Download Euclidean Geometry Postulates_ Theorem_ Definitions Only

Euclidean Geometry Post Theorem Def only.notebook
March 13, 2009
Euclidean Geometry
Undefined terms: point, line, plane
A postulate is a statement whose truth is accepted without proof.
Postulates
Reflexive postulate: Any quantity is equal to itself (a=a). Symmetric postulate: An equality may be expressed in either order
(If a=b then b=a).
Transitive postulate: If quantities are equal to the same quantity, they are equal to each other (If a=b and b=c then a=c).
Substitution postulate: A quantity may be substituted for its equal in any expression.
The whole is equal to the sum of its parts.
If equals are added to equals the results are equal.
If equals are subtracted from equals the results are equal.
If equals are multiplied by equals the results are equal.
If equals are divided by equals the results are equal.
Squares of equal quantities are equal.
Halves of equal quantities are equal.
Positive square roots of equal quantities are equal.
Definitions
Collinear points are points that lie on the same line.
Coplanar points are points that lie in the same plane.
A midpoint is the point that divides a segment into two congruent segments,each one­
half the length of the original segment.
A segment bisector passes through the midpoint and divides a segment into two congruent segments.
An angle bisector is the ray that divides an angle into two congruent angles.
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Euclidean Geometry Post Theorem Def only.notebook
March 13, 2009
Definitions
Perpendicular Lines are two lines that intersect and form right angles.
An isosceles triangle is a triangle having two congruent sides.
A median of a triangle is a segment from a vertex to the midpoint of the opposite side.
An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side.
Vertical angles are non­adjacent angles formed by intersecting lines.
Complementary angles are two angles whose sum is 90ο.
Supplementary angles are two angles whose sum is 180ο.
Two polygons are congruent if and only if (iff) their corresponding angles and sides are congruent.
(Corresponding parts of congruent triangles are congruent.)
A theorem is a statement that is proved by deductive reasoning.
Theorems
All right angles are congruent.
All straight angles are congruent.
Complements of the same angle or congruent angles are congruent.
Supplements of the same angle or congruent angles are congruent.
If two adjacent angles form a linear pair(a straight line) then they are supplementary angles.
If two adjacent angles form a right angle then they are complementary angles.
Vertical angles are congruent.
Postulates Pg 135
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Euclidean Geometry Post Theorem Def only.notebook
March 13, 2009
Triangle Congruence
If two sides and the included angle of a triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
If two angles and the included side of a triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
If three sides of a triangle are congruent to three sides of another triangle, then the triangles are congruent.
If two angles and a side of a triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
If the hypotenuse and a leg of a right triangle are congruent to the corresponding parts of another right triangle then the triangles are congruent.
Corresponding parts of congruent triangles are congruent.
The Isosceles Triangle Theorem
If two sides of a triangle are congruent then the angles opposite those sides are congruent ( Base angles of an isosceles triangle are congruent).
Corollaries
The bisector of the vertex angle of an isosceles triangle bisects the base.
The bisector of the vertex angle of an isosceles triangle is
perpendicular to the base.
The median to the base of an isosceles triangle bisects the vertex angle.
The median to the base of an isosceles triangle is perpendicular to the base.
Every equilateral triangle is
equiangular.
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Euclidean Geometry Post Theorem Def only.notebook
March 13, 2009
Perpendicular Lines
Def: Perpendicular lines are two lines that intersect and form right angles.
Th: If two intersecting lines form congruent adjacent angles then the lines are perpendicular
Perpendicular Bisectors
Def: A perpendicular bisector of a segment is a line that is perpendicular to the segment at it's midpoint.
Th: If two points are equidistant from the endpoints of a line segment, the points determine the perpendicular bisector of the segment.
Th: Any point on the perpendicular bisector of a line segment is equidistant from the endpoints of the line segment.
The whole is equal to the sum of its parts.
Time and again these men and women struggled and sacrificed and worked till their hands were raw so that we might live a better life. They saw America as greater than the sum of our individual ambitions..
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Euclidean Geometry Post Theorem Def only.notebook
March 13, 2009
Inequality Postulates and Theorems
The whole is greater than any of its parts.
The shortest distance between two points is the line segment determined by the points.
Trichotomy postulate: Given any two quantities, a and b, either:
1) a > b
2) a < b
3) a = b
Transitive Postulate of Inequality: If a > b and b > c then a > c.
Substitution Postulate: A quantity may be substituted for its equal in any inequality.
If equals are added to or subtracted from unequals the results are unequal in the same order.
If unequals are multiplied or divided by positive equals the results are unequal in the same order (Inequality reverses when the equals are negative).
TH: The sum of the lengths of two sides of a triangle is greater than the length of the third side.
TH: An exterior angle of a triangle is greater than either nonadjacent
(remote) interior angle.
Inequalities
TH. In a triangle, the larger angle is opposite the longer side.
TH. In a triangle, the longer side is opposite the larger angle.
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Euclidean Geometry Post Theorem Def only.notebook
March 13, 2009
6­2 Proving Lines Parallel Def. Parallel Lines are coplanar lines that either have no points in common or have all points in common.
Postulate: Through a point not on a given line, there exists one and only one line parallel to the given line.
Th: If a line intersects one of two parallel lines, it intersects the other.
