Geometry

A plane is an undefined term. We can describe it as

a surface such that if any two points in the surface are joined by a line, then the line lies entirely in the surface. (definition)

If points, lines, segments lie in the same plane, we call them … If they do not lie in the same plane, they are …

coplanar; not coplanar (definition)

If a line passes through two points that lie in a plane, then the line lies…

in the plane (postulate)

If a line intersects a plane not containing it, then they intersect at…

one point (postulate)

If two planes intersect, then their intersection is…

one line (postulate)

What are the ways to determine a plane?

3 non-collinear points, a line and a point not on the line, two intersecting lines

A line is perpendicular to a plane if…

it is perpendicular to every line in the plane that passes through its foot. (definition)

If a line is perpendicular to … then it is perpendicular to the plane.

at least two distinct lines that lie in a plane and pass through its foot (theorem)

Given a plane and a point on the plane, there is … passing through the given point that is perpendicular to the plane.

exactly one line (postulate)

Given a plane and a point not on the plane, there is … passing through the given point that is perpendicular to the plane.

exactly one line (postulate)

A line is … to a plane if it intersects the plane at exactly one point and is not perpendicular to the plane.

oblique (definition)

Law of Trichotomy

For real numbers, exactly one of the following is true:
a<b, a=b, a>b

A …. is a line that intersects two lines in two distinct points.

transversal (definition)

Two … lines are parallel if …

coplanar, they do not intersect (definition)

What is the format for proving lines parallel because of pairs of angles?

If two lines are cut by a transversal and they form a pair of … angles, then they are parallel. (theorems)

Through a point not on a given line, there exists … that is parallel to the given line.

one unique line (postulate, specifically the Parallel Postulate)

What are each of the undefined terms?

Point, line, plane (definition)

A … has neither length nor width but indicates a position. We represent it with a dot.

point (definition)

A … has no width and can be extended as far as desired in either direction.

line (definition)

If 2 or more points belong to the same … they are said to be …

line, collinear (definition)

If 3 or more lines contain the same … they are said to be …

point, concurrent (definition)

If point B is between A and C, then A, B, and C are distinct … points and …

collinear, AB+BC=AC (definition)

Points B and C are said to be on the same side of point A if …

A, B, and C are distinct collinear points and A is not between B and C. (definition)

A line segment is a set of … points of a line and all the points … them.

2, between (definition)

A … is a set of points consisting of a fixed point of a line and all the points of that line on the … of the fixed point.

ray, same side (definition)

… are two rays of the same line that have a … and no other point in …

Opposite rays, common endpoint, common (definition)

An … is a set of points consisting of a union of two … not lying on the same line that have a common …

rays, endpoint (definition)

A point P lies in the … of an angle if there exist two points, one on each …, neither at the … such that the point P is between said two points.

interior, side, vertex (definition)

Congruent line segments are segments that are …

equal in length (definition)

Congruent angles are angles that are …

equal in measure (definition)

The … of a line segment is a point of the line segment such that the two segments formed are …

midpoint, congruent (definition)

The trisection points of a line segment are the … points of the line segment such that the … segments formed are …

2, 3, congruent (definition)

A segment bisector is a line, line segment, or ray that intersects the line segment at its …

midpoint (definition)

A right angle is angle measuring …

90 degrees (definition)

An acute angle is an angle measuring

>0 degrees and <90 degrees (definition)

Anobtuse angle is an angle measuring …

>90 degrees and <180 degrees (definition)

Complementary angles are two angles the sum of whose measures is …

90 degrees (definition)

Supplementary angles are two angles the sum of whose measures is …

180 degrees (definition)

Two lines are said to intersect if they have a … in common

point (definition)

Perpendicular lines are two lines that … and form …

intersect, right angles (definition)

If two lines in the same plane do not … then they are parallel.

intersect (definition)

An angle bisector is the … whose endpoint is the … of the angle and that divides it into two … angles

ray, vertex, congruent (definition)

Angle trisectors are the … whose common endpoint is the… of the angle and that divide it into … angles

rays, vertex, 3 congruent (definition)

The … of two objects is the set of all points that are contained in at least one of the two objects.

union (∪) (definition)

The … of two objects is the set of all points that are contained in at least one of the two objects.

intersection (∩) (definition)

The point that divides a line segment into two congruent segments is the … of the line segment.

midpoint (definition)

A line that intersects only the midpoint of a segment is a …

segment bisector (definition)

An angle whose measure is 90 degrees is a … angle.

right (definition)

A ray whose endpoint is the vertex of an angle and that forms two congruent angles is an …

angle bisector (definition)

Angles that are equal in measure are …

congruent (definition)

Line segments that are equal in measure are …

congruent (definition)

What can you assume from the diagram?

