Properties and Theorems for Geometry

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Rectangle: quadrilateral with four right angles
Theorem 5-12: the diagonals of a rectangle are congruent
Theorem 3-13: the sum of the measures of the angles of a convex polygon with n sides is (n-2)180
Theorem 4-1: opposite sides of a parallelogram are congruent
Isosceles Triangle: at least two sides congruent
Ways to prove two lines parallel: in a plane show that both lines are perpendicular to a third line
Theorem 5-13: the diagonals of a rhombus are perpendicular
Postulate 7: through any three points there is at least one plane, and through any three non collinear points there is exactly one plane.
Theorem 3-14: the sum of the the measures of the exterior angles of any convex polygon, one angle at each vertex, is 180
HL theorem: if the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another triangle, then the triangle are congruent
Ways to prove right triangles congruent: HL
Theorem 3-8: through a point outside a line, there is exactly one line parallel to the given line
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 3-2: if two parallel lines are cut by a transversal, then alternate interior angles are congruent
Theorem 5-4: if both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram
A way to prove two segments or two angles are congruent: prove that the triangles are congruent
Corollary 2: each angle of an equiangular triangle has measure of 60
SAS postulate: if 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
Ways to prove that a quadrilateral is a parallelogram: show that BOTH pairs of opposite sides are parallel
Ways to prove two lines parallel: show that a pair of alternate interior angles are congruent
Theorem 3-3: if two parallel lines are cut by a transversal, then same-side interior angles are supplementary
Obtuse triangle: one obtuse angle
Transversal: lines that intersects two or more coplanar lines in different points
Corollary 4-3: the bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint
Theorem 4-2: if two angles of a triangle are congruent, then the sides opposite those angles are congruent
Theorem 4-5: if a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment
Postulate 9: if two planes intersect, then their intersection is a line.
Congruent: having the same size and shape
Ways to prove two lines parallel: show that both lines are parallel to a third line
Parallel lines: coplanar lines that do not intersect
Theorem 2-3: Vertical angles are congruent
Midpoint Theorem: if M is the midpoint of segment AB, then AM=1/2AB and MB=1/2AB
Theorem 5-9: if three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal
Theorem 5-8: if two lines are parallel, then all points on one line are equidistant from the other line
Theorem 5-11: the segment that joins the midpoints of two sides of a triangle is parallel to the third side and is half as long as the third side
Reasons Used in Proofs: Given info, Definitions, Postulates (and properties from algebra), Proven Theorems
Theorem 4-7: if a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle
Segment Addition Postulate: if B is between A and C, then: AB+BC=AC
Corollary 3: in a triangle, there can be at most one right angle or obtuse angle
A way to prove two segments or two angles are congruent: state that the two parts are congruent, using the reason CPCTC
Theorem 5-19: the median of a trapezoid is parallel to the bases and has a length equal to the average of the base lengths
Angle Addition Postulate: if angle AOC is a straight and and B is any point not on line AC, then angle AOB = angle BOC = 180
Corollary 1: if two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent
Theorem 2-6: if the exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary
Theorem 3-9: through a point outside a line, there is exactly one line perpendicular to the given line
Deductive Reasoning: conclusion based on accepted statements (definitions, postulates, previous theorems, corollaries, and given information) conclusion MUST be true if hypotheses are true
Polygon: many angles
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 a way that: ray OA is paired with 0, and ray OB with 180, or, if ray OP is paired with X, and ray OQ with y, then angle POQ = l x-y l.
