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Theorems and Postulates for Geometry Final > Theorems and Postulates > Flashcards

Theorems and Postulates Flashcards

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1
Q

5 ways to prove a quadrilateral is a parallelogram

A
  1. Both pairs of opposite sides are parallel
  2. Both pairs of opposite sides are congruent
  3. One pair of opposite sides is BOTH parallel and congruent
  4. Diagonals bisect each other
  5. Both pairs of opposite angles are congruent
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2
Q

2 ways to prove a quadrilateral is a rectangle

A
  1. First prove it is a parallelogram, then either
    a. Prove it has ONE right angle or
    b. Prove the diagonals are congruent
  2. Prove it has FOUR right angles
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3
Q

2 ways to prove a quadrilateral is a rhombus

A
  1. First prove its is a parallelogram, then either
    a. Prove two consecutive sides are congruent or
    b. Prove either diagonal bisects two angles
  2. Prove that both diagonals are perpendicular bisectors
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4
Q

1 way to prove a quadrilateral is a square

A
  1. Prove that it is both a rectangle and a rhombus, which means:
    a. ­it is a parallelogram
    ­b. it has 1 right angle
    ­c. it has 2 consecutive sides congruent
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5
Q

3 ways to prove a quadrilateral is a kite

A
  1. Prove that two disjoint pairs of consecutive sides are congruent
  2. Prove that ONE of the diagonals is the perpendicular bisector of the other
  3. Prove that one of the diagonals bisects a pair of opposite angles
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6
Q

1 way to prove a trapezoid is a isosceles trapezoid

A
  1. Prove one pair of sides parallel. Then either:
    a. Prove the nonparallel sides are congruent or
    b. Prove either the lower or upper base angles are congruent or
    c. Prove the diagonals are congruent
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7
Q

4 ways to determine a plane

A
  1. Three noncollinear points determine a plane.
  2. A line and a point not on the line determine a plane.
  3. Two intersecting lines determine a plane.
  4. Two parallel lines determine a plane.
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8
Q

If a line intersects a plane not containing it then the intersection is…

A

exactly one point (called the foot)

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9
Q

If two planes intersect their intersection is…

A

exactly one line

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10
Q

A line is perpendicular to a plane if…

A

it is perpendicular to every one of the lines in the plane that pass through its foot

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11
Q

If a line is perpendicular to 2 distinct lines in a plane which pass through its foot, then it is…

A

perpendicular to the plane

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12
Q

A line and a plane are parallel when…

A

they do not intersect

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13
Q

Two planes are parallel if…

A

they do not intersect

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14
Q

If a plane intersects two parallel planes, the lines of intersection are…

A

parallel

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15
Q

If two planes are perpendicular to the same line, they are…

A

parallel to each other

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16
Q

If a line is perpendicular to one of two parallel planes, it is…

A

perpendicular to the other plane as well

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17
Q

If two planes are parallel to the same plane, they are…

A

parallel to each other

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18
Q

If two lines are perpendicular to the same plane, they are…

A

parallel to each other

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19
Q

If a plane is perpendicular to one of two parallel lines, it is…

A

perpendicular to the other line as well

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20
Q

The sum of the internal angles in a triangle add up to…

A

180°

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21
Q

The measure of an exterior angle of a triangle is equal to the sum of the measures of…

A

the remote interior angles

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22
Q

Midline Theorem

A

A segment joining the midpoints of 2 sides of a triangle is parallel to the 3rd side, and its length is 1/2 the length of the 3rd side

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23
Q

No Choice Theorem

A

If two angles of one triangle are congruent to two angles of a second triangle, then the third angles are congruent

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24
Q

AAS

A

Angle-Angle-Side: used for proving the congruence of triangles using the corresponding congruence of 2 angles and 1 non-included side

