geo hna thm posts n defs

Converse of Conditional

Swapping the hypothesis and conclusion: (q->p)

Point

A location in space with no dimensions, only position (named with capital letters like A, B, C; coordinates).

Line

Defined by two points, infinitely thin, and extends infinitely in both directions.

Collinear

Points that lie on the same line.

Plane

Defined by 3 points, infinitely thin, and extends in all directions.

Coplanar

Points that lie on the same plane.

Space

A boundless, 3D set of all points.

Angle

Defined when two rays share the same endpoint.

Segment Addition Postulate

If A, B, and C are collinear and B is between A and C, then AB + BC = AC.

Angle Addition Postulate

If ray BD is in between angle ABC, then m<ABD + m<DBC = m<ABC.

Line Postulate

There is always a line between any 2 points.

Parallel Postulate

If there is a given line and a given point not on that line, there is only one line that passes through that point that is parallel to the given line.

Perpendicular Postulate

If there is a given line and a given point, there is only one line that intersects that point that is perpendicular to the given line.

Corresponding Angles Postulate

If two parallel lines are intersected by a transversal, then the corresponding angles are congruent.

Converse of Corresponding Angles Postulate

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

Linear Pair Postulate

If two angles form a linear pair, then they are supplementary.

Reflexive Property of Equality/Congruence

A value or angle is equal/congruent to itself (e.g., m<A = m<A).

Symmetric Property of Equality/Congruence

You can flip sides on an equal sign (if x = 4, then 4 = x).

Transitive Property of Equality/Congruence

If a = b and b = c, then a = c.

Distributive Property

2(x + 4) = 2x + 8.

Substitution Property

If a = b, then one can replace the other in an equation (plugging in).

Addition Property of Equality

If a = b, then a + c = b + c (or if c = d, a + c = b + d).

Subtraction Property of Equality

If a = b, then a - c = b - c.

Multiplication Property of Equality

If a = b, then ac = bc.

Division Property of Equality

If a = b, then a/c = b/c.

Alternate Interior Angle Theorem

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

Alternate Exterior Angle Theorem

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

Same Side Interior Angle Theorem

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

Same Side Exterior Angle Theorem

If two parallel lines are cut by a transversal, then same side exterior angles are supplementary.

Converse of Alternate Interior Angles Theorem

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

Properties of Angles

Properties that govern angle relationships when lines intersect.

Transversal

A line that intersects two or more lines at different points.

Acute Angle

An angle that measures less than 90 degrees.

Right Angle

An angle that measures exactly 90 degrees.

Obtuse Angle

An angle that measures more than 90 degrees but less than 180 degrees.

Straight Angle

An angle that measures exactly 180 degrees.

Complementary Angles

Two angles whose measures add up to 90 degrees.

Supplementary Angles

Two angles whose measures add up to 180 degrees.

Vertical Angles

Angles that are opposite each other when two lines intersect; they are congruent.

Adjacent Angles

Two angles that share a common side and a common vertex.

Circumference of a Circle

The distance around a circle, calculated as C = $2\text{π}r$.

Area of a Circle

The space inside a circle, calculated as A = $\text{π}r^2$.

Volume of a Cylinder

The space inside a cylinder, calculated as V = $\text{π}r^2h$.

Pythagorean Theorem

For a right triangle, the relationship $a^2 + b^2 = c^2$, where c is the hypotenuse.

Surface Area of a Sphere

Calculated as SA = $4\text{π}r^2$.

Volume of a Sphere

Calculated as V = $\frac{4}{3}\text{π}r^3$.