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Math > Geometry > Flashcards

Geometry Flashcards

1
Q

Dimensions of a point

A

0

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2
Q

Dimensions of a line

A

1

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3
Q

Dimensions of a plane

A

2

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4
Q

Types of 1 dimensional shapes

A

line, ray, segment

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5
Q

Collinear

A

points on the same line (any 2 points are collinear)

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6
Q

Coplanar

A

points or lines on the same plane. Any 3 points are coplanar

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7
Q

2 possibilities of coplanar lines

A

either parallel or intersecting (coplanar rays and segments do not have to be one of the two)

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8
Q

Perpendicular lines, segments, or rays

A

intersect at 90

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9
Q

Oblique lines, segments, or rays

A

intersect at any angle except for 90

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10
Q

Skew lines, segments, or rays

A

noncoplanar

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11
Q

2 possibilities of 2 planes

A

parallel or intersecting

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12
Q

acute angle

A

less than 90

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13
Q

right angle

A

90

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14
Q

obtuse angle

A

greater than 90

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15
Q

Straight angle

A

180

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16
Q

Reflex angle

A

More than 180 (the other side of an ordinary angle)

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17
Q

Adjacent angles

A

Neighboring angles that have the same vertex and share a side, and neither angle can be inside the other

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18
Q

Complementary angle

A

two angles that add up to 90

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19
Q

Supplementary angle

A

two angles that add up to 180

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20
Q

Vertical angles

A

at an intersection of two lines, the two angles opposite of each other

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21
Q

Congruent segments

A

2 segments that are the same length

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22
Q

Segment with out the line above it

A

referring to the distance, so use = sign instead of congruent sign

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23
Q

Congruent angles

A

angles that are equal

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24
Q

Bisect / trisect

A

2 / 3 equal parts of the original (divide doesn’t have to be equal)

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25
Q

If a side of a triangle is trisected by rays from the opposite vertex, the vertex angle can’t be…

A

trisected (the same goes for when rays trisect an angle of a triangle, the opposite side of the triangle is never trisected by these rays)

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26
Q

Marking for congruent angle

A

an arc with two dashed lines through it

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27
Q

Like Multiples / Like Divisions

A

if two angles or segments are congruent, then multiplying or dividing by a constant gives congruent results

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28
Q

Vertical angles are always…

A

congruent

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29
Q

Transitive Property

A

a = b and b = c, then a = c

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30
Q

Substitution Property

A

a = b and b

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31
Q

Scalene triangle

A

no congruent sides (all angles are not equal) (none of the altitudes are equal)

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32
Q

Isosceles Triangle

A

at least two congruent sides (which means two congruent angles) (two of the altitudes are equal)

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33
Q

Equilateral /equiangular triangle

A

three congruent sides and three congruent angles (all triangles are either scalene or isosceles) (all of the altitudes are equal)

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34
Q

Angle to side ratios

A

Remember that in a triangle, just because an angle is twice as large as another, does not mean the side is twice as long

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35
Q

Name the angles of an isosceles triangle

A

The two congruent angles are called the base angles. The vertex angle is the other one

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36
Q

The triangle inequality principle

A

any two sides of a triangle’s sum will always be greater than the length of the third side

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37
Q

Acute triangle

A

All three angles are less than 90 (all three altitudes are inside the triangle)

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38
Q

Obtuse triangle

A

One of the angles is more than 90 (only one altitude is inside the triangle, the other two are outside)

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39
Q

Right triangle

A

One of the angles is 90 (one altitude is inside the triangle, the other two altitudes are legs of the triangle)

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40
Q

Altitude of a triangle

A

The distance of the segment that goes from the vertex that is orthogonal to the base (every triangle has three altitudes, to make the line orthogonal, it can be extended outside the boundaries of the triangle)

Go to a vertex, and make a right angle on the opposite side

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41
Q

Area of a triangle

A

Or: A = 1/2 * ab sin (Ø)

