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Geometry Postulates, Theorems and Corollaries

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Postulate 2 Segment Addition PostulateIf C is between A and B, then AC + CB = AB.
Postulate 3Any segment has exactly one midpoint
Postulate 4 Protractor PostulateIn a given plane, select any line AB and any point C between A and B. Also select any two points R and S on the same side of AB such that S is not on CR. Then there is a pairing of rays to real numbers from 0 to 180 as follows: 1. CA is paired with 0 and CB is paired with 180.
Postulate 5 Angle Addition PostulateIf D is in the interior of angle ABC, then m angle ABC = m angle ABD + m angle LDBC.
Postulate 6Every angle, except a straight angle, has exactly one bisector.
Postulate 7If the outer rays of two adjacent angles form a straight angle, then the sum of the measures of the angles is 180.
Theorem 1.1If the outer rays of two acute adjacent angles are perpendicular, then the sum of the measures of the angles is 90.
Theorem 2.1If two angles are supplements of congruent angles, then they are congruent. (Supplements of congruent angles are congruent.)
Theorem 2.1 - CorollaryIf two angles are supplements of the same angle, then they are congruent. (Supplements of the same angle are congruent.)
Theorem 2.2If two angles are complements of congruent angles, then they are congruent. (Complements of congruent angles are congruent.)
Theorem 2.2 - CorollaryIf two angles are complements of the same angle, then they are congruent. (Complements of congruent angles are congruent.)
Theorem 2.3If two angles are right angles, then they are congruent.
Theorem 2.4Vertical Angles Theorem: Vertical angles are congruent.
Theorem 2.4 - CorollaryIf two lines are perpendicular, then four right angles are formed.
Postulate 8A line contains at least two points. A plane contains at least three noncolinear points. Space contains at least four noncoplaner points
Postulate 9For any two points, there is exactly one line containing them.
Postulate 10If two points of a line are in a given plane, then the line itself is in the plane.
Theorem 2.6If a line intersects a plane, but is not contained in the plane, then the intersection is exactly one point.
Postulate 11If two planes intersect, then they intersect in exactly one line.
Postulate 12Three noncollinear points are contained in exactly one plane.
Theorem 2.7A line and a point not on the line are contained in exactly one plane.
Theorem 2.8Two intersecting lines are contained in exactly one plane.
Postulate 13Alternate Interior Angles Postulate: If a transversal intersects two lines such that alternate interior angles are congruent (equal in measure), then the lines are parallel.
Theorem 3.1If a transversal intersects two lines such that corresponding angles are congruent, then the lines are parallel.
Theorem 3.2If two lines are intersected by a transversal such that interior angles on the same side of the transversal are supplementary, then the lines are parallel.
Theorem 14Parallel Postulate: Through a point not on a line, there is exactly one line parallel to the given line.
Theorem 3.4If two parallel lines are intersected by a transversal, then alternate interior angles are congruent.
Theorem 3.5If two parallel lines are intersected by a transversal, then corresponding angles are congruent.
Theorem 3.6If two parallel lines are intersected by a transversal, then interior angles on the same side of the transversal are supplementary.
Theorem 3.7If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.
Theorem 3.8In a plane, if two lines are parallel to the line, then they are parallel to each other.
Theorem 3.9The sum of the measures of the angles of a triangle is 180.
Theorem 3.10Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.
Theorem 3.11If two parallel planes are intersected by a third plane, then the lines of intersection are parallel.
Theorem 4.1In a right triangle, the two angles other than the right angle are complementary and acute.
Postulate 15 SAS Postulate for Congruence of TrianglesIf two sides and the included angle of one triangle are congruent to the corresponding two sides and included angle of a second triangle, then the triangles are congruent.
Postulate 16 SSS Postulate for Congruence of Triangles:If the three sides of one triangle are congruent to the corresponding three sides of a second triangle, then the triangles are congruent.
Postulate 17 ASA Postulate for Congruence of Triangles:If two angles and the included side of one triangle are congruent to the corresponding two angles and included side of a second triangle, then the triangles are congruent.
Theorem 4.2 Third Angle Theorem:If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent.
Theorem 4.3 AAS Theorem:If two angles and a nonincluded side of one triangle are congruent to the corresponding two angles and side of a second triangle, then the triangles are congruent.
Theorem 5.1If two sides of a triangle are congruent, then angles opposite these sides are congruent. (The base angles of an isosceles triangle are congruent.)
Theorem 5.1 - CorollaryIf a triangle is equiangular, then it is also equilateral.
Theorem 5.2If two angles of a triangle are congruent, then the sides opposite these angles are congruent.
Theorem 5.2 - CorollaryIf a triangle is equiangular, then it is also equilateral.
Theorem 5.3 Hypotenuse-Leg (HL) Theorem:Two right triangles are congruent if the hypotenuse and a leg of one are congruent, respectively, to the hypotenuse and corresponding leg of the other.
Theorem 5.4The altitude from the vertex angle to the base of an isosceles triangle is a median. (The altitude bisects the base.)
Theorem 5.5Corresponding medians of congruent triangles are congruent.
Theorem 5.6Corresponding altitudes of congruent triangles arc congruent.
Theorem 5.7The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base.
Theorem 5.7 - CorollaryThe bisector of the vertex angle of an isosceles triangle Is also a median and an altitude of the triangle.
Theorem 5.8A line containing two points, each equidistant from the endpoints of a given segment, is the perpendicular bisector of the segment.
Theorem 5.9Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment.
Theorem 5.10Exterior Angle Inequality Theorem: The measure of an exterior angle of a triangle is greater than the measure of either of its remote interior angles.
Theorem 5.11If one side of a triangle is longer than another side, then the measure of the angle opposite the longer side is greater than the measure of the angle opposite the shorter side.
Theorem 5.12If one angle of a triangle has a greater measure than a second angle, then the side opposite the greater angle is longer than the side opposite the smaller angle.
Theorem 5.13In a scalene triangle, the longest side is opposite the largest angle and the largest angle is opposite the longest side.
Theorem 5.14The perpendicular segment from a point to a line is the shortest segment from the point to the line.
Theorem 5.14 - CorollaryThe longest side of a right triangle is the hypotenuse.
Theorem 5.15 Triangle Inequality Theorem:The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Theorem 5.16 SAS Inequality Theorem:If two sides of