Theorem 3-11: the sum of the measures of the angles of a triangle is 180
Corollary 4-1: an equilateral triangle is also equiangular
Postulate 11: if two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel
Isosceles Triangle Theorem: if two sides of a triangle are congruent, then the angles opposite those sides are congruent
Theorem 5-15: the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices
Ways to prove that a quadrilateral is a parallelogram: show that the diagonals bisect each other
Theorem 2-7: if two angles are supplements of congruent angles (or of the same angle) then the two angles are congruent
Ways to prove two lines parallel: show that a pair of corresponding angles are congruent
Equilateral triangle: all sides congruent
Altitude: perpendicular segment from a vertex to the line that contains the opposite side in a triangle
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 5-14: each diagonal of a rhombus bisects two angles of the rhombus
Angle Bisector Theorem: If ray BX is the bisector of angle ABC, then angle ABX = 1/2angleABC and angle XBC = 1/2 angle ABC
Corollary 4-4: an equiangular triangle is also equilateral
Skew lines: noncoplanar lines
Ways to prove that a quadrilateral is a parallelogram: show that BOTH pairs of opposite sides are congruent
Theorem 1-2: through a line and a point not in the line there is exactly one plane
Median: segment from a vertex to the midpoint of the opposite side in a triangle
Corollary 4: the acute angles of a right triangle are complementary
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
Ange Addition Postulate: If point B lies in the Interior of angle AOC, then angle AOB + angle BOC = angle AOC
Same-side interior angles: two interior angles on the same side of the transversal
Parallelogram: quadrilateral with both pairs of opposite sides parallel
Ways to prove two lines parallel: show that a pair of same-side interior angles are supplementary
Triangle: figure formed by three segments joining three noncollinear points
Postulate 6: through any two points there is exactly one line
Theorem 3-7: in a plane two liens perpendicular to the same lines are parallel
Ways to prove that a quadrilateral is a parallelogram: show that ONE pair of opposite sides are both congruent and parallel
Rhombus: quadrilateral with four congruent sides
Perpendicular bisector: line (or ray or segment) that is perpendicular to the segment at its midpoint
Scalene Triangle: no sides congruent
Theorem 4-6: if a point is equidistant from the endpoints of a segment, then the point is on the perpendicular bisector of the segment
SSS postulate: if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent
Theorem 1-3: If two lines intersect, then exactly one plane contains the lines
Right triangle: one right angle
Postulate 5: a line contains at least two points; a plane contains at least three points not all in one line; space contains at least four points not all in one plane
Theorem 3-1: if two parallel planes are cut by a third plane, then the lines of intersection are parallel
Ruler Postulate: the pints on a line can be paired with the real numbers in such a way that any two points can have coordinates 0 and 1
Trapezoid: quadrilateral with exactly one pair of parallel sides
Theorem 5-5: if one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram
Ways to prove that a quadrilateral is a parallelogram: show that both pairs of opposite angles are congruent
Theorem 5-16: if an angle of a parallelogram is a right angle, then the parallelogram is a rectangle
Postulate 10: if two parallel lines are cut by a transversal, then corresponding 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
Theorem 5-7: if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram
A way to prove two segments or two angles are congruent: identify two triangles in which the two segments or angles are corresponding parts
Corresponding angles: two angles in corresponding positions relative to the two lines
Equiangular: all angles congruent
Theorem 5-3: diagonals of a parallelogram bisect each other
Inductive Reasoning: reasoning that is widely used in science and everyday life
Theorem 1-1: if two lines intersect, then they intersect in exactly one point
Theorem 5-18: base angles of an isosceles trapezoid are congruent
Theorem 5-6: if both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram
Theorem 3-10: two lines parallel to a third line are parallel to each other
Inductive Resoning: Conclusion based on several past observations. conclusion is PROBABLY true, but not necessarily true
Isosceles trapezoid: trapezoid with congruent legs
ASA postulate: if two angles and the included sides of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent
Theorem 3-6: if two lines are cut by a transversal and same-side interior angles are supplementary, then the lines are parallel
Corollary 4-2: an equilateral triangle has three 60° angles
Theorem 3-4: if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other ones also
Alternate interior angles: two nonadjacent interior angles on opposite sides of the transversal
Theorem 5-17: if two consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus
Theorem 3-5: if two lines are cut by a transversal and alternate interior angles are congruent, then the liens are parallel
Postulate 10: if two parallel lines are cut by a transversal, then corresponding angles are congruent
Theorem 5-2: opposite sides of a parallelogram are congruent
Theorem 2-5: if two lines form congruent adjacent angles, then the lines are perpindicular
Convex Polygon: polygon such that no line containing a side of the polygon contains a point in the interior of the polygon
Postulate 8: if two points are in a plane, then the line that contains the points is in that plane
Theorem 2-4: if two lines are perpendicular, then they from congruent adjacent angles
Acute triangle: three acute angles
Ways to prove triangles congruent: SSS, SAS, ASA, AAS
Square: quadrilateral with four right angles and four congruent sides

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