Th: If two coplanar lines, cut by a transversal form congruent alternate interior (exterior) angles, then the lines are parallel.
Th: If two coplanar lines, cut by a transversal form congruent corresponding angles, then the lines are parallel.
Th: If two coplanar lines, cut by a transversal form supplementary,same side interior (exterior) angles, then the lines are parallel.
Th: If two coplanar lines are perpendicular to the same line, then they are parallel.
Th: Two lines parallel to the same line are parallel to each other.
Th: Two lines perpendicular to the same line are parallel to each other.
6­3 Properties of Parallel Lines
If two parallel lines are cut by a transversal, then the alternate interior (exterior) angles are congruent.
If two parallel lines are cut by a transversal, then the corresponding angles are congruent.
If two parallel lines are cut by a transversal, then the same side interior (exterior) angles are supplementary.
If a line is perpendicular to one of two parallel lines, then it is perpendicular to the other.
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Euclidean Geometry Post Theorem Def only.notebook
March 13, 2009
6­4 Angles of Triangles
Th: The sum of the degree measures of the angles of a triangle is 180ο.
Cor: If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.
Cor: The acute angles of a right triangle are complementary.
Cor: Each acute angle of an isosceles right triangle measures 45ο.
Cor: Each angle of an equilateral triangle measures 60ο.
Cor: The sum of the degree measures of the angles of a quadrilateral is 360ο.
6­5 Angle ­ Angle ­ Side
Th: If two angles and a side of a triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
6­6 Converse of Isosceles Triangle Theorem
If two angles of a triangle are congruent, then the sides opposite these angles are congruent.
Cor: If a triangle is equiangular, then it is equilateral.
6­7 Hypotenuse Leg
If the hypotenuse and leg of one right triangle are congruent to the hypotenuse and leg of another right triangle then the triangles are congruent (Hy. Leg or HL).
* Must state that the triangles are right triangles in the proof.
Th: The measure of an exterior angle of a triangle is equal to the sum of the measures of the non­ adjacent interior angles.
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Euclidean Geometry Post Theorem Def only.notebook
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Chapter 7 Quadrilaterals
7­1 & 7­2 Properties of Parallelograms
Def. A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.
Th. A diagonal of a parallelogram divides the parallelogram into two congruent triangles.
Cor: The opposite sides of a parallelogram are congruent.
Cor. The opposite angles of a parallelogram are congruent.
Cor. The consecutive angles of a parallelogram are supplementary.
Cor. The diagonals of a parallelogram bisect each other.
7­3 Ways to Prove a Quad. is a Parallelogram
Def. If both pairs of opposite sides of a quad. are parallel then the quad is a parallelogram.
Th. If both pairs of opposites sides of a quad. are congruent then the quad. is a parallelogram.
Th. If both pairs of opposite angles of a quad. are congruent then the quad is a parallelogram.
Th. If the diagonals of a quad. bisect each other then the quad. is a parallelogram.
Th. If one pair of sides of a quad. are congruent and parallel then the quad. is a parallelogram.
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Euclidean Geometry Post Theorem Def only.notebook
March 13, 2009
7­4 The Rectangle Def. A rectangle is a parallelogram containing a right angle.
Th. All angles of a rectangle are right angles
(A rectangle is equiangular).
Th. The diagonals of a rectangle are congruent.
Ways to Prove a Quad. is a Rectangle.
If a parallelogram contains a right angle then the parallelogram is a rectangle.
If a parallelogram has congruent diagonals then the parallelogram is a rectangle.
If a quad. is equiangular then the quad. is a rectangle.
Rhombii (Plural of Rhombus)
A rhombus is a parallelogram having two consecutive sides congruent.
All sides of a rhombus are congruent.
The diagonals of a rhombus are perpendicular.
The diagonals of a rhombus bisect its angles.
Ways to prove a quad. is a rhombus
If a parallelogram has two congruent,consecutive sides then the parallelogram is a rhombus.
If the diagonals of a parallelogram are perpendicular then the parallelogram is a rhombus.
If a quadrilateral is equilateral then the quadrilateral is a rhombus.
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Euclidean Geometry Post Theorem Def only.notebook
March 13, 2009
Squares
Def. A square is a rectangle having two consecutive sides congruent.
Th. A square is a regular (equilateral and equiangular) quadrilateral. Proving a Quadrilateral is a Square
If a rectangle has two consecutive sides congruent then it is a square.
If a rhombus contains a right angle then it is a square.
Trapezoids
A trapezoid is a quadrilateral having exactly one pair of parallel sides.
An isosceles trapezoid is a trap. in which the non­parallel sides are congruent.
Th. The diagonals of an isosceles trapezoid are congruent.
Th. Each pair of base angles of an isosceles trapezoid are congruent. Def. A median of an isosceles trapezoid is the line segment joining the midpoints of the nonparallel sides.
TH. The median of an isosceles trap. is:
a) parallel to the bases.
b) equal to the average of the lengths of the bases.
Proving a Trap. is Isosceles
Def. If the non­parallel sides of a trapezoid are congruent then it is an isosceles trapezoid.
Th. If the base angles of a trapezoid are congruent then it is isosceles.
Th. If the diagonals of a trapezoid are congruent then it is isosceles.
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Euclidean Geometry Post Theorem Def only.notebook
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