Ch. 3 Flashcards by Renee L

synthetic proof

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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justifications of synthetic proof

given statements

definitions

postulates

previously proved theorems

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Bisecting Diagonals Th

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

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implications

if-then statements that can be represented by symbols

Ex: p → q reads “If p, then q”

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logically equivalent

either both true or both false

original & contrapositive

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not logically equivalent

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

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converse

q → p

If q, then p

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inverse

~p → ~q

If not p, then not q

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contrapositive

~q → ~p

If not q, then not p

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median (of triangle)

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

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coordinate proof

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

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justifications for coordinate proof

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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distance formula

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midpt formula

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Isosceles Median Theorem

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

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isosceles trapezoid

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

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Isosceles Trapezoid Theorem

In an isosceles trapezoid:

the legs r equal in measure
the diagonals r equal in measure
the 2 angles @ each base r equal in measure

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inclusive definition

a definition that includes all possibilites

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exclusive definition

a definition that excludes some possibilities

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quadrilateral chart

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past postulates & theorems

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 180^o

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 180^o

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 60^o 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 = 360^o

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

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Interior Angle Measures in Polygons Th

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

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

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exterior angle measures in polygons th

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

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regular polygon

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

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inscribed

drawn inside the figure

circumscribed

drawn outside the figure

chords

segments whose endpoints are on the circle

Perpendicular bisector of a chord th

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

central angle

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

minor arc

\< 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)

semicircle

= 180 named w/ 3 letters outside letters = diameter; Ex: arcSTU ⇒ SU is a diameter

major arc

180 \< arc \< 360 named w/ 3 letters

inscribed angle

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

intercepted arc

the arc that lies w/in an inscribed angle

inscribed angle measure th

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

inscribed right angle th

equal inscribed angles th

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

intersecting chords theorem

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)

secants & tangents th

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)

tips for finding angles

continue radius to diameter use systems of equations remember perp rule

tangent

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

secant

a line intersecting the circle in 2 pts

Short Point Postulate

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

Perpendicular Tangent Th

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.

Converse of Perpendicular Tangent Th

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.

Equal Tangents Th

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

semiperimeter

half the perimeter

how to find area of circumscribed polygons using trig

1. total degrees = 360^o 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. tan36^o = x / 30; x = ~ 21.8 5. 21.8 • 2 = 43.6; 1/2bh = 1/2(30)(43.6) = 654cm² 6. 654 • 5 = 3,270cm³

area of a circumscribed polygon

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

polyhedron

a space figure whose faces are all polygons

semiregular polyhedron

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

regular polyhedron

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

tetrahedron

4 equilateral triangles

hexahedron

6 squares

octahedron

8 equilateral triangles

dodecahedron

12 regular pentagons

icosahedron

20 equilateral triangles

Convex Polyhedron Postulate

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

net

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

defect

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

Euler's formula

F + V = E + 2 f - # of faces v - # of vertices e - # of edges

Ch. 3 Flashcards

(61 cards)