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25
Formula: sum of the interior angles of a polygon
sum=(n-2)180
26
The sum of exterior angles of a polygon will always be...
360°
27
Formula: measure of each exterior angle in a regular polygon
angle=360°/n
28
Formula: number of diagonals that can be drawn in a polygon
no. of diagonals=n(n-3)/2
29
Formula: measure of each interior angle in a regular polygon
angle=(n-2)180/n
30
Rules of Ratios:
* ALWAYS REDUCE to lowest terms * Convert/cancel UNITS so ratio is dimensionless * Fractions NEVER contain decimals OR other fractions OR mixed numbers
31
Means-Extremes Product Theorem
In a proportion, the product of the means = product of the extremes i.e. cross multiplication
32
Means-Extremes Ratio Theorem
Means may be interchanged and/or extremes may be interchanged (without invalidating the equality)
33
Property of Proportions
An equation is still valid if the same multiple of each denominator is added to its numerator on BOTH sides
34
AAA
Angle-Angle-Angle: If the three angles of one triangle are congruent to the three corresponding angles of another triangle, then the two triangles are similar
35
AA
Angle-Angle: If two angles of one triangle are congruent to the two corresponding angles of another triangle, then the two triangles are similar
36
SSS (for similar triangles)
Side-Side-Side: If the ratios of the measures of three pairs of corresponding sides in two triangles are the same, then the triangles are similar
37
SAS (for similar triangles)
Side-Angle-Side: If the ratios of the measures of two pairs of corresponding sides are equal AND the included angles of each triangle are congruent, then the triangles are similar
38
CASTC
Corresponding Angles of Similar Triangles are Congruent: If two triangles are similar, then corresponding angles are congruent used after SAS and SSS
39
CSSTP
Corresponding Sides of Similar Triangles are Proportional: If two triangles are similar, then corresponding sides are proportional used after AA and SAS
40
Side-Splitter Theorem
If a line is parallel to one side of a triangle then it splits the other two sides proportionally
41
If two or more parallel lines are cut by two transversals, then the parallel lines divide the transversals...
proportionally
42
If a ray bisects an angle of a triangle, it divides the opposite side into segments that are...
proportional to the adjacent sides
43
If an altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are...
similar to the given right triangle and to each other
44
Geometric Mean (Altitude) Theorem
If the altitude is drawn to the hypotenuse of a right triangle, then its length is the geometric mean of the lengths of segments of the hypotenuse
45
Geometric Mean (Leg) Theorem
If the altitude divides the hypotenuse of a right triangle into two segments, then either leg is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to that leg
46
Pythagorean Theorem
The square of the measure of the hypotenuse of a right triangle is equal to the sum of the squares of the measures of its legs (a^2)+(b^2)=(c^2)
47
Converse of Pythagorean Theorem
If (a^2)+(b^2)=(c^2), then the triangle is right and the angle opposite the longest side is a right angle
48
Corollary of Pythagorean Theorem
If (a^2)+(b^2)>(c^2), then the triangle is acute If (a^2)+(b^2)
49
Distance Formula
distance=√((x2-x1)^2+(y2-y1)^2)
50
Distance Formula in 3-D
distance=√(length^2+width^2+height^2)
51
Formula: area of a circle
A=πr^2
52
Formula: circumference of a circle
C=2πr=πd
53
The total arc measure of a circle is...
360°
54
Formula: the length of an arc
The fraction of the circle's circumference occupied by the arc; expressed in linear units such as feet, centimeters, or inches Length=m
55
Formula: area of a sector
A=m
56
If two circles are congruent, then they have...
congruent radii
57
If the distance to a point is less than the radius, then the point is on the...
interior of the circle
58
If the distance to a point is greater than the radius, then the point is on the...
exterior of the circle
59
If the distance to a point is equal to the radius, then the point is...
on the circle
60
If a radius is perpendicular to a chord, then it (blank) the chord.
bisects
61
If a radius of a circle bisects a chord that is not a diameter, then it is (blank) to that chord
perpendicular
62
The perpendicular bisector of a chord passes through the...
center of the circle
63
If two chords of a circle are equidistant from the center, then they are...
congruent
64
If two chords of a circle are congruent, then they are (blank) from the center of the circle
equidistant
65
If the central angles of a circle (or congruent circles) are congruent, the corresponding (blank and blank) are also congruent.
arcs and chords
66
If two chords of a circle (or congruent circles) are congruent, the corresponding (blank and blank) are also congruent.
central angles and arcs
67
If two arcs of a circle (or congruent circles) are congruent, the corresponding (blank and blank) are also congruent.
central angles and chords
68
A tangent line is perpendicular to the radius drawn to the...
point of tangency
69
If a line is perpendicular to a radius at its point of contact with the circle, then it is (blank) to the circle
tangent
70
Two Tangent Theorem
If two tangent segments are drawn to a circle from an external point, then those segments are congruent
71
the 4 places a vertex of an angle can be on a triangle
* The center of the circle * On the circle * Inside the circle but not at the center * Outside the circle
72
Formula: angle with vertex on the circle
Equal to half the measure of its intercepted arc Ө=1/2(intercepted arc)
73
Formula: Tangent-Chord Angles
Formed be a tangent and a chord; also equal to half the measure of the intercepted arc Ө=1/2(intercepted arc)
74
Formula: Chord-Chord Angles
Formed by two chords that intersect inside (but not at the center of) a circle; equal to half the sum of the arcs intercepted by the chord­-chord angle and its vertical angle Ө=a+b/2
75
Formula: Secant/Tangent-Secant/Tangent Angles
Formed by 2 secants, a secant and a tangent, or 2 tangents. Equal to half the difference of the intercepted arc. Ө=b-a/2
76
If two inscribed or tangent­-chord angles intercept the same or congruent arc(s), then they are...
congruent
77
An angle inscribed in a semicircle is...
right
78
The sum of the measures of a tangent-­tangent angle and its minor arc is...
180°
79
If a quadrilateral is inscribe in a circle, its opposite angles are...
supplementary
80
If a parallelogram is inscribed in a circle, it must be a...
rectangle
81
Chord-Chord Power Theorem
If two chords intersect inside a circle, then the product of the measure of the segments of one chord is equal to the product of the measures of the other chord
82
Tangent-Secant Power Theorem
If a tangent segment and a secant segment are drawn from an external point to a circle, then the square of the measure of the tangent segment is equal to the product of the measures of the entire secant segment and its external part
83
Secant-Secant Power Theorem
If two secant segments are drawn from an external point to a circle, then the product of the measures of one secant segment and its external point is equal to the product of the other secant segment and its external part
84
Formula: Area of Rectangle
A=bh
85
Formula: Area of Parallelogram
A=bh
86
Formula: Area of Square
A=s^2
87
Formula: Area of Triangle
A=1/2bh
88
Formula: Area of Equilateral Triangle(with side s)
A=(√3a^2)/4 | not important
89
Formula: Area of Trapezoid
A=(b1+b2/2)h
90
Formula: Area of a Kite
A=pq/2 | p and q are diagonals (also not an important formula)
91
Formula: Area of any Regular Polygon
A=1/2(apothem)(perimeter)
92
Formula: Volume of Rect. Prism
V=lwh
93
Formula: Volume of any Prism
V=(Area of base)h
94
Formula: Volume of any Pyramid/Cone
V=1/3(Area of base)h
95
Formula: Volume of a Sphere
V=4/3πr^3

Theorems and Postulates for Geometry Final (1 decks)

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