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Math II > Ch. 3 > Flashcards

Ch. 3 Flashcards

1
Q

synthetic proof

A

proof built using a system of postulates & theorems in which the prop’s of figures, but not their actual measurements r studied

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2
Q

justifications of synthetic proof

A

given statements

definitions

postulates

previously proved theorems

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3
Q

Bisecting Diagonals Th

A

If the 2 diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

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4
Q

implications

A

if-then statements that can be represented by symbols

Ex: p → q reads “If p, then q”

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5
Q

logically equivalent

A

either both true or both false

original & contrapositive

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6
Q

not logically equivalent

A

just cuz original = true, doesn’t mean converse & inverse r too

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7
Q

converse

A

q → p

If q, then p

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8
Q

inverse

A

~p → ~q

If not p, then not q

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9
Q

contrapositive

A

~q → ~p

If not q, then not p

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10
Q

median (of triangle)

A

a segment joining a vertex to the midpt of the opp. side

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11
Q

coordinate proof

A

a proof based on a coord. system in which all pts r represented by ordered pairs of #’s

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12
Q

justifications for coordinate proof

A

distance & midpt formulas

parallel lines have the same slope

perp. lines have slopes tht r neg. reciprocals of each other

a geometric figure may be placed anywhere in the coord. plane

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13
Q

distance formula

A
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14
Q

midpt formula

A
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15
Q

Isosceles Median Theorem

A

in an isosceles triangle, the medians drawn to the legs r equal in measure

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16
Q

isosceles trapezoid

A

a trapezoid w/ a line of symmetry that passes through the midpts of the bases

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17
Q

Isosceles Trapezoid Theorem

A

In an isosceles trapezoid:

  1. the legs r equal in measure
  2. the diagonals r equal in measure
  3. the 2 angles @ each base r equal in measure
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18
Q

inclusive definition

A

a definition that includes all possibilites

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19
Q

exclusive definition

A

a definition that excludes some possibilities

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20
Q

quadrilateral chart

A
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21
Q

past postulates & theorems

A

Addition, Subtraction, Mult, Div Prop’s of Eq Post’s

Reflexive Prop of Eq Post

Substitution Property Post

Distributive Prop. Post

If 2 angles r supp’s of same angle, then r equal in measure

if 2 angles r complements of same angle, then equal in measure

Straight Angle Post - if the sides of an angle form a straight line, then the angle is a straight angle w/ measure 180o

Angle/Segment Addition Post - For any seg or angle, the measure of the whole is equal to the sum of the measures of its non-overlapping parts

vertical angles r equal in measure

the sum of the measures of the angles of a triangle is 180o

an exterior angle of a triangle is equal in measure to the sum of the measures of its 2 remote interior angles

if 2 sides of a triangle r equal in measure, then the angles opp those sides r = in measure

if 2 angles of a triangle r =, then the sides opp those angles r =

If a tri is equilateral, then is also equiangular, w/ three 60o angles

If a tri is equiangular, then its also equilateral

if 2 parallel lines r intersected by a trans, then corr angles r =, vv

If 2 parallel lines r intersected by a trans, then alt int angles r =, vv

If 2 parallel lines r intersected by a trans, then co-int angles r supp, vv

If 2 lines r perp to the same trans, then they r parallel

If a trans = perp to one of 2 parallel lines, then its perp to the other one also

Thru a pt not on a given line, there’s 1 and only 1 parallel line to the given line

If a pt is the same distance from both endpts of a segment, then it lies on the perp bisector of the seg

A seg can be drawn perp to a given line from a pt not on the line

AA similarity - if 2 angles of 1 tri r = to 2 angles of another tri, then the 2 tri’s r similar

If a line is drawn from a pt on 1 side of a tri parallel to another side, then it forms a tri similar to the original tri

In a tri, a segment that connects the midpts of 2 sides is parallel to the 3rd side & half as long

ASA, AAS th’s

SAS, SSS Post’s

If the alt = drawn to the hyp of a right tri, then the 2 triangles formed r similar to the original tri & to each other

Pythagorean Th

If the alt is drawn to the hyp of a right tri, then the measure of the alt is the geometric mean b/w the measures of the parts of the hyp

the sum of the lengths of any 2 sides of a tri is greater than the length of the 3rd side

in an isosc tri, the medians drawn to the legs r equal in measure

in a parallelogram, the diagonals have the same midpt

In a rectangle, the diagonals r equal in measure

In a kite, the diagonals r perp to each other

in a parallelogram, opp sides r equal in measure

If a quadrilateral is a parallelogram, then consecutive angles r supp

If a quad is parallelogram, then opp angles r =

the sum of the measures of the angles of a quad = 360o

if both pairs of opp angles of a quad r equal in measure, then the quad = a parallelogram

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22
Q

Interior Angle Measures in Polygons Th

A

the sum of the angle measures of an n-gon is given by the formula

S(n) = (n - 2)180o

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23
Q

exterior angle measures in polygons th

A

the sum of the exterior angle measures of an n-gon, 1 angle at each vertex, is 360o

