Geometry Definitions, Postulates, Axioms, Theorems and Corollaries – Flashcards
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Def. of Space
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The set of all points.
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Def. of Collinear Points
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Points that lie on the same line.
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Def. of Coplanar Points
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Points that lie on the same plane.
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Def. of Intersection
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A set of points that lie on 2 figures.
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Def. of Segment
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Consists of two endpoints and all points that in between.
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Def. of Ray
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Consists of an endpoint and a line going away from the endpoint.
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Def. of Opposite Rays
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Rays that share an endpoint and goes in opposite directions.
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Ruler Post.
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It shows that points on a line can be paired with real numbers in such a way that 2 points can have coordinates 0 & 1 and also guarantees unshowed points exists.
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Segment Addition Post.
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If B is between A & C, then AB + BC = AC.
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Def. of Congruent Segments
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Segments that have the same length.
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Def. of Midpoint of a Segment
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Point that divides segment into two congruent segments.
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Def. of Bisector of a Segment
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Line, segment, ray, or plane that intersect segment at midpoint.
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Def. of Angle
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Figure formed by 2 sides (rays, lines, segments) that have the same endpoint.
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Def. of Congruent Angles
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Angles that have same measures.
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Def. of Adjacent Angles
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Two angles in a plane that have a common vertex and a common side but no common interior points.
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Def. of Angle Bisector
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Ray that divides the angle into two congruent adjacent angles.
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Protractor Post.
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All rays drawn from O on Line AB can be paired with real numbers from 0 to 180.
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Angle Addition Post.
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If point B lies in the interior of Angle AOC, then the measure of Angle AOB + the measure of Angle BOC = the measure of Angle AOC. Special case: when one angle and another angle adds up to 180 degrees.
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Point Line Post.
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Through any 2 points there is exactly 1 line.
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Line Point Post.
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A line contains at least 2 points.
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Point Plane Post.
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Through 3 noncollinear points there is exactly 1 plane.
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Plane Point Post.
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A plane contains at least 3 noncollinear points.
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Space Post.
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Space contains at least 4 noncollinear points.
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Line in Plane Post.
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If 2 points lie in a plane, then the line through them lies in the plane.
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Plane Intersection Post.
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If 2 planes intersect, then they intersect in exactly 1 line.
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Line Intersection Thm.
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If 2 lines intersect, then they intersect in exactly 1 point.
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Line and Point in Plane Thm.
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A line and a point not on the line lie in exactly 1 plane.
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2 Lines in Plane Thm.
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2 lines that intersect lie in exactly 1 plane.
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Line Plane Intersection Thm.
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If a plane and line not in the plane intersect, then they intersect in exactly 1 point.
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Def. of Conditional
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If p (hypothesis), then q (conclusion) or p only if q.
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Def. of Biconditional
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p if and only if q, which means if p, then q and if q, then p.
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Def. of Converse
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If q, then p.
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Reflexive Axiom
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a = a
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Addition Axm.
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If a = b, then a + c = b +c. Also, if a = b and c = d, then a + c = b + d.
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Multiplication Axm.
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If a = b, then ac = bc.
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Substitution Axm.
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If a = b, then they may be substituted for each other.
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Midpoint Thm.
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If M is the midpoint of Segment AB, then AM = 1/2AB and MB = 1/2AB.
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Angle Bisector Thm.
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If Ray BX is the bisector of Angle ABC, then the measure of Angle ABX = 1/2 the measure of Angle ABC and the measure of Angle XBC = 1/2 the measure of Angle ABC.
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Def. of Complementary Angles
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2 angles whose measures have the sum of 90 degrees.
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Def. of Supplementary Angles
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2 angles whose measures have the sum of 180 degrees.
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Def. of Vertical ANgles
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2 angles that have sides that are opposite rays of each other.
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Vertical Angle Thm.
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Vertical angles are congruent.
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Def. of Perpendicular Lines
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Lines that form 1 right (90 degrees) angle
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Perpendicular Lines Cor.
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Perpendicular lines form 4 right angles.
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Complementary Angles Thm.
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If 2 angles are complements of congruent angles (or of the same angle), then the 2 angles are congruent.
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Supplementary Angles Thm.
