Important Geometry Stuff Flashcards
Parallel Lines
Same Slope
Perpendicular Lines
Opposite Reciprocal Slope
Distance Formula
d = square root of (x2-x1) squared + (y2-y1) squared
Midpoint Formula
m = (x1 + x2 / 2 , y1 + y2 / 2)
Slope Formula
m = y2 - y1 / x2 - x1
OR
run
Quadratic Formula
x= -b plus or minus the square root of b^2 minus 4ac all over 2a
5 methods to prove congruent triangles
- SSS - Side Side Side
- SAS - Side Angle Side
- AAS - Angle Angle Side
- ASA - Angle Side Angle
- HL - Hypotenuse Leg
CPCTC
Corresponding Parts of Congruent Triangles are Congruent
Prove Triangles Congruent First
What are Complementary Angles
angles that add up to 90
What are Supplementary Angles?
Angles that add up to 180
Parallel Lines that form Angles Theorems (5 of them)
- Corresponding Angles Postulate
- Alternate Interior Angle Theorem
- Same Side Interior Angle Theorem
- Alternate Exterior Angle Theorem
- Same Side Exterior Angle Theorem
Proving Lines Parallel (5 of them)
- Converse of Corresponding Angles Postulate
- Converse of Alternate Interior Angle Theorem
- Converse of Same Side Interior Angle Theorem
- Converse of Alternate Exterior Angle Theorem
- Converse of Same Side Exterior Angle Theorem
All triangles add up to…
180
All quadrilaterals (and circles) add up to…
360
Polygon Angle-Sum Theorem
in a n-gon…
180 (n-2)
the exterior angles of a polygon add up to…
360 degrees
In isosceles triangles…
two sides are congruent and two base angles are congruent
the bisector of the vertex angle of an isosceles triangle is…
the perpendicular bisector of the base
For overlapping or two sets of triangles, what should you try and remember?
What triangles can you prove first?
What part of those triangles is also a part of the second set?
What is important about midsegments?
It connects the midpoints of two sides of a triangle
The midsegment of a triangle is parallel to the third side and half its length
what side is opposite the largest angle?
what side is opposite the middle angle?
what side is opposite the smallest angle?
the largest side
the middle side
the short side
Properties of a Parallelogram (4 of them)
- Opposite sides of a parallelogram are congruent
- Opposite angles of a parallelogram are congruent
- The diagonals of a parallelogram bisect each other
- Consecutive Angles of a parallelogram are supplementary
If three or more parallel lines cut of congruent segments of one transversal…
then they cut of congruent segments on every transversal
5 ways to prove a quadrilateral is a parallelogram
- BOTH pairs of opposite sides are parallel
- BOTH pairs of opposite sides are congruent
- BOTH pairs of opposite angles are congruent
- The diagonals bisect each other
- ONE PAIR of opposite sides is parallel and congruent
important rhombus stuff (3 things)
- parallelogram with 2 congruent consecutive sides
- the diagonal bisects 2 angles of a rhombus
- the diagonals are perpendicular
important rectangle stuff (2 things)
- parallelogram with 1 right angle
- diagonals are congruent
important square stuff (3 things)
- both a rectangle and rhombus
- has all properties of a parallelogram, rhombus and rectangle
- diagonals of a square are congruent, bisect each other, each bisect two angles, are perpendicular
important trapezoid stuff (both regular and isosceles) (6 things)
- parallel sides are bases; non-parallel sides are legs
- 2 angles that share a base are called base angles
- 2 angles that share a leg are supplementary
- isosceles*
- legs are congruent
- base angles congruent
- diagonals congruent
important kite stuff (2 things)
- 2 pairs of consecutive/adjacent congruent sides
- perpendicular diagonals
trapezoid midsegment theorem
the midsegment of a trapezoid is parallel to the bases and the length of the midsegment is half the sum of the lengths of the bases
