geometry regents

sum of interior angles

180(number of triangles-2)

each interior angle of a regular polygon

180(number of triangles-2)/number of triangles

sum of exterior angles

360

each exterior angle

360/number of triangles

scalene triangle

no congruent sides

equilateral triangle

all sides are congruent

isosceles traingle

2 congruent sides

equiangular

3 congruent angles

exterior angle theorem

The exterior angle is equal to the sum of the two non-adjacent interior angles

midsegment

a segment that joins two midpoints

• always parallel to the third side

• 1/2 the length of the 3rd side

• splits the triangle into 2 similar triangles

slope intercept form of a line

y = mx + b, where m is the slope and b is the y-intercept.

point slope form of a line

**y-y1=m(x-x1).** where m is the slope and x1 and y1 are the values of a given point on the line

slope formula

(y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on a line. rise/run

parallel lines

have the same slope

perpendicular lines

have neg reciprocal slopes (flip the fraction and change the sign into its opposite)

collinear points

are points that lie of the same line

mid point formula

It involves averaging the x and y coordinates of the endpoints.

Formula: M =(x₁+x₂)/2, (y₁+y₂)/2.

distance formula

**d = √((x2 - x1)^2 + (y2 - y1)^2)**, where (x1, y1) and (x2, y2) are the coordinates of the two points and d is the distance between them.

segment ratios

the ratio of the lengths of two segments that share an endpoint.

triangle theorems

• the sum of 2 sides must be greater than the 2 sides

• the difference of 2 sides must be less than the 3rd side

• the longest side of the triangle is opposite the largest angle

• the shortest side of the triangle is opposite the smallest angle

isosceles triangle properties

• 2 congruent sides and 2 congruent base angles

• the altitude drawn from the vertex is also the median and angle bisector

• if two sides of a triangle are congruent, then the angles opposite those congruent angles congruent

alternate interior amgles

are congruent

alternate exterior angles

corresponding angles

same side interior angles

supplemantary

side splitter theorem

if a line is parallel to one side of a triangle and intersects the other two sides, it divides those sides proportionally.

triangle congruency theorems

CPCTC

If two triangles are congruent, then their corresponding parts are congruent.

similar triangle theorems

• Similar figures have congruent angles and proportional sides

• CSSTP-Corresponding Sides of Similar Triangles are in Proportion

• In a proportion, the product of the means equals the product of the extremes

altitude theorem SAAS

The altitude is the geometric mean between the 2 segments of the hypotenuse

leg theorem

The leg is the geometric mean between the segment it touches and the whole hypotenuse.

confunctions

sine and cosine are confunctions which are complemntary (add up to 90)

**Ex: sin60=cos(90-60) and vice versa**

if angle a and angle b r the acute angles of a right trinaglke, then sin a = cos b

rigid motion

A transformation that preserves distance and angle measures between objects. Examples include translations, rotations, and reflections.

rotations

\

dilation enlargement/reduction

• create similar figures where the corresponding sides are in proportion and the corresponding angles are congruent

• do not preserve distance or congruency

multiply the x and y values by the number

ex: (2,4) reduced by a half = (2,4)x1/2=(1,2)

composition of transformations

types of compisition trnadsfomagition

A composition of 2 reflections over 2 intersecting lines is equivalent to a ROTATION.

A composition of 2 reflections over 2 parallel lines is equivalent to a TRANSLATION.

rotational symmetry theorem

a regular polygon with 3 or more sides always has rotational symmetry , with rotations in increments. commonly referred as “mapping the figure onto itself”.

**360/n n=3 of sides**

circle equations

standard equation of a circle: where C,D and E are constants

center-radius equation of a circle: where (h, k) is the center and r is the radius

Chord

A line segment connecting two points on a curve or circle.

Secant

A line that intersects the circle in two or more points. This is a different secant than the secant in trigonometry.

Tangent

A line from outside a circle that intersects the border of a circle at one point but does not pass through the border to the circle’s interior. This is a different tangent than the tangent in trigonometry.

Central Angle

An angle whose vertex is at the center of a circle and whose legs are radii of the circle.

Inscribed Angle

An angle created when two chords of a circle share a point on that circle.

Circumscribed Angle

An angle created when two tangent lines to a circle intersect outside of that circle.