Collinearity of Points, Betweenness of Points, Incidence of Lines, Coplanarity of the diagram

Points draw on one line really lie on one line

Collinearity of Points

A point drawn on the same line as two other points that is in between them is actually between them.

Betweenness of Points

Two lines shown as intersecting a point do intersect at that point.

Incidence of Lines

Everything shown in the diagram lies in the same plane unless stated otherwise.

Coplanarity of the diagram

A statement that is unproven, but we assume it to be true.

postulate or axiom

Given any two distinct points, … line contains them both.

exactly one (Line Axiom)

Each line contains … points.

infinitely many (Line Axiom)

Given a line, there exists a point … on the line.

not (Line Axiom)

To every pair of distinct points there corresponds a unique positive number. This number is called the distance between the two points, The distance between two points is …, if and only if the points are not distinct.

0 (Distance Assignment Postulate)

Given ray XY and segment AB, there exists exactly one point P on ray XY such that segment XP is congruent to …

segment AB (Segment Construction Postulate)

If point B is between points A and C,then segment AB _ segment BC is congruent to segment AC, and AB+BC=A

Line Segment Partition Postulate (The whole is equal to the sum of its parts)

To every angle there corresponds a unique real number between 0 and 180. This numberis called its measure.

Angle Measure Assignment Postulate

Given ray XY and a real number k between 0 and 180, there exists exactly one ray XR on a given side of line XY, such that measure of angle RXY=k

Angle Construction Postulate

If P lies in the interior of angle ABC, then angle ABP + angle PBC is congruent to angle ABC, and measure of angle ABP + measure of angle PBC = measure of angle ABC

Angle Partition Postulate (The whole is equal to the sum of its parts)

Any two points have a … midpoint.

unique

Each angle has a … angle bisector.

unique

Two distinct lines intersect in … one point.

at most

If equals are added to equals, then their sums are equal.

The Addition Property of Equality (postulate)

If congruent segments (or angles) are added to congruent segments (or angles) then their sums are congruent.

The Addition Property of Congruence (Postulate)

IF equals are subtracted from equals, then their differences are equal

The Subtraction Property of Equality (Postulate)

If congruent segments (or angles) are subtracted from congruent segments (or angles) then their differences are congruent.

The Subtraction Property of Congruence (Postulate)

A … is a statement that has been proven by a chain of reasoning.

theorem

Adjacent angles are two coplanar angles that share a … vertex and a … but share no …points.

common, side, interior

Two … angles whose noncommon sides are opposite rays form a linear pair.

adjacent

If two angles are right angles, then they are …

congruent/supplementary (theorem)

If two angles are complementary to the … angle, then they are …

same, congruent (theorem)

if two angles are supplementary to the … angle, then they are …

same, congruent (theorem)

If two angles are complementary to … angles, then they are …

congruent, congruent (theorem)

If two angles are supplementary to … angles, then they are …

congruent, congruent (theorem)

If two angles form a linear pair, then they are …

supplementary (postulate)

Two angles whose sides form two pairs of … rays are a …

opposite, vertical pair (definition)

If two angles form a … then they are congruent

vertical pair

If a=b then ca=cb

Multiplication Property of Equality

If a=b and c≠0, a/c=b/c

Division Property of Equality

If two segments (or angles) are congruent, then their like divisions are congruent.

Division Property of Congruence

If two segments (or angles) are congruent, then their like multiples are congruent.

Multiplication Property of Congruence

Construct the line joining two points

Two points determine a line (from Line Axiom)

Extend a line segment or ray

A line can be extended as far as desired

Construct a segment of a desired length on a ray

Segment construction postulate

Given a ray, construct an angle of a desired size

Angle construction postulate

Construct the angle bisector of a given angle

Every angle has a unique angle bisector

Erect the line perpendicular to a given line at a point on the given line

Through a point on a line exactly one line can be constructed perpendicular to the given line

Drop the line perpendicular to a given line passing through a point not on the given line

Through a point not on a line exactly one line can be constructed perpendicular to the given line

Construct the perpendicular bisector of a given line segment

Every segment has a unique perpendicular bisector

Two triangles are congruent if and only if there exists a correspondence between the vertices of one triangle and the vertices of the other triangle such that each … of … are congruent

pair, corresponding parts

To prove two triangles congruent, we do not need to show all … pairs of parts are congruent. We only need to show … carefully chosen pairs of corresponding parts are congruent.

six, three

Corresponding parts of congruent triangles are congruent

CPCTC

If two sides and the included angle of one triangle are congruent to the corresponding two sides and included angle of another triangle, then the two triangles are congruent.

Side-Angle-Side (SAS) Postulate of Congruence