Math Proof Toolkit-5.3

Definitions:

Collinear points

Points are collinear if and only if they are on the same line.

Distance between two points

If you take the absolute value of the difference of the coordinates of the 2 points, then you get the distance between them.

Congruent segments

Two segments are congruent if and only if they have the same length (measure).

Bisect

An object bisects a segment if and only if it divides it into two congruent parts (in half).

Midpoint of a line segment

A point is a midpoint if and only if it divides the segment into 2 congruent parts.

Segment bisector

A line or part of a line is a segment bisector if and only if it intersects a segment at its midpoint and divides it into 2 congruent parts.

Angle

An object is an angle if and only if it is the union of two rays with a common endpoint called the vertex.

Acute Angle

An angle is acute if and only if it's measure is greater than 0° and less than 90°.

Right Angle

An angle is a right angle if and only if it measures 90°.

Obtuse Angle

An angle is obtuse if and only if it is greater than 90° and less than 180°.

Straight Angle

An angle is straight if and only if it measures 180°.

Perpendicular lines

Lines are perpendicular if and only if they intersect to form right angles.

Perpendicular bisector

A line is a perpendicular bisector if and only if it is perpendicular to a line segment at its midpoint.

Congruent angles

Angles are congruent if and only if they have the same measure.

Angle bisector

A ray is an angle bisector if and only if it divides an angle into two congruent parts.

Adjacent angles

Angles are adjacent if and only if they are an angle pair that has a common vertex, common side, and no interior points in common.

Vertical angles

Two angles are vertical if and only if they are two nonadjacent angles formed by intersecting lines.

Complementary angles

Two angles are complementary if and only if the sum of their measures is 90°.

Supplementary angles

Two angles are supplementary if and only if the sum of their measures is 180°.

Linear pair

Angles are a linear pair if and only if they are two adjacent angles whose non-common sides form a straight line.

Perpendicular pair

Angles are a perpendicular pair if and only if they are two adjacent angles whose non-common sides are perpendicular.

Regular polygon

A polygon is regular if and only if it has all sides and angles congruent.

Congruent triangles

Two triangles are congruent if and only if all corresponding sides and angles are congruent.

Axioms/Postulates:

For any two points, there exists one and only one line containing both points.

The intersection of two lines is a point.

For any three non-collinear points, there exists one and only one plane containing all three points.

The intersection of two planes is a line.

If two points are in a plane then the line containing them is also in the plane.

A line segment has one and only one midpoint.

Segment Addition

The whole is equal to the sum of its parts.

Angle Addition

The whole is equal to the sum of its parts.

An angle has one and only one angle bisector.

Supplement Axiom

The angles of a linear pair are supplementary.

Complement Axiom

The angles of a perpendicular pair are complementary.

Addition Property

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

Subtraction Property

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

Multiplication Property

If a=b, then a×c=b×c.

Division Property

If a=b and c≠0, then a÷c=b÷c.

Reflexive Property

a=a.

Symmetric Property

If a=b, then b=a.

Transitive Property

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

Substitution Property

If A+t=10 and t=6, then A+6=10.

Euclid's Parallel Axiom

Through a point not on a given line, there is exactly one parallel line to the given line.

SAS Axiom

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 triangles are congruent.

Theorems:

Midpoint Theorem

If M is the midpoint of AB, then AM = ½AB, and AB = 2AM.

Angle Bisector Theorem

If BD bisects ∠ABC, then m∠ABD = ½m∠ABC, and m∠ABC = 2m∠ABD.

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

If two angles are supplementary to congruent angles, then they are congruent.

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

If two angles are complementary to congruent angles, then they are congruent.

If two angles are right angles, then they are congruent to each other.

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

If two lines intersect to form congruent adjacent angles, then they are perpendicular.

If two lines are perpendicular, then they intersect to form congruent adjacent angles.

Vertical angles are congruent.

If lines are parallel, then corresponding angles are congruent.

If corresponding angles are congruent, then the lines are parallel.

If lines are parallel, then alternate interior angles are congruent.

If alternate interior angles are congruent, then lines are parallel.

If lines are parallel, then same side interior angles are supplementary.

If same side interior angles are supplementary, then lines are parallel.

If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.

If a transversal is perpendicular to each of two lines, then the lines are parallel.

If two lines are perpendicular to the same line, then they are parallel to each other.

If two lines are parallel to the same line, then they are parallel to each other.

The sum of the interior angles of a triangle is 180°.

The exterior angle of a triangle equals the sum of the remote interior angles.

If two angles of one triangle are congruent to two angles of another triangle, the third angles are congruent.

Each angle of an equilateral triangle is 60°.

The acute angles of a right triangle are complementary.

Corresponding parts of congruent triangles are congruent (CPCTC).