two polygons are similar if both:
- corresponding angles are congruent
2. corresponding sides are proportional
3 ways to prove triangles similar:
AA ~ - Angle Angle
SAS ~ - Side Angle Side
SSS ~ - Side Side Side
The length of the altitude to the hypotenuse is the GEOMETRIC MEAN between the SEGMENTS of the HYPOTENUSE
piece / alt = alt / piece
EACH LEG of the triangle is the GEOMETRIC MEAN between the LENGTH of the HYPOTENUSE and the ADJACENT SEGMENT of the HYPOTENUSE
hypotenuse / leg = leg / piece
Side Splitter
makes proportions of the triangles
Theorems 8-2, 8-3, and 8-4 (right, obtuse, acute)
8-2 : If a^2 + b^2 = c^2, then the triangle is right
8-3 : If a^2 + b^2 < c^2, then the triangle is obtuse
8-4 : If a^2 + b^2 > c^2, then the triangle is acute
SOH - CAH - TOA
Sin = opposite/hypotenuse Cos = adjacent/hypotenuse Tan = opposite/adjacent
area of triangle formula
1/2 bh
or
1/2 absinc
area of regular polygon
1/2 ap
1/2 (apothem) (perimeter)
4 types of transformations and definitions
- translation - slide
- reflection - flip
- rotation - turn
- dilation - bigger or smaller
what do you change in a…
- x-axis reflection
- y-axis reflection
- across line y = x
- across line y = -x
- change sign of y
- change sign of x
- switch x and y coordinates
- switch x and y coordinates AND change their signs
when rotated around the origin counterclockwise, what do you need to remember?
90 degree rotation - (x,y) –> (-y,x)
180 degree rotation - (x,y) –> (-x,-y)
270 degree rotation - (x,y) –> (y, -x)
What do you do in a rotation when center is not 0?
- make new axis
2. rename points and use the regular rule around new axis
In a dilation when the center is (0,0), what do you do?
- multiply points by scale factor
2. plot points
how to plot a dilation when center is not (0,0)
- plot center
- find how far away a point is from the center
- multiply that by scale factor
- plot point
how to find center of dilation
- draw lines connecting each point to its image
- find meeting point
finding segment lengths
- 2 chords
- 2 secants
- Secant and Tangent
- (piece 1) (piece 2) = (piece 3) (piece 4)
- (whole secant) (outside piece) = (ws) (op)
- (ws) (op) = (tangent)^2
finding angles
- intersects inside
- intersects outside
- angle = 1/2 (large angle + small angle)
2. angle = 1/2 (large angle - small angle)
standard form for circle:
(x-h)^2 + (y-k)^2 = r^2
How do you complete the square and get a circle in Standard Form?
- Gather the x terms and y terms so they are next to each other in the equation. Get the constant on the other side.
- Look at the coefficients of the x term and the y term. Divide the coefficients by 2 and square them. Add those numbers to BOTH sides
- Factor the x-trinomial and the y-trinomial. Both must factor into the same parenthesis twice. ex: (x-4)^2
The circle is now in the correct form.
Arc Length formula
the length of an arc of a circle is the product of the ratio of the measure of the arc and the circumference of the circle
length =
n
—— x 2πr
360
Area of a Sector of a Circle
Sector Area =
n
—— x πr^2
360
ratio of perimeter
scale factor
ratio of area
(scale factor)^2
ratio of volume
(scale factor)^3
what is a median?
a segment whose endpoints are a vertex and the midpoint of the opposite side
the medians of a triangle are concurrent at a point that is…
2/3 the distance from each vertex to the midpoint of the opposite side
THIS IS THE CENTROID
if you have a smaller segment than you need in a proof, what do you do? (what are the three steps)
- Segment Addition Postulate
- Addition Property
- Substitution
if you have bigger segments than you need in a proof, what do you do? (what are the three steps)
- Segment Addition Postulate
- Substitution
- Subtraction