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24
Q

regular polygon

A

iff all its sides r equal in measure & all its angles r = in measure

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25
Q

inscribed

A

drawn inside the figure

26
Q

circumscribed

A

drawn outside the figure

27
Q

chords

A

segments whose endpoints are on the circle

28
Q

Perpendicular bisector of a chord th

A

The perp bisector of a chord of a circle passes thru the center of the circle

29
Q

central angle

A

an angle w/ its vertex @ the center of the circle

measure of an arc intercepted (cut off) by a central angle = the measure of that central angle

30
Q

minor arc

A

< 180

can be named with 2/3 letters (just remember that a major arc is named w/ 3 letters to distinguish it from a minor arc w/ the same endpts)

31
Q

semicircle

A

= 180

named w/ 3 letters

outside letters = diameter; Ex: arcSTU ⇒ SU is a diameter

32
Q

major arc

A

180 < arc < 360

named w/ 3 letters

33
Q

inscribed angle

A

an angle formed by 2 chords that intersect at a point ON a circle

34
Q

intercepted arc

A

the arc that lies w/in an inscribed angle

35
Q

inscribed angle measure th

A

the measure of an inscribed angle of a circle = 1/2 the measure of its intercepted arc

36
Q

inscribed right angle th

A
37
Q

equal inscribed angles th

A

If 2 inscribed angles in the same circle intercept the same arc, then they r = in measure

38
Q

intersecting chords theorem

A

the measure of an angle formed by 2 chords that intersect INSIDE a circle = 1/2 the sum of the measures of the intercepted arcs

.5 (a + b)

39
Q

secants & tangents th

A

the measure of an angle formed by 2 secants, 2 tangents, or a secant & a tangent drawn from a pt outside the circle = 1/2 the diff of the measures of the intercepted arcs

.5 (big - small)

40
Q

tips for finding angles

A

continue radius to diameter

use systems of equations

remember perp rule

41
Q

tangent

A

a line in the plane of a circle & intersecting the circle in exactly 1 pt

42
Q

secant

A

a line intersecting the circle in 2 pts

43
Q

Short Point Postulate

A

a segment can be drawn perpendicular to a given line from a point not on the line

the length of this segment is the shortest distance from the point to the line

44
Q

Perpendicular Tangent Th

A

If a line is tangent to a circle, then the line is perpendicular to the radius drawn from the center to the point of tangency.

45
Q

Converse of Perpendicular Tangent Th

A

If a line in the plane of a circle is perp to a radius at its outer endpt, then the line is tangent to the circle.

46
Q

Equal Tangents Th

A

If 2 tangent segments are drawn from the same pt to the same circle, then they r equal in measure.

47
Q

semiperimeter

A

half the perimeter

48
Q

how to find area of circumscribed polygons using trig

A
  1. total degrees = 360o
  2. divide 360 by # of vertices to find all internal angles
  3. divide internal angles by 2 to find angle in new right triangle
  4. use trigonometry to find sides (already have radius & angle & know its right triangle)
  5. find area of original triangle
  6. multiply by # of vertices

Ex: pentagon w/ radius 30cm

  1. total degrees = 360
  2. 360 / 5 = 72
  3. 72 / 2 = 36
  4. tan36o = x / 30; x = ~ 21.8
  5. 21.8 • 2 = 43.6; 1/2bh = 1/2(30)(43.6) = 654cm2
  6. 654 • 5 = 3,270cm3
49
Q

area of a circumscribed polygon

A

The area of any circumscribed polygon is the product of the radius (r) of the inscribed circle & the semiperimeter (s)

A = rs

50
Q

polyhedron

A

a space figure whose faces are all polygons

51
Q

semiregular polyhedron

A

a polyhedron w/ faces that r all regular polygons, & w/ the same # of faces of each type @ each vertex

52
Q

regular polyhedron

A

a polyhedron w/ faces tht r all the same type of regular polygon, & w/ the same # of faces @ each vertex

5 regulars; tetrahedron, hexahedron, octahedron, dodecahedron, icosahedron

53
Q

tetrahedron

A

4 equilateral triangles

54
Q

hexahedron

A

6 squares

55
Q

octahedron

A

8 equilateral triangles

56
Q

dodecahedron

A

12 regular pentagons

57
Q

icosahedron

A

20 equilateral triangles

58
Q

Convex Polyhedron Postulate

A

In any convex polyhedron, the sum of the measures of the angles at each vertex is less than 360o

59
Q

net

A

a 2D drawing showing the connected faces of a space figure & how they r connected.

can be cut out & “folded up” to form the space figure

60
Q

defect

A

the angle measure of the gap @ a vertex on a net for a polyhedron

can be found by subtracting the sum of the angles @ that vertex from 360

61
Q

Euler’s formula

A

F + V = E + 2

f - # of faces

v - # of vertices

e - # of edges

Math II (17 decks)

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