Though today these terms are easily understood and known, in Euclids day they were not. You should keep that in mind as you go through these Theorems. Euclid also didn't rely on measurement of any kind so you should keep that in mind as well.

Primitives: Line, Point, and Congruent, Contains or lies on, will remain undefined.

The distance from a point P to a line L is defined by first finding the line that is perpendicular to L through the Point P and Q be the intersection of this line with L. Then the length of the line segment PQ is the distance from P to L.

Angle: the figure formed by two rays who share a common end point

...............................................................................................................................................................................

Right Angle: an angle that is congruent to its supplementary angle

Acute Angle: an angle that is less than a right angle

Obtuse Angle: an angle greater than a right angle

Right Triangle: a polygon of three sides that contains a right angle

Acute Triangle: a polygon of three sides where all interior angles are smaller than a right angle

Obtuse Triangle: a polygon of three sides where one of the interior angles is greater than a right angle

Vertical Angles: opposite angles of two lines that only share one point

Equilateral Triangle: a polygon of three sides where all of the interior angles are congruent

Isosceles Triangle: a polygon of three sides where two of the sides are congruent

Scalene Triangle: a polygon of three sides where none of the interior angles are congruent

Perpendicular Lines: two lines that share only one point and that form four right angles

Parallel Lines: two lines that never touch

Transversal: a line that intersects two other lines

Alternate Interior/Exterior Angles: an angles corresponding vertical angle on the other parallel line when two parallel lines are intersected by a transversal line

Circle: an infinite set of points that are equidistant from a single point

Radius: the length from any point on the perimeter of a circle and its center.

Diameter: the width of the circle

Circumference: the length of the perimeter of a circle

Theorem: a mathematical statement that can be proven

Proof: a set of justified steps that validate a mathematical statement

Lemma: a previously proven mathematical statement used to prove another mathematical statement

Corollary: a theorem that fallows another theorem

Hypothesis: the if part of an “if-then” statement

Line segment: a line with two endpoints

Ray: a line with one end point

Linear pair of angles: two angles that share a side and whose sum equals 180

Complementary angles: two angles who create a right triangle

Supplementary angles: two angles whose sum equals 180

Conjecture: a mathematical statement that isn’t proven

Reflexive Property: everything is congruent to itself. AB=AB

CPCTC: Corresponding Parts of Congruent Triangles are Congruent

The distance from a point P to a line L is defined by first finding the line that is perpendicular to L through the Point P and Q be the intersection of this line with L. Then the length of the line segment PQ is the distance from P to L.

Angle: the figure formed by two rays who share a common end point

...............................................................................................................................................................................

Right Angle: an angle that is congruent to its supplementary angle

Acute Angle: an angle that is less than a right angle

Obtuse Angle: an angle greater than a right angle

Right Triangle: a polygon of three sides that contains a right angle

Acute Triangle: a polygon of three sides where all interior angles are smaller than a right angle

Obtuse Triangle: a polygon of three sides where one of the interior angles is greater than a right angle

Vertical Angles: opposite angles of two lines that only share one point

Equilateral Triangle: a polygon of three sides where all of the interior angles are congruent

Isosceles Triangle: a polygon of three sides where two of the sides are congruent

Scalene Triangle: a polygon of three sides where none of the interior angles are congruent

Perpendicular Lines: two lines that share only one point and that form four right angles

Parallel Lines: two lines that never touch

Transversal: a line that intersects two other lines

Alternate Interior/Exterior Angles: an angles corresponding vertical angle on the other parallel line when two parallel lines are intersected by a transversal line

Circle: an infinite set of points that are equidistant from a single point

Radius: the length from any point on the perimeter of a circle and its center.

Diameter: the width of the circle

Circumference: the length of the perimeter of a circle

Theorem: a mathematical statement that can be proven

Proof: a set of justified steps that validate a mathematical statement

Lemma: a previously proven mathematical statement used to prove another mathematical statement

Corollary: a theorem that fallows another theorem

Hypothesis: the if part of an “if-then” statement

Line segment: a line with two endpoints

Ray: a line with one end point

Linear pair of angles: two angles that share a side and whose sum equals 180

Complementary angles: two angles who create a right triangle

Supplementary angles: two angles whose sum equals 180

Conjecture: a mathematical statement that isn’t proven

Reflexive Property: everything is congruent to itself. AB=AB

CPCTC: Corresponding Parts of Congruent Triangles are Congruent

Axiom 1: To draw a straight line from any point to any point.

Axiom 2: To produce a straight line continuously in a straight line.

Axiom 3: To describe a circle with any centre and radius.

Axiom 4: That all right angles are congruent.

Euclid’s Fifth Axiom: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

SAS Axiom: If two triangles have a pair of sides on one triangle congruent to a pair of sides on the other triangle and the included angles are congruent then the triangles are congruent.

Axiom 2: To produce a straight line continuously in a straight line.

Axiom 3: To describe a circle with any centre and radius.

Axiom 4: That all right angles are congruent.

Euclid’s Fifth Axiom: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

SAS Axiom: If two triangles have a pair of sides on one triangle congruent to a pair of sides on the other triangle and the included angles are congruent then the triangles are congruent.

Common Notion 1: Things which equal the same thing also equal each other.

Common Notion 2: If equals are added to equals then the wholes are equal.

Common Notion 3: If equals are subtracted from equals the remainders are equal.

Common Notion 4: Things which coincide with one another equal one another.

Common Notion 5: The whole is greater than the part.

The term*equal* in the Common Notions needs some explanation. We are very familiar with the idea of equality of numbers and thus what equals means in arithmetic and algebra. In this context the Common notions are straight forward. When we are talking about two geometric objects being equal we need to use a bit more care. We usually use the phrase *congruent* when talking about geometric objects that are *the same* in some sense. Thus congruent would be another use of the word equal in the Common Notions. If we are talking about a line segment with endpoints A and B we denote it as . The length of segment is denoted *AB.* So is an object while *AB *is a number. We say that two line segments and are congruent and write if and only if *AB=CD.* In this context Common Notion 1 would mean if and then . Similarly we say that two angles are congruent if their measures are equal. We denote this by if and only if .

Common Notion 2: If equals are added to equals then the wholes are equal.

Common Notion 3: If equals are subtracted from equals the remainders are equal.

Common Notion 4: Things which coincide with one another equal one another.

Common Notion 5: The whole is greater than the part.

The term

Powered by

✕