Where “theta” is the angle between any sides, AB

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42
Q

Hero’s formula for the area of a triangle

A

where a,b,c are the lengths of the three sides, and S is half of the perimeter of the triangle

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43
Q

Area of an equilateral triangle

A
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44
Q

Median of a triangle

A

A segment that goes from one of the vertices to the midpoint of the opposite side

For each median, the distance from the vertex to the centroid is twice as long as the distance from the centroid to the midpoint

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45
Q

Centroid

A

The point at which the three medians intersect

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46
Q

Incenter of a triangle

A

The point where the three angular bisectors of a triangle meet (a circle around this point will create an inscribed circle within the triangle)

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47
Q

Circumcenter of a triangle

A

The point where three perpendicular bisectors of the sides intersect (90 deg with the side and splits the side in half) (this results in the center of a circle that is circumscribed abound the triangle) (circumcenters are inside all acute triangles, outside all obtuse triangles, and on all right triangles (at the midpoint of the hypotenuse)

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48
Q

Orthocenter of a triangle

A

The point where the triangle’s three altitudes intersect. An obtuse triangle’s orthocenter is outside of it. The altitudes are from the vertex, to the other side and creates a right angle. (The orthocenter of a right triangle is the vertex of the right angle part)

Basically, go to each vertex, and make a right triangle out of it and extend the line. Intersection is the orthocenter. THIS IS WHAT AN ALTITUDE IS

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49
Q

The first four pythagorean triple triangles

A

3-4-5

5-12-13

7-24-25

8-15-17

(These are never 30-60-90 triangles)

50
Q

Side ratios of a 45-45-90 triangle

A

leg : leg : hypotenuse

51
Q

Side rations of a 30-60-90 triangle

A

short leg : long leg : hypotenuse

52
Q

How to prove triangles are congruent (5 cases)

A

SSS (Side-Side-Side) (all sides are equal, so congruent)

SAS (Side-Angle-Side) (two sides and the included angle are equal, so congruent)

ASA (Angle-Side-Angle) (two angles and the included side are equal, so congruent)

AAS (Angle-Angle-Side) (two angles and a side not between them mean congruent)

HLR (Hypotenuse-Leg-Right angle) (in a right angle triangle, you only need congruent hypotenuse and leg)

53
Q

How to prove isosceles triangle

A

If two angles are congruent, then the opposite sides are congruent

If two sides are congruent, then the opposite angles are congruent

54
Q

What are the 7 quadrilaterals (4 sides)

A

Kite

Parallelogram (2 pairs of parallel sides)

Rhombus (4 congruent sides, both a kite and a parallelogram)

Rectangle (a parallelogram with 4 right angles)

Square (a rhombus with four right angles) (also a type of rectangle)

Trapezoid (exactly one pair of parallel sides) (parallel sides are called bases)

Isosceles Trapezoid (when the nonparallel sides are congruent)

55
Q

Quadrilateral Hierarchy

A
56
Q

Properties of a parallelogram

A

opposite sides are parallel, opposite sides are congruent, opposite angles are congruent, consectuive angles are supplementary, diagonals bisect each other (the diagonals are not congruent!!!)

57
Q

Properties of a rhombus

A

all the properties of a parallelogram, all sides congruent, the diagonals bisect the angles, diagonals are perpendicular bisectors of each other

58
Q

Properties of a rectangle

A

properties of a parallelogram, all angles are right, the diagonals are congruent

59
Q

Properties of a square

A

properties of a rhombus, properties of a rectangle, all sides are congruent, all angles are right

60
Q

Remember the “Z” angles

A

The inner and outer parts of the z are congruent to each other (the two lines must be parallel)

61
Q

Properties of a Kite

A

Diagonals are perpendicular

The main diagonal is a perpendicular bisector of the cross diagonal

The main diagonal bisects the two angles

The two angles at the ends of the cross are congruent

62
Q

Properties of the trapezoid

A

The bases are parallel

Each lower base angle is supplementary to the upper base angle on the same side

For an isosceles trapezoid:

