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Geometry Postulates, Theorems, and Proofs
About this deck
By: Christopher Hall
Created: 2011-09-15
Size: 162 flashcards
Views: 61
Created: 2011-09-15
Size: 162 flashcards
Views: 61
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Ruler Postulate
1. The points on a line can be paired with the real numbers such a way that any two points can have coordinates 0 and 1
2.Once a coordinate system has been chosen in this way, the distance between any two points equals the absolute value of the difference of their coordinates.
Segment Addition Postulate
If B a is between A and C, then
AB+BC=AC
Protractor Postulate
On line AB in a given plane, choose any point O between A and B. Consider ray OA and ray OB and all the rays that can be drawn from O on one side of line AB. These rays can be paired with the real numbers from 0 to 180 in such away that:
a. Ray OA is Paired with 0 and ray OB with 180.
b. If ray OP is paired with x, and ray OQ with y, then the m∠POQ= the absolute value of x-y.
Angle Addition Postulate
If point B lies in the interior of ∠AOC then
m∠AOB + m∠BOC = m∠AOC
If ∠AOC is a straight line and B is any point not on line AC, then
m∠AOB + m∠BOC = 180
Postulate 5
A line contains at least two points: a plane contains at least thee points not all one one line; space contains at least four points not all in one plane.
Postulate 6
Through any two points there is exactly one line.
Postulate 7
Through any three points there is at least one plane, and through any three noncollinear points there is exactly one plane.
Postulate 8
If two points are in a plane, then the line that contains the points is in that plane.
Postulate 9
If two planes intersect, then their intersection is a line
Postulate 10
If two parallel lines are cut by a transversal, then corresponding angles are congruent.
Postulate 11
If two lines are cut by a tansversal and corresponding angles are congruent, then the lines are parallel.
SSS Postulate
If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
SAS Postulate
Iif two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
ASA Postulate
If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.
Theorem 1-1
If two lines intersect, then they intersect in exactly one point.
Theorem 1-2
Through a line and a point not in the line there is exactly one plane.
Theorem 1-3
If two lines intersect, then exactly one plane contains the lines.
Midpoint Theorem
If M is the midpoint of line segment AB, then AM=1/2 AB and MB=1/2 AB
Angle Bisector Theorem
If ray BX is the bisector of angle ABC, then m∠ABX=1/2 m∠ABC and m∠XBC=1/2m∠ABC
Theorem 2-3
Vertical angles are congruent
Theorem 2-4
If two lines are perpendicular, then they form congruent adjacent angles.
Theorem 2-5
If two lines form congruent adjacent angles, then the lines are perpendicular.
Theorem 2-6
If the exterior sides of two adjacent acute angles are perpendiculat, the the angles are complementary.
Theorem 2-7
If two angles are supplements of congruent angles (or of the same angle), then the two angles are congruent.
Theorem 2-8
If two angles are complements of congruent angles (or of the same angle), then the two angles are congruent.
Acute Angle
An angle with a measure between 0 and 90.
Adjacent angles
Two angles in plane that have a common vertex and common side but no common interior points
Angle
A figure formed by two rays that have the endpoint. the two rays are called the sides of the angle. their common endpoint is the vertex.
Axiom
A statement that is accepted without proof
Biconditional
A statement that contains the words if and only if.
Bisector of an Angle
The ray that divides the angle into two congruent adjacent angles.
Bisector of a Segment
A line, segment, ray, or plane that intersects the segment at its midpoint.
Collinear Points
Points all in one line.
Complementary Angles
Two angles whose measures have the sum 90.
Congruent Angles
Angles that have equal measures.
Congruent Segments
Segments that have equal lengths.
Converse
The converse of the statement if p,the q is the statement if q, then p
Coplanar Points
Points all in one plane.
Counterexample
An counterexample used to prove that an if-then statement is false. For that counterexample, the hypothesis is true and the conclusion is false.
Deductive Reasoning
Proving statements by reasoning from accepted postulates, definitions, theorems, and given information.
If-Then Statement
A statement whose basic form is If p, then q. Statement p is the hypotheses and statement q is the conclusion.
Intersection of two Figures
The set of pints that are in both figures.
Length of a Segment
The distance between its endpoints.
Measure of an Angle
A unique positive number, less than or equal to 180, that is paired with the angle.
Midpoint of a Segment
The pint that divides the segment into two congruent segments.
Obtuse Angle
An angle with a measure between 90 and 180
Perpendicular Lines
Two lines that intersect to form right angles.
Postulate
A statement that is accepted without proof.
Ray
The ray AC consists of segment AC and all other points P such that C is between A and P. The point named first, here A, is the endpoint of ray AC.
Right Angle
An angle with a measure 90.
Segment of a Line
Two points on the line and all points between them. The two points are called the endpoints of the segment.
Straight Angle
An angle with a measure of 180.
Supplementary Angles
Two angles whose measures have the sum 180.
Theorem
A statement that can be proved.
Vertical Angles
Two angles whose sides form two pairs of opposite rays.
