Geometry Flashcards
Dimensions of a point
0
Dimensions of a line
1
Dimensions of a plane
2
Types of 1 dimensional shapes
line, ray, segment
Collinear
points on the same line (any 2 points are collinear)
Coplanar
points or lines on the same plane. Any 3 points are coplanar
2 possibilities of coplanar lines
either parallel or intersecting (coplanar rays and segments do not have to be one of the two)
Perpendicular lines, segments, or rays
intersect at 90
Oblique lines, segments, or rays
intersect at any angle except for 90
Skew lines, segments, or rays
noncoplanar
2 possibilities of 2 planes
parallel or intersecting
acute angle
less than 90
right angle
90
obtuse angle
greater than 90
Straight angle
180
Reflex angle
More than 180 (the other side of an ordinary angle)
Adjacent angles
Neighboring angles that have the same vertex and share a side, and neither angle can be inside the other
Complementary angle
two angles that add up to 90
Supplementary angle
two angles that add up to 180
Vertical angles
at an intersection of two lines, the two angles opposite of each other
Congruent segments
2 segments that are the same length
Segment with out the line above it
referring to the distance, so use = sign instead of congruent sign
Congruent angles
angles that are equal
Bisect / trisect
2 / 3 equal parts of the original (divide doesn’t have to be equal)
If a side of a triangle is trisected by rays from the opposite vertex, the vertex angle can’t be…
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)
Marking for congruent angle
an arc with two dashed lines through it
Like Multiples / Like Divisions
if two angles or segments are congruent, then multiplying or dividing by a constant gives congruent results
Vertical angles are always…
congruent
Transitive Property
a = b and b = c, then a = c
Substitution Property
a = b and b
Scalene triangle
no congruent sides (all angles are not equal) (none of the altitudes are equal)
Isosceles Triangle
at least two congruent sides (which means two congruent angles) (two of the altitudes are equal)
Equilateral /equiangular triangle
three congruent sides and three congruent angles (all triangles are either scalene or isosceles) (all of the altitudes are equal)
Angle to side ratios
Remember that in a triangle, just because an angle is twice as large as another, does not mean the side is twice as long
Name the angles of an isosceles triangle
The two congruent angles are called the base angles. The vertex angle is the other one
The triangle inequality principle
any two sides of a triangle’s sum will always be greater than the length of the third side
Acute triangle
All three angles are less than 90 (all three altitudes are inside the triangle)
Obtuse triangle
One of the angles is more than 90 (only one altitude is inside the triangle, the other two are outside)
Right triangle
One of the angles is 90 (one altitude is inside the triangle, the other two altitudes are legs of the triangle)
Altitude of a triangle
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
Area of a triangle
Or: A = 1/2 * ab sin (Ø)
Where “theta” is the angle between any sides, AB
Hero’s formula for the area of a triangle
where a,b,c are the lengths of the three sides, and S is half of the perimeter of the triangle
Area of an equilateral triangle
Median of a triangle
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
Centroid
The point at which the three medians intersect
Incenter of a triangle
The point where the three angular bisectors of a triangle meet (a circle around this point will create an inscribed circle within the triangle)
Circumcenter of a triangle
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)
Orthocenter of a triangle
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
The first four pythagorean triple triangles
3-4-5
5-12-13
7-24-25
8-15-17
(These are never 30-60-90 triangles)
Side ratios of a 45-45-90 triangle
leg : leg : hypotenuse
Side rations of a 30-60-90 triangle
short leg : long leg : hypotenuse
How to prove triangles are congruent (5 cases)
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)
How to prove isosceles triangle
If two angles are congruent, then the opposite sides are congruent
If two sides are congruent, then the opposite angles are congruent
What are the 7 quadrilaterals (4 sides)
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)
Quadrilateral Hierarchy
Properties of a parallelogram
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!!!)