The legs are congruent, the lower base angles are congruent, the upper base angles are congruent, the upper and lower base angles are supplementary on the same side, the diagonals are congruent

63
Q

Area of a rectangle

A

A = B * H

64
Q

Area of a Parallelogram

A

A = B * H

65
Q

Area of a rhombus

A

A = B * H

66
Q

Area of a kite

A

1/2 * main diagonal * cross diagonal

67
Q

Area of a square

A

A = side^2, or A = 1/2 * diagonal^2

68
Q

Area of a trapezoid

A

A = 1/2 * (base1 + base2) * height

69
Q

Area of a regular polygon

A

A = 1/2 * perimeter * apothem

(regular polygons are equilateral and equiangular)

(apothem is the distance from the center to a side)

70
Q

Equations for angles of polygons

A

Sum of the interior angles of n sides

sum = (n-2)*180

if you count one exterior angle at each vertex, the sum of them is 360 (measure of each is 360/n)

number of diagonals (diagonals are lines that connect non-adjacent vertices):

(n*(n-3)) / 2

71
Q

Similarity

A

Same shape, different size

Both must be true: corresponding angles are congruent, corresponding sides are proportional (results in proportional perimeter)

(notated with a “ ~ “ )

72
Q

Ways to prove two triangles similar (3 cases)

A

AA (if two angles are congruent)

SSS (if the ratios of the three pairs are equal)

SAS (two equal ratios and the angle between the two sides congruent)

73
Q

Midline Theorem

A

Draw a segment joining two midpoints of two sides results in the segment being:

parallel to the third side, and 0.5 the length of the third side

74
Q

Altitude on hypotenuse theorem

A

Draw the altitude from the right angle vertex… The 2 triangles created are similar to each other, and similar to the master triangle

75
Q

Side Splitter Theorem

A

Draw a line parallel to a side. It will divide both sides proportionally

Extension:

If three or more parallel lines are intersected by two or more tranversals, the parallel lines divide the transversals proportionaly

76
Q

Side Splitter Theorem Extension

A
77
Q

Angle-Bisector Theorem

A

Bisect a vertex. The result will be:

The ratio of the two corresponding side to that vertex will be equal to the ratio of the segments on the opposite side

78
Q

What is a chord in a circle

A

Just a segment that connects two points on a circle (can be offset)

79
Q

Quick circle theorems

A

If a radius is perpendicular to a chord, then it bisects the chord

If a radius bisects a chord (that isn’t a diameter) then it’s perpendicular to the chord

If two chords of a circle are equidistant from the center of the circle, then they’re congruent (the reverse logic applies: if congruent; they’re equidistant)

80
Q

Arc definition

A

The curve created by two points on the circle (there is a minor and major arc created everytime (except when on diameter endpoints))

81
Q

Central Angle

A

An angle whose vertex is at the center of a circle (the measure of the resulting arc in degrees is equal to the central angle measurement)

82
Q

Chords created by central angles

A

The resulting chord (created by connecting the two endpoints of the arc) is congruent to another chord if created from the same size central angle

83
Q

Tangent line to a circle

A

A line that kisses a point on the edge of a circle. Perpendicular to the radius segment at that point

84
Q

Dance cup theorem

A

From any point in space, two tangent lines from that point will be congruent

85
Q

Equation for arc length

A

degrees of the central angle / 360 times the circumference

86
Q

What is a sector vs. a segment of a circle?

A

A sector is like a slice of pizza. A segment is a sector with the triangle taken out of it

87
Q

Area of a sector

A

Sector ratio * total area of the circle

88
Q

Area of a segment

A

Find the area of a sector, then subtract out the area of the triangle

89
Q

What are the two types of angles that are on a circle edge

A

Inscribed and tangent-chord

Both angles are equal to half of the resulting arc angle

90
Q

What is an inscribed angle

A

When the vertex is on the edge of the circle, and the two sides are chords of the circle

91
Q

What ia a tangent chord angle

A

Vertex on the edge of the circle, one side tangent, one side a chord

92
Q

What is the interesting property of inscribed and tangent chord angles

A

the resulting sector angle is twice as large as the original angle

93
Q

What is the relationship between angles and sector angles, when the vertex is inside the circle?