∠'s 1 and 2 are vertical angles, as are ∠'s 3 and 4
AA Similarity Postulate
If two angles of one triangle are congruent to two angles of another triangle, the the triangles are similar.
Arc Addition Postulate
The measure of the arc formed by two adjacent arcs is the sum of the measures of these two arcs.
Postulate 17
The area of a square is the square of the length of side.
Area Congruence Postulate
If two figures are congruent, then they have the some area.
Area Addition Postulate
The area of region is the sum of the areas of its non-overlapping parts.
Theorem 3-1
If two parallel planes are cut by a third plane, then the lines of intersection are parallel.
Theorem 3-2
If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
Theorem 3-3
If two parallel lines are cut by a transversal, the same-side interior angles are supplementary.
Theorem 3-4
If a transversal is perpendicular to one of two parallel line, then it is perpendicular to the other one also.
Theorem 3-5
If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel.
Theorem 3-6
If two lines are cut by a transversal and same-side interior angles are supplementary.
Theorem 3-7
In a plane two lines perpendicular to the same line are parallel.
Theorem 3-8
Through a point outside a line, there is exactly one parallel to the the given line.
Theorem 3-9
Through a point outside a line, there is exactly one line perpendicular to the given line.
Theorem 3-10
Two lines parallel to a third line are parallel to each other.
Theorem 3-11
The sum of the measures of the angles of a triangle is 180.
3-11 Corollary 1
If two angles of one triangle are congruent to two angles of another triangle then the third angles are congruent.
3-11 Corollary 2
Each angle of a equiangular triangle has the measure of 60°.
3-11 Corollary 3
In a triangle, there can be at most one right or obtuse angle.
3-11 Corollary 4
The acute angles of a right triangle are complementary.
Theorem 3-12
The measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles.
Theorem 3-13
the sum of the measures of the angles of a convex polygon with n sides is:
(n-20)180.
Theorem 3-14
The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360.
Theorem 4-1 The Isosceles Triangle Theorem
If two sides of a triangle are congruent then the angles opposite those sides ate congruent.
4-1 Corollary 1
An equilateral triangle is also equiangular.
4-1 Corollary 2
An equilateral triangle has three 60° angles.
4-1 Corollary 3
The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at it's midpoint..
Theorem 4-2
If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
4-2 Corollary
An equiangular triangle is also equilateral.
AAS Theorem
If two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
HL Theorem
If the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, the the triangles are congruent.
Theorem 4-5
If a point lies on the perpendicular bisector of a segment, then the point is equidistant form the endpoints of the segment
Theorem 4-6
If a point is equidistant from the endpoints of a segment, the the point lies on the perpendicular bisector of the segment.
Theorem 4-7
If a pint lies on the bisector of an angle, then the point is equidistant from the sides of the angle.
Theorem 4-8
If a point is equidistant from the sides of an angle, the the point lies on the bisector of the angle.
Alternate Interior Angles
Two nonadjacent interior angles on opposite sides of a transversal.
Angles 1 and 2 are alternate interior angles.
Auxiliary Line
A line (or ray or segment) added to a diagram to help in a proof.
Congruent Figures
Figures having the sames size and shape.
Congruent Polygons
Polygons whose vertices can be matched up so that the corresponding parts (angles and sides) of the polygons are congruent.
Corollary of a Theorem
A statement that can be proved easily by applying the theorem.
Equiangular Triangle
A triangle with all angles congruent.
Equilateral Triangle
A triangle with all sides congruent.
Exterior Angle of a Triangle
The angle formed when one side of the triangle is extend.
∠DAC is an exterior angle of ∧ ABC, and ∠s B and C are remote interior angles with respect to ∠DAC.
Exterior angle is also applied to other polygons.
Hypotenuse
In a right triangle the side opposite the right angle. The other two sides are called legs.
Inductive Reasoning
A king of reasoning in which the conclusion is based on several pas observations
Isosceles Triangle
A triangle with at least two sides congruent.
Legs of an Isosceles Triangle
The two congruent sides. The third side is the base.
Obtuse Triangle
A triangle with one obtuse angle
Parallel Line and Plane
A line and a plane that do not intersect.
Parallel Lines
Coplanar Lines that do not intersect.
Parallel Planes
Planes that do not intersect.
Perpendicular Line and Plane
A line and a plane are perpendicular if and only if they intersect and the line is perpendicular to all lines in the plane that pass through the point of intersection.
Perpendicular Lines
Two lines that intersect to from right angles.
Right Triangle
A triangle with one right angle.
Same-side Interior Angles
Two interior angles on the same side of a transversal
Scalene Triangle
A triangle with no sides congruent.
Skew Lines
Lines that are not coplanar.
Transversal
A line that intersects two or more coplanar lines in different points.
Triangle
The figure formed by three segments joining three noncollinear points.
Theorem 5-1
Opposite sides of a parallelogram are congruent.
Theorem 5-2
Opposite angles of a parallelogram are congruent
Theorem 5-3
Diagonals of a parallelogram bisect each other.
