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Betweeness Theorem: If B is between A and C, then AB+BC=AC Linear Pair Theorem: If two angles form a linear pair, then they are supplementary Vertical Angle Theorem: If two angles are vertical angles, they have equal measures Parallel Lines and Slopes Theorem: Two nonverical lines are parallel if and only if they have the same slope Trasitivity of Parallelism Theorem: In a plane, if L is parallel to M, and M is parallel to N, then L is parallel to N Two Perpendiculars Theorem: If two coplanar lines L and M are each perpendicular to the same line, then they are parallel to each other Perpendicular to Parallels Theorem: In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other Perpendicular Lines and Slopes Theorem: Two nonvertical lines are perpendicular if and only if the product of their slopes is -1 Figure Reflection Theorem: If a figure is determined by certain points, then its reflection image is the corresponding figure determined by the reflection images of those points Perpendicular Bisector Theorem: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment Flip-Flop Theorem: If F and F' are points or figures and r(F)=F', then r(F')=F Segment Symmetry Theorem: A segment has exactly two symmetry lines: its perpendicular bisector and its line containing the segment Side Switching Theorem: If one side of an angle is reflected the line containing the angle bisector, its image is the other side of the angle Angle Symmetry Theorem: The line containing the bisector of the angle is a symmetry line of the angle Isosceles Triangle Symmetry Theorem: The line containing the bisectors of the vertex angle of an isosceles triangle is a symmetry line for the triangle Isosceles Triangle Theorem: If a triangle has two equal sides, then the angles opposite them are equal Kite Symmetry Theorem: The line containing the ends of a kite is a symmetry line for the kite Kite Diagonal Theorem: The symmetry diagonal of a kite is the perpendicular bisector of the other diagonal and bisects the two angles at the ends of the kite Rhombus Symmetry Theorem: Every rhombus has two symmetry lines: the bisectors of its diagonals Trapezoid Angle Theorem: In a trapezoid consecutive angles between a pair of parallel sides are supplementary Isosceles Trapezoid Symmetry Theorem: The perpendicular bisector of one base of an isosceles trapezoid is the perpendicular bisector of the other base and a symmetry line for the trapezoid Isosceles Trapezoid Theorem: In an isosceles trapezoid, the non-base sides are equal in measure Rectangle Symmetry Theorem: Every rectangle has two symmetry lines: the perpendicular bisectors of its bases Parallel Lines/Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then alternate interior angles are equal in measure Alternate Interior Angles/Parallel Lines Theorem: If two lines are cut by a transversal and form equal alternate interior angles, then the lines are parallel Quadrilateral Hierarchy Theorem: If a figure is of any type on the hierarchy, it is also of all types connected above it Triangle-Sum Theorem: The sum of the measures of the angles of a triangle is 180 degrees Quadrilateral-Sum Theorem: The sum of the measures of the angles of a convex quadrilateral is 360 degrees Polygon-Sum Theorem: The sum of the measures of the angles of a convex polygon of n sides is (n-2)180 ABCD Theorem: Every isometry preserves angle measure, betweenness, collinearity(lines), and distance(length of segments) Segment Congruence Theorem: Two segments are congruent if and only if they have the same length Angle Congruence Theorem: Two angles are congruent if and only if they have equal measures CPCF Theorem: If two figures are congruent, then any pair of corresponding parts is congruent SSS Congruence Theorem: If, in two triangles, three sides of one are congruent to three sides of the other, then the triangles are congruent SAS Congruence Theorem: If, in two triangles, two sides and the included angle of one, are congruent to two sides and the included anle of the other, then the triangles are congruent ASA Congruence Theorem: If, in two triangles, two angles and the included side of one, are congruent to two sides and the included angle of the other, then the triangles are congruent AAS Congruence Theorem: If, in two triangle, two angles and a non- included side of one, are congruent respectively to two angles and a non-included side of the other then the two triangles are congruent HL Congruence Theorem: If, in two right triangles, the hypotenuse and leg of one are congruent to the hypotenuse and leg of the other, then the two triangles are congruent SsA Congruence Theorem: If, in two triangles, two sides and the angle opposite the longer of the two sides in one are congruent respectively to two sides and the angle opposite the longer of the two sides in the other, then the two triangles are congruent. Properties of a Parallelogram Theorem: In any parallelogram each diagonal forms two congruent triangles, opposite sides are congruent, and the diagonals intersect at their midpoints Center of a Regular Polygon Theorem: In any regular polygon there is a point (its center) that is equidistant from all its verticies Sufficient Conditions for a Parallelogram Theorem: If, in a quadrilateral: (a) both pairs of opposite sides are congruent, or (b) both pairs of opposite angles are congruent, or (c) the diagonals bisect each other, or (d) one pair of sides is parallel and congruent, then the quadrilateral is a parallelogram SAS Inequality Theorem: If two sides of a triangle are congruent to two sides of a second triangle, and the measure of the included angle of the first triangle is less than the measure of the included angle of the second, then the third side of the first triangle is shorter than the third side of the second Pythagorean Theorem: In any right triangle with legs a and b and hypotenuse