Geometry Regents

Sum of Interior Angles in a Polygon

The sum of the interior angles of a polygon is given by the formula 180(n-2), where n is the number of sides.

Each Interior Angle of a Regular Polygon

Each interior angle of a regular polygon is calculated using the formula 180(n-2)/n.

Each Exterior Angle of a Polygon

Each exterior angle of a polygon can be calculated using the formula 360/n.

Midpoint Formula

The midpoint between two points (x1, y1) and (x2, y2) is given by the formula (x1+x2/2 , y1+y2/2).

Distance Formula

The distance between two points (x1, y1) and (x2, y2) is represented by d = √((x2-x1)² + (y2-y1)²).

Slope-Intercept Form

The slope-intercept form of a linear equation is given as y = mx + b, where m is the slope and b is the y-intercept.

Point-Slope Form

The point-slope form of a linear equation can be expressed as y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.

Pythagorean Theorem

The Pythagorean theorem states that for a right triangle, a² + b² = c², where c is the length of the hypotenuse.

Sine Ratio

In a right triangle, the sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse (SOH).

Cosine Ratio

In a right triangle, the cosine function is defined as the ratio of the length of the adjacent side to the length of the hypotenuse (CAH).

Tangent Ratio

In a right triangle, the tangent function is defined as the ratio of the length of the opposite side to the length of the adjacent side (TOA).

Quadrilateral Definition

A quadrilateral is defined as a polygon with four sides.

Trapezoid Definition

A trapezoid is defined as a quadrilateral with at least one pair of parallel sides.

Parallelogram Properties

A parallelogram has opposite sides that are parallel and congruent, opposite angles that are congruent, and diagonals that bisect each other.

Area of a Circle

The area of a circle is calculated using the formula A = πr², where r is the radius.

Area of a Triangle

The area of a triangle is calculated using the formula A = 1/2 * base * height.

Area of a Rectangle

The area of a rectangle is calculated using the formula A = base * height.

Angle Pairs with Transversals

When parallel lines are cut by a transversal, alternate interior angles, alternate exterior angles, and corresponding angles are congruent.

Exterior Angle Theorem

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

Congruent Triangles

Two triangles are congruent if all corresponding sides and angles are equal, commonly proven by SSS, SAS, ASA, AAS, and HL.

Dilations

Dilations are transformations that alter the size of a figure while maintaining its shape.

Isosceles Trapezoid Definition

A quadrilateral with exactly one pair of parallel sides.

Rhombus Definition

A parallelogram with four congruent sides.

Rectangle Definition

A parallelogram with four right angles.

Square Definition

A parallelogram with four congruent sides and four right angles.

Perimeter Definition

The total distance around the outside of a two-dimensional shape.

Diagonal Definition

A line segment that connects two vertices of a polygon that are not adjacent.

Reflection Definition

A transformation that "flips" a figure over a line.

Translation Definition

A transformation that "slides" a figure along a straight line.

Rotation Definition

A transformation that "turns" a figure around a fixed point.

Perpendicular Lines Definition

Lines that intersect at a right angle (90 degrees).

Parallel Lines Definition

Lines in a plane that never intersect; they have the same slope.

Vertical Angles Theorem

If two angles are vertical angles, then they are congruent.

Supplementary Angles

Two angles are supplementary if the sum of their measures is 180 degrees.

Complementary Angles

Two angles are complementary if the sum of their measures is 90 degrees.

Corresponding Angles Postulate

If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.

Alternate Interior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.

Alternate Exterior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.

Same-Side Interior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of same-side interior angles are supplementary.

Altitude of a Triangle

A perpendicular segment from a vertex to the opposite side or to the line containing the opposite side.

Median of a Triangle

A segment from a vertex to the midpoint of the opposite side.

Vertical Angles Theorem

If two angles are vertical angles, then they are congruent.

Supplementary Angles

Two angles are supplementary if the sum of their measures is 180 degrees.

Complementary Angles

Two angles are complementary if the sum of their measures is 90 degrees.

Corresponding Angles Postulate

If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.

Alternate Interior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.

Alternate Exterior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.

Same-Side Interior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of same-side interior angles are supplementary.

Altitude of a Triangle

A perpendicular segment from a vertex to the opposite side or to the line containing the opposite side.

Median of a Triangle

A segment from a vertex to the midpoint of the opposite side.