Geometry Regents
Polygons
Sum Interior Angles: 180 (n-2)
Each Interior Angle Regular Polygon:
180 (n-2) / n
Each Exterior Angle: 360/n
Interior Angles in a Polygon Add to 180, their Exterior Angles Add to 360
An Interior + It’s Exterior is 180
Lines
slope-intercept: y=mx+b
point slope: y-y1 = m(x-x1)
slope formula: change in y/change in x (y2-y1/x2-x1)
parallel lines have the same slope
parallel lines have the same slope when dilated and all transformations
midpoint: (x1+x2/2 , y1+y2/2)
distance: d (x2-x1)^2 + (y2-y1)^2 under radical
Angles
parallel lines cut by a transversal create angle pairs
alternate interior angles are congruent
Alternate exterior angles are congruent
Corresponding angles are congruent
Same side interior angles are congruent
Triangles
scalene triangles have no congruent sides
isosceles triangles have two congruent sides
Equilateral triangles have three congruent sides
acute angles and angle smaller than 90°
A right angle is equal to 90°
An obtuse angle is anything larger than 90°
equiangular triangles have three congruent angles, all equal to 60°
all triangles have a measure of 180°
exterior angle theorem is that the sum of an exterior angle is equal to the sum of two non-adjacent interior angles
A mid segment joints to mid points on opposite side of triangle, the mid segment is parallel to the third side and is half of the length of the third side, it creates similar triangles
The sum of two side sides must be greater than the third side
The difference of two sides must be less than the third side
The longest side of the triangle is opposite the largest angle, and the shortest side of the triangle is opposite the smallest angle
two best angles in an isosceles triangle are congruent alongside their two sides
an altitude drawn from the vertex of an isosceles triangle is also median and an angle bisector
if a line is parallel to a side of the triangle and intersects two sides, then this line divides those two sides proportionally
triangle congruence can be proved by: sss, sas, asa, aas, and HL
corresponding parts of congruent triangles are congruent or CPCTC
triangles can be proven similar by: aa, sas, sss
similar figures have congruent, angles, and proportional sites
corresponding sides of somewhat triangles are proportional or CSSTP
mean proportional theorems are used to find different parts of triangles with an altitude
segment over altitude equals altitude over segment or SAAS
hypotenuse over leg is equal to leg over segment or HLLS
The Pythagorean theorem only works for right triangles and says that a squared plus B squared is equal to C squared
The Pythagorean theorem can be used to find the measure of a missing side and a right triangle
trigonometric ratios are used to solve a side in a right triangle when angles are given or find an angle when sides are given
sine is equal to opposite over hypotenuse (SOH)
Cosine is equal to adjacent over hypotenuse (CAH)
Tangent is equal to opposite over adjacent (TOA)
when solving for an side, buttons, as is, but when solving first side, use the negative function
sine and cosine are co-functions
Transformations
reflection over x-axis (x,-y)
reflection over y-axis (-x,y)
reflection over y=x (y,x)
reflection y=-x (-y, -x)
reflection over origin (-x,-y)
rotation 90° clockwise or 270° counterclockwise (-y, x)
rotation 180° (-x,-y)
rotation 270° clockwise or 90° counterclockwise (y, -x)
translation shifts points
dilations multiply given side measures
to dilate from another point, do rise over run and times it by the scale factor
in a composition, do the first one second
rotational symmetry means how many degrees a figure rotates until it maps onto itself (360/n)
Circles
standard formula for circles: X squared plus Y squared plus CX plus DY plus E equals zero
center-radius: X subtract H squared plus Y subtract case equals R squared
A.m. method means to find a factor that adds to one number and multiplies to another
central angle: x = arc it intercepts
inscribed angle: x = half it’s intercepted arc
tangent-chord angle : x = half it’s intercepted arc
two chord angles: x = arc1 + arc2/2
big arc-little arc/2 = x
A tangent is perpendicular to its radius and forms and 90° angle
an angle inscribed in a semi circle equals 90°
if a quadrilateral is described in a circle then it’s opposite angles are equal 180°
segement in a circle: pxp=pxp (or part-time part equals part times part)
parallel chords intercept congruent arcs
(w)(e)= (w)(e) (or whole Times exterior equals whole times exterior)
(w)(e)=t^2 (or whole exterior equals tangent squared)
PPPPWEWEWET^2 is used to find measures of lines in circles
if two segments are congruent, then the arc they intercept are congruent
if a diameter or radius is perpendicular to a cord, then the diameter bisect the cord and it’s arc
area of a sector n/360 pie r^2
legnth of a sector n/360 pie D
angles that intercept the same arc are congruent
Quadrilaterals
A quadrilateral is a four sided polygon
A trapezoid has at least one pair of parallel sides
trapezoids have median, median equals half base one plus base two
and I saw these trapezoid has base angles that are congruent diagonals that are congruent, and one pair of congruent sides which are the non-parallel sides
A parallelogram has opposite sides that are parallel and congruent opposite angles that are congruent, consecutive angles that are supplementary and diagonals that bisect each other
rectangle all angles right angles, and the diagonals are congruent
in a rhombus, all sides are congruent diagonals are perpendicular diagonals bisect opposite angles form, four congruent, right triangles, and two pairs of congruent isosceles triangles
A square has died was a form four congruent isosceles right triangles. All of its side measures are equal, and all of its angle measures are equal.
Other Formulas
Area of a circle is pie r squared
Area of a triangle is half base x side
Area of a rectangular square is base x height
volume formulas are given, but make sure base is area and not just the measure of the base