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If 2 angles are supplements of congruent angles (or of the same angle), then the 2 angles are congruent.
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Def. of Circle
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The set of points in a plane at a given distance from a given point in that plane.
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Def. of Center
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The given point of the circle.
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Def. of Radius
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The given distance of the circle.
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Def. of Chord
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Segment whose endpoints lie on circle.
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Def. of Diameter
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Chord that contains center.
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Def. of Secant
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Line that contains chord.
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Def. of Tangent
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Line in plane of circle that intersects circle in 1 pt.
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Def. of Congruent Circles
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Circles that have congruent radii.
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Def. of Concentric Circles
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Circles that have same center.
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Def. of Circumscribed Circle
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A circle is circumscribed about a polygon when each vertex of the polygon lies on the circle.
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Def. of Inscribed Circle
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A circle is inscribed to a polygon when each side of the polygon is tangent to the circle.
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Def. of Sphere
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A sphere with a given point at the center and a radius is the set of all points in space at a distance of the radius from the center.
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Congruent Radii Cor.
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All radii of a circle are congruent.
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Circle Tangency Thm.
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If a line is tangent to a circle, the it is perpendicular to the radius at the point of tangency.
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Tangent Segment Cor.
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Tangents to a circle from a point are concruent
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Conv. Circle Tangency Thm.
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If a line in the plane of a circle is perpendicular to a radius at its outer endpoint, the the line is tangent to the circle.
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Def. of Common Internal Tangents
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A tangent that intersects the segment joining the centers of two coplanar circles.
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Def. of Common External Tangents
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A tangent that does not intersect the segment joining the centers of two coplanar circles.
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Def. of Externally Tangent Circles
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Two coplanar circles, neither of which is inside the other, that are tangent to the same line at the same point.
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Def. of Internally Tangent Circles
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Two coplanar circles, one of which is inside the other, that are tangent to the same line at the same point.
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Def. of Central Angle
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An angle of a circle with its vertex at the center of the circle.
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Def. of Minor Arc
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An arc made from a central angle that has an angle degree measure of less than 180.
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Def. of Major Arc
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An arc made from a central angle that has an angle degree measure of more than 180.
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Def. of Semicircle
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The arc made by a diameter.
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Def. of Adjacent Arcs
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Arcs that share one common endpoint.
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Def. of Congruent Arcs
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Arcs on the same circle or congruent circles that have the same measure.
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Arc Addition Post.
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The measure of the arc formed by two adjacent arcs it the sum of the measures of these two arcs.
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Congruent Arcs and Angles Thm.
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In the same circle or in congruent circles, two minor arcs are congruent if and only if their central angles are congruent.
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Congruent Arcs and Chords Thm.
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In the same circle or in congruent circles, congruent arcs have congruent chords and congruent chords have congruent arcs.
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Chord Perpendicular Bisector Thm.
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A diameter that is perpendicular to a chord bisects the chord and its arc.
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Chord Distance Thm.
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In the same circle or in congruent circles, chords equally distant from the center(s) are congruent and congruent chords are equally distant from the center(s).
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Def. of Inscribed Angle
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An angle whose vertex is on a circle and whose sides contain chords of the circls.
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Inscribed Angle Thm.
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The measure of an inscribe angle is equal to half the measure of its intercepted arc.
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Congruent Inscribed Angles Cor.
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If two inscribed angles intercept the same arc, then the angles are congruent.
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Right Inscribed Angle Cor.
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An angle inscribed in a semicircle is a right angle.
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Inscribed Quad Cor.
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If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.
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Chord and Tangent Angle Thm
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The measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc.
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Half of Sum Angle Thm.
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The measure of an angle formed by two chords that intersect inside a circle is equal to half the sum of the measures of the intercepted arcs.
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Half of Difference Angle Thm
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The measure of an angle formed by two secants, two tangents, or ta secant and a tangent drawn from a point outside a circle is equal to half the difference of the measures of the intercepted arcs.
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Chord Segment Thm.
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When two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord.
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Secant Segment Thm.
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When two secant segments are drawn to a circle from an external point, the product of one secant segment and its external segment equals the product of the other secant segment and its external segment.
