euclid

Collinear

If points are collinear, then they lie on the same line.

Congruent

IF two segments/angles are congruent, then they are equal in measure.

Midpoint

If a point is a midpoint, then it divides the segment into two congruent parts.

Segment Bisector

If a line (or part of a line) is a segment bisector, then it intersects the segment at its midpoint.

Angle Bisector

If a line (or part of a line) is an angle bisector, then it divides the angle into two congruent angles.

Complementary Angles

If two angles are complementary angles, then the sum of their measures is 90 degrees.

Supplementary Angles

If two angles are supplementary angles, then the sum of their measures is 180 degrees.

Right Angle

If an angle is a right angle, then its measure is 90 degrees.

Vertical Angles (Vertical Pair)

If two angles are vertical angles, then they share a vertex and the sides of one angle form opposite rays with the sides of the other angle.

Linear Pair

If two angles form a linear pair, then they are adjacent and their non-shared sides form opposite rays.

Perpendicular

If two lines are perpendicular, then they intersect to form a right angle.

Reflexive Property of Equality

A quantity is equal to itself

Symmetric Property of Equality

Equal quantities may be expressed in either order.

Transitive Property of Equality

If two quantities are equal to the same or equal quantaties, then the two quantities are equal to each other. (It's always transitive unless it's not.)

Substitution Property of Equality

A quantity may be substituted for its equal in any expression.

Addition Property of Equality

If equal quantities are added to equal quantities, then the sums are equal

Subtraction Property of Equality

If equal quantities are subtracted from equal quantities, the differences are equal.

Reflexive Property of Congruence

A segment/angle is congruent to itself.

Symmetric Property of Congruence

Congruent segments/angles may be expressed in either order.

Transitive Property of Congruence

If two segments/angles are congruent to the same or congruent segments/angles, then the two segments/angles are congruent to each other.

Addition Theorem of Congruence

If congruent segments/angles are added to congruent segments/angles, then the resulting segments/angles (sums) are congruent

Subtraction Theorem of Congruence

If congruent segments/angles are subtracted from congruent segments/angles, then the resulting segments/angles (difference) are congruent

The Multiplication Property of Equality

If equal quantities are multiplied by a constant, then their products are equal.

The Division Property of Equality

If equal quantities are divided by an non-zero constant, then their quotients are equal.

Multiplication Theorem of Congruence

Multiples of congruent segments/angles are congruent.

Division Theorem of Congruence

Quotients of congruent segments/angles are congruent.

Polygon

A polygon is the union of a set of points (called vertices) and segments (called sides) such that 1. No two sides intersect at a point other than a vertex, 2. Each vertex must be the endpoint of exactly two sides, and 3. No three vertices are collinear.

Triangle

A triangle is a polygon with exactly three sides.

Correspondence

The correspondence of two polygons maps one vertex of one polygon to a vertex of the second polygon. ORDER MATTERS. ( <EF)

Congruent Polygons

Two polygons are congruent if and only if there exists a correspondence between the vertices of one polygon and the vertices of the other polygon such that each pair of corresponding sides/angles are congruent.

Congruent Triangles

IF two triangles are congruent, then there exists a correspondence between the vertices of one triangle and the vertices of the other triangle such that each pair of corresponding parts are congruent.

Side-Angle-Side (SAS) Postulate of Congruence

If there exists a correspondence between the vertices of one triangle and the vertices of the other triangle such that two pairs of corresponding sides and the included angles are congruent, then the two triangles are congruent.

Ways to prove triangles congruent

Side-Angle-Side (SAS) postulate of congruence, Angle-Side-Angle (ASA) Theorem of congruence, and Side-Side-Side (SSS) Theorem of Congruence.

Postulate 1

2 points determine a line

Postulate 2

Two lines intersect in at most one point

Postulate 3

Given a line, there exists a point not on it

Postulate 4

Each line contains at least two points

Postulate 5

Every distance is a non-negative number

Postulate 6

The distance between two points is zero if and only if the two points are coincident/overlapping

Distance Assignment Postulate

Given a positive real number, d, and a ray AB, there exists AC such that AC = d and point C is on AB

Segment Existence Postulate

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

Segment Extension Postulate

A line segment can be extended to any length in either direction

Postulate 10

A line segment has exactly one midpoint

Postulate 11

Every angle measure is a real number between 0 degrees and 360 degrees.

Postulate 12

An angle has exactly one angle bisector

Partition Postulate

A whole is equal to the sum of all its parts

Angle existence postulate/A line divides a plane into two half-planes

Given a real number, r, between 0 degrees and 180 degrees and a ray AB, there exists two angles (angle CAB and angle DAB) such that the measure of angle CAB equals measure of angle DAB equals r, where C and D lie on the opposite half planes of AB.

Postulate 15

The sum of the degree measures of all the angles about a given point is 360 degrees.