c, a(sq) + b(sq) = c(sq) Pythagorean Converse Theorem: If a triangle has sides of lengths a, b, and c, and a(sq) + b(sq) = c(sq), then the triangle is a right triangle Midpoint Connector theorem: The segment connecting the midpoints of two sides of a triangle is parallel to and half the length of the third side. Size Change Distance Theorem: Under a size change with magnitude k > 0, the distance between any two image points is k times the distance between their preimages Size Change Theorem: Under a size transformation: angles and their measures are preserved: betweenness is preserved; collinearity is preserved: and lines and their images are parallel Figure Size Change Theorem: If a figure is determined by certain points, then its size change image is the corresponding figure determined by the size change images of those points Means-Extremes Property: If a/b = c/d, then ad = bc Means Exchange Property: If a/b = c/d, then a/c = b/d Reciprocals Property: If a/b = c/d, then b/a = d/c Similar Figures Theorem: If two figures are similar, then: Corresponding angles are congruent Corresponding lengths are proportional Fundamental Theorem of Similarity: If G ~ G’ and k is the ratio of similtude, then: Perimeter (G’) = k times Perimeter (G) Area (G’) = k(sq) times Area(G) Volume (G’) = k(cu) times Volume (G) SSS Similarity Theorem: If the three sides of one triangle are proportional to the three sides of a second triangle, then the triangles are similar AA Similarity Theorem: If two triangles have two angles of one congruent to two angles of the other, then the triangles are similar SAS Similarity Theorem: If, in two triangles, the ratios of two pairs of corresponding sides are equal and ith included angles are congruent, then the triangles are similar. Side-Splitting Theorem: If a line intersects ray OP and ray OQ in distince points X and Y so that OX/XP = OY/YQ, then line XY is parallel to line PQ Radius-Tangent Theorem: A line is tangent to a circle if and only if it is perpendicular to a radius at the radius’s endpoint on the circle Uniqueness of Parallels Theorem: Through a point not on a line, there is exactly one parallel o the given line Exterior Angle Theorem In a triangle, the measure of an exterior angle is equal to the sum of the measures of the two nonadjacent interior angles Exterior Angle Inequality: In a triangle, the measure of an exterior angle is equal to the sum of the measures of the two nonadjacent interior angles Unequal Sides Theorem: If two sides of a triangle are not congruent, then the angles opposite them are not congruent, and the larget angle is opposite the longer side Unequal Angles Theorem: If two anles of a triangle are not congruent, then the sides opposite them are not congruent, and the longer side is opposite the larger angle Exterior Angles of a Polygon Sum Theorem: In any convex polygon, the sum of the measures of the exterior angles, one at each vertex, is 360 Isosceles Right Triangle Theorem: In an isosceles right triangle, if a leg is x then the hypotenuse is x times the square root of 2 30-60-90 Triangle Theorem: In a 30-60-90 right triangle, if the short leg is x then the longer leg is x times the square root of 3 and the hypotenuse is 2x Isoperimetric Theorem: Of all solids with the same volume, the sphere has the least surface area Isoperimetric Theorem: Of all solids with the same surface area, the sphere has the most volume Isoperimetric Theorem: Of all plane figures withthe same area, the circle has the least perimeter Isoperimetric Inequality: If a plane figure has area A and perimeter p, then A is greater then or equal to p(sq)/4(Pi) Isoperimetric Theorem: Of all plane figures with the same perimeter, the circle has the most area Tangent Square Theoerm: The power of point P for circle O is square of the length of a segment tangent to circle O Secant Length Theorem: Suppose one secant intersects a circle at A and B, and a second secant intersects the circle at C and D. If the secants intersect at P, then AP times BP equals CP times DP Tangent-Segment Theorem: The measure of an angle between two tangents, or between a tangent and a secant, is half the distance of the intercepted arcs Tangent-Chord Theorem: The measure of an angle formed by a tangent and a chord is half the measures of the intercepted arc Angle-Secant Theorem: The measure of an angle formed by two secants intersecting outside a circle is half the difference of the arcs intercepted by it Angle-Chord Theorem: The measure of an angle formed by two intersecting chords is one-half the sum of the measures of the arcs intercepted by it and its vertical angle Inscribed Angle Theorem: In a circle the measure of an inscribed angle os one-half the measure of its intercepted arc Arc-Chord Congruence Theorem: In a circle or in congruent circles: (a.) If two arcs have the same measure, they are congruent and their chords are congruent (b.) If two chords have the same length, their minor arcs have the same measure Chord-Center Theorem: (a.) The line containing the center of a circle perpendicular to a chord bisects the chord (b.) The line containing the center of a circle and the midpoint of a chord bisects the central angle determined by the chord (c.) The bisector of the central angle of a chord is perpendicular to the chord and bisects the chord (d.) The perpendicular bisector of a chord of a circle contains the center of the circle Vector Addition Theorem: The sum of the vectors (a,b) and (c,d) is the vector (a +c, b +d) Properties of Vector Addition Theorem: 1. Vector addition is communative 2. Vector addition is associative 3. (0,0) is an identity for vector addition 4. Every vector (a,b) has an additive inverse (-a,-b) Right Triangle Altitude Theorem: In a right triangle: The altitude to the hypotenuse is the geometric mean of the segments into which it divides the hypotenuse: And each leg is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to the leg