Answer 25

a = a

Question 26

Symmetric Property of Equality

Answer 26

a = b

b = a

Question 27

Transitive Property of Equality

Answer 27

a = b, b = c

a = c

Question 28

Substitution Property of Equality

Answer 28

If a = b, we can replace any occurrance of a with b

Question 29

Conditional Statements

Answer 29

If [hypothesis], then [conclusion]

Question 30

Converse of a Statement

Answer 30

Swtich the hypothesis and conclusion

If [conclusion], then [hypothesis]

Question 31

Biconditional Statements

Answer 31

Combines a statement and its converse

Question 32

Inverse of a Statement

Answer 32

Negate the statement

If not [hypothesis], then not [conclusion]

Question 33

Contrapositive

Answer 33

Switch and negate the hypothesis and conclusion

If not [conclusion], then not [hypothesis]

Question 34

Venn Diagrams

if p, then q

if not q, then not p

if q, then p

if not p, then not q

Answer 34

if p, then q = if not q, then not p

if q, then p = if not p, then not q

Question 35

Logically Equivalent

Answer 35

Statements that have the same truth value