Postulate 16

The sum of the degree measures of all the angles of one side of a given line, whose common vertex is a given point on the line, is 180 degrees.

Theorem 1

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

Things we can assume

Straight lines and angles, collinearity of points, betweenness of points, and relative positions of points.

Things we cannot assume

Right angles, congruent segments, congruent angles, and relative sizes of segments/angles.

Postulate

An unproved assumption.

Theorem

A statement that can be proven.

Multiplication theorem of congruence- additional conclusions

Doubles of congruent segments/angles are congruent, and

Triples of congruent segments/angles are congruent

Division theorem of congruence- additional conclusions

Halves of congruent segments/angles are congruent

Thirds of congruent segments/angles are congruent.

Theorems regarding angles (#1)

If two angles are complementary to the same angle, then the two angles are congruent to each other.

Theorems regarding angles (#2)

If two angles are complementary to congruent angles, then the two angles are congruent to each other.

Theorems regarding angles (#3)

If two angles are supplementary to the same angle, then the two angles are congruent to each other.

Theorems regarding angles (#4)

If two angles are supplementary to congruent angles, then the two angles are congruent to each other.

Theorems regarding angles (#5)

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

When to use Multiplication/Division theorem of congruent

When you are given (1) Two segments/angles that are congruent, or (2) a double midpoint or double bisect on a segment/angle

Theorems regarding angles (#6)

If the non-shared sides of two adjacent angles are perpendicular, then the two angles are complementary.

Theorems regarding angles (#7)

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

Theorems regarding angles (#8)

If two angles form a vertical pair, then the two angles are congruent, or If two angles are vertical angles, then the two angles are congruent.

Stating Linear Pairs in Proofs

Since we can assume 1. Lines are straight and 2. Adjacent angles,

We can state linear pairs in proof without explicitly stating opposite rays or adjacent angles.

However, we must still justify linear pair! (If two angles are adjacent and their non-shared sides form opposite rays, then the angles form a linear pair.)

Stating Vertical Pairs in Proofs

Since we can assume 1. Lines are straight and 2. Relative position of points,

We can state vertical pairs in proofs without explicitly stating opposite rays or common vertex.

However, we must still justify vertical pair! (If two angles have a common vertex such that the sides of one angle are opposite rays of the sides of the other angle, then the angles form a vertical pair.)

A polygon is the union of…

A set of points {A, B, C,…} (called vertices)

And segments {AB, BC,…} (called sides)

Such that

1. no two sides intersect at a point other than a vertex,

2. Each vertex must be the endpoint of exactly two sides, and

3. no three vertices are collinear.

Theorems about polygons

• Two polygons are congruent if and only if there exists a correspondence between the vertices of one polygon and the vertices of the other polygon such that each pair of corresponding parts are congruent.

• If two triangles are congruent, then there exists a correspondence between the vertices of one triangle and the vertices of the other triangle such that each pair of corresponding parts are congruent.

Side-Angle-Side (SAS) Postulate of Congruence

If there exists a correspondence between the vertices of one triangle and the vertices of the other triangle such that the two pairs of corresponding sides and the included angles are congruent, then the two triangles are congruent.

Ways to prove triangles congruent

• SAS postulate of congruence (two pairs of sides, and a pair of the included angle)

• ASA theorem of congruence (Two pairs of angles, and a pair of the included side)

• SSS Theorem of congruence (Three Paris of sides)

Linear Pair -> Supplementary

If tow angles form a linear pair, then they are supplementary

Vertical Pair -> Congruent

If two angles form a vertical pair, then the two angles are congruent

Vertical angles -> Congruent

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

CPCTC

If two angles are congruent, then each park of corresponding parts are congruent, or corresponding parts of congruent triangles are congruent.

Side-Angle theorem

If two sides of a triangle are congruent, then the angles opposite these sides are congruent

Auxiliary lines

-Lines that are constructed in proofs are called auxiliary lines.

-They are to assist in drawing a conclusion. They are always drawn dashed.

-You must state in your proof that auxiliary lines have been constructed, and unless the problem states otherwise, don't use auxiliary lines in proofs.

Converse of Side-Angle Theorem

If two angles of a triangle are congruent , then the sides opposite to these angles are congruent.

Isosceles triangle

-If a triangle is an isosceles triangle then at least two sides of the triangle are congruent.

-If at least two sides of a triangle are congruent, then the triangle is an isosceles triangle.

Scalene triangle

If a triangle is a scalene triangle, then no two sides are congruent

Equilateral triangle

If a triangle is an equilateral triangle, then all three sides are congruent.

Acute triangle

If a triangle is an acute triangle, then each angle of the triangle is an acute angle.

Obtuse triangle

If a triangle is an obtuse triangle, then one angle of the triangle is an obtuse angle.