Properties of a rhombus
all the properties of a parallelogram, all sides congruent, the diagonals bisect the angles, diagonals are perpendicular bisectors of each other
Properties of a rectangle
properties of a parallelogram, all angles are right, the diagonals are congruent
Properties of a square
properties of a rhombus, properties of a rectangle, all sides are congruent, all angles are right
Remember the “Z” angles
The inner and outer parts of the z are congruent to each other (the two lines must be parallel)
Properties of a Kite
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
Properties of the trapezoid
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
Area of a rectangle
A = B * H
Area of a Parallelogram
A = B * H
Area of a rhombus
A = B * H
Area of a kite
1/2 * main diagonal * cross diagonal
Area of a square
A = side^2, or A = 1/2 * diagonal^2
Area of a trapezoid
A = 1/2 * (base1 + base2) * height
Area of a regular polygon
A = 1/2 * perimeter * apothem
(regular polygons are equilateral and equiangular)
(apothem is the distance from the center to a side)
Equations for angles of polygons
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
Similarity
Same shape, different size
Both must be true: corresponding angles are congruent, corresponding sides are proportional (results in proportional perimeter)
(notated with a “ ~ “ )
Ways to prove two triangles similar (3 cases)
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)
Midline Theorem
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
Altitude on hypotenuse theorem
Draw the altitude from the right angle vertex… The 2 triangles created are similar to each other, and similar to the master triangle
Side Splitter Theorem
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
Side Splitter Theorem Extension
Angle-Bisector Theorem
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
What is a chord in a circle
Just a segment that connects two points on a circle (can be offset)
Quick circle theorems
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)
Arc definition
The curve created by two points on the circle (there is a minor and major arc created everytime (except when on diameter endpoints))
Central Angle
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)
Chords created by central angles
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
Tangent line to a circle
A line that kisses a point on the edge of a circle. Perpendicular to the radius segment at that point
Dance cup theorem
From any point in space, two tangent lines from that point will be congruent
Equation for arc length
degrees of the central angle / 360 times the circumference
What is a sector vs. a segment of a circle?
A sector is like a slice of pizza. A segment is a sector with the triangle taken out of it
Area of a sector
Sector ratio * total area of the circle
Area of a segment
Find the area of a sector, then subtract out the area of the triangle
What are the two types of angles that are on a circle edge
Inscribed and tangent-chord
Both angles are equal to half of the resulting arc angle
What is an inscribed angle
When the vertex is on the edge of the circle, and the two sides are chords of the circle
What ia a tangent chord angle
Vertex on the edge of the circle, one side tangent, one side a chord
What is the interesting property of inscribed and tangent chord angles
the resulting sector angle is twice as large as the original angle
What is the relationship between angles and sector angles, when the vertex is inside the circle?
The angle is equal to 1/2 the sum of the sector angle and the opposit sector angle
What are the properties of angles outside of a circle
The angle is equal to 1/2 of the difference of the larger sector angle and the smaller sector angle
What is the definition of a ray that is secant to a circle
It hits two points of the circle edge
What are the three different types of angles that are outside a circle
secant-secant
senat-tangent
tangent-tangent
Quick way to remeber outside, on, inside angle calculations
to get a small, subtract
to get a big, add
to get a medium, do nothing
All of them are 1/2
Chord-Chord Power Theorem
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
Tangent-Secant Power theorem
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)
Secant-Secant Power Theorem
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)
Line-Plane Perpendicularity Theorem
If a line is perpendicular to two lines, that lie in the same plane, then that original line is perpendicular to the plane
Four ways to determine a plane
Three noncollinear points
A line and a point not on the line
Two intersecting lines
Two parallel lines
A plane that intersects two parallel planes theorem
Creates two lines that are parallel
What are the two types of flat-top figures
Prism and cylinder
What is a prism
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
What is a cylinder
same as a prism, but the top and bottom are rounded (circle, ellipse)
What are the equations for volume and surface area of a flat top figure?
Vol = areabase/top * height
SA = 2 * areabase/top * lateral area of the wrapping rectangles
What are the two types of pointy top figures
Pyramid and cone
What are the equations for volume and surface area of pointy top figures
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
Equations for volume and surface area of a sphere
vol = (4/3)πr3
SA = 4πr2
The xy plane coordinate system labeled 1-4
1-4 counter clockwise
Equation for slope between two points
slope = rise / run
rise = y2 - y1
run = x2 - x1
Slope relations between parallel and perpendicular lines
parallel: slopes are equal
perpendicular: opposite reciprocal
Distance formula
Midpoint Formula
Slope intercept form and point-slope form equations for a line
y = mx + b
y - y1 = m( x - x1 )
Equation for horizontal line
Equation for vertical line
horizontal: y = b (b is the y intercept)
vertical: x = a (a is the x intercept)
General equation for a circle
( x - h )2 + ( y - k )2 = r2
(h, k) is the center
What are four types of isometries (a type of transformation)
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
What is a locus problem?
Also known as a set, it’s creating a geometric shape that follows the given rules