Theorem 5-4
If both pairs of opposite sides of a quadrilateral are congruent, the the quadrilateral is a parallelogram.
Theorem 5-5
If one pair of apposite sides of a quadrilateral are both congruent and parallel, the the quadrilateral is a parallelogram.
Theorem 5-6
If both pairs of opposite angles of a quadrilateral congruent, then the quadrilateral is a parallelogram.
Theorem 5-7
If the diagonals of a quadrilateral bisect each other the the quadrilateral is a parallelogram.
Theorem 5-8
If two lines are parallel, then all points on one line are equidistant from the other line.
Theorem 5-9
If three parallel lines cut off congruent segments on one transversal, the they cut off congruent segments on every transversal.
Theorem 5-10
A line that contains the midpoint of one side of a triangle and is parallel to another side passes through the midpoint of the third side.
Theorem 5-11
The segment that joins the midpoints of two sides of a triangle
(1) is parallel to the third side
(2) is half as long as the third side.
Theorem 5-12
The diagonals of a rectangle are congruent.
Theorem 5-14
The diagonals of a rhombus are perpendicular.
Theorem 5-14
Each diagonal of a rhombus bisects two angles of the rhombus.
Theorem 5-15
The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices.
Theorem 5-16
If and angle of a parallelogram is a right angle, the the parallelogram is a rectangle.
Theorem 5-17
If two consecutive sids of a parallelogram are congruent, the the parallelogram is a rhombus.
Theorem 5-18
Bas angle of an isosceles trapezoid are congruent.
Theorem 5-19
The median of a trapezoid
(1) is parallel to the basses
(2) Has a length equal to the average of the base lengths
Isosceles trapezoid
A trapezoid with congruent legs.
Kite
A quadrilateral that has two pairs of congruent sides, but opposite sides are not congruent.
Parallelogram
A quadrilateral with both pairs of opposite sides parallel.
Rectangle
A quadrilateral with four right angles.
Rhombus
A quadrilateral with four congruent sides.
Square
A quadrilateral with four right angles and four congruent sides.
Trapezoid
A quadrilateral with exactly one pair of parallel sides, called bases. The other sides are called legs.
Theorem 6-1 The exterior Angle Inequality Theorem
The measure of an exterior angle of a triangle is greater then the measure of either remote interior angles.
6-3 Corollary 1
The perpendicular segment from a point to a line is the shortest segment from the point to the line.
6-3 Corollary 2
The perpendicular segment from a poiint to a plane is the shortest segment grom the point to the plane.
The Triangle Inequality
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
SAS Inequality Theorem
If two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first is larger then the included angle of the second, then the third side of the first triangle is longer thatn the third side of the second triangle.
SSS inequality Theorem
If two sides of one triangle are congruent to two sides of another triangle, but the third side of the first triangle is longer thant the third side of the second, then the included angle of the first triangle is larger than the included angle of the second.
Contrapositive of a conditional
The contrapositive of the statement If p, the q is the statement If q, then p
Indirect proof
A proof in which you assume temporarily that the conclusion is not true, and then deduce a contradiction
Inverse of a conditional
The inverse of the statement If p, then q is If not p, then not q.
Logically Equivalent statements
Statements that are either both true or both false.
Venn Diagroam
A circle diagram that may be used to represent a conditional.
SAS Similarity Theorem
If an angle of one triangle is congruent to an angle off another triangle and the sides including those angles are i proportion, then the triangles are similar.
SSS Similarity Theorem
If the sides of two triangles are in proportion, the the triangles are similar.
Triangle Proportionality Theorem
If a line parallel to one side of a trianlge intersects the other two sides, then it divides those sides proportionally
Triangle Proportionality Theorem Corollary
If three parallel lines intersect two transversals, the they divide the transversals proportionally.
Triangle Angle bisector Theorem
IF a ray bisects an angle of a triangle, then it divides the opposite side into segment proportional to the other two sides.
Golden Rectangle
A rectangle such that its length l and width w satisfy the equation l/w=(l+w)/w. The ration l:w is called the golden ratio.P
Proportion
An equation stating that two ratios are equal. the first and last terms are the exstremes; the middle terms are the means.
Ratio
The ratio of x to y (y≠0) is x/y and is sometimes written x: y.
Segments divided proportionally
AB and CD are divided proportionally if points L and M lie on AB and CD, respectively, and AL/LB=CM
About this deck
By: Christopher Hall
Created: 2011-09-15
Size: 162 flashcards
Views: 61
Created: 2011-09-15
Size: 162 flashcards
Views: 61
About StudyBlue
STUDYBLUE makes things that make you better at school.
Things like online flashcards with photos and audio.
Things like personalized quizzes and friendly reminders about when (and what) to study next.
Think of it as a digital backpack™: access to all of your study materials online and on your phone.
STUDYBLUE exists to make studying efficient and effective for every student, for free. Join us.
“Simply amazing. The flash cards are smooth, there are many different types of studying tools, and there is a great search engine. I praise you on the awesomeness.”
Dennis
Dennis