A

The angle is equal to 1/2 the sum of the sector angle and the opposit sector angle

94
Q

What are the properties of angles outside of a circle

A

The angle is equal to 1/2 of the difference of the larger sector angle and the smaller sector angle

95
Q

What is the definition of a ray that is secant to a circle

A

It hits two points of the circle edge

96
Q

What are the three different types of angles that are outside a circle

A

secant-secant

senat-tangent

tangent-tangent

97
Q

Quick way to remeber outside, on, inside angle calculations

A

to get a small, subtract

to get a big, add

to get a medium, do nothing

All of them are 1/2

98
Q

Chord-Chord Power Theorem

A

If two chords intersect, the product of the two segments of one chord is equal to the product of the two segments of the other chord

99
Q

Tangent-Secant Power theorem

A

If a tangent and a secant are drawn from an external point to a circle, then the square of the measure of the tangent is equal to the product of the measures of the secant’s external part and the entire secant

picture example: 82 = 4(4+12)

100
Q

Secant-Secant Power Theorem

A

If two secants are drawn, then the product of the measures of one secant’s external part and that entire secant is equal to the same thing as the other secant line. (the secant lines have to originate from the same point)

101
Q

Line-Plane Perpendicularity Theorem

A

If a line is perpendicular to two lines, that lie in the same plane, then that original line is perpendicular to the plane

102
Q

Four ways to determine a plane

A

Three noncollinear points

A line and a point not on the line

Two intersecting lines

Two parallel lines

103
Q

A plane that intersects two parallel planes theorem

A

Creates two lines that are parallel

104
Q

What are the two types of flat-top figures

A

Prism and cylinder

105
Q

What is a prism

A

A solid figure created by two congruent and parallel top and bottom polygonal shapes

(a normal prism top and bottom can be parallel, but offset. When they are not offset, it is called a “right prism”)

Made of faces, edges, and vertices

106
Q

What is a cylinder

A

same as a prism, but the top and bottom are rounded (circle, ellipse)

107
Q

What are the equations for volume and surface area of a flat top figure?

A

Vol = areabase/top * height

SA = 2 * areabase/top * lateral area of the wrapping rectangles

108
Q

What are the two types of pointy top figures

A

Pyramid and cone

109
Q

What are the equations for volume and surface area of pointy top figures

A

vol = 1/3 * areabase * height

SA = areabase + lateral areatriangular side/sides

For a cone, the lateral area is equal to 1/2 * 2πr * slant height

110
Q

Equations for volume and surface area of a sphere

A

vol = (4/3)πr3

SA = 4πr2

111
Q

The xy plane coordinate system labeled 1-4

A

1-4 counter clockwise

112
Q

Equation for slope between two points

A

slope = rise / run

rise = y2 - y1

run = x2 - x1

113
Q

Slope relations between parallel and perpendicular lines

A

parallel: slopes are equal
perpendicular: opposite reciprocal

114
Q

Distance formula

A
115
Q

Midpoint Formula

A
116
Q

Slope intercept form and point-slope form equations for a line

A

y = mx + b

y - y1 = m( x - x1 )

117
Q

Equation for horizontal line

Equation for vertical line

A

horizontal: y = b (b is the y intercept)
vertical: x = a (a is the x intercept)

118
Q

General equation for a circle

A

( x - h )2 + ( y - k )2 = r2

(h, k) is the center

119
Q

What are four types of isometries (a type of transformation)

A

reflection, translation (moving), rotation, glide reflection

Even number of reflections gives you the original

Odd number gives you opposite orientation

The reflecting lines are found by getting midpoints and the orthogonal line slope

120
Q

What is a locus problem?

A

Also known as a set, it’s creating a geometric shape that follows the given rules

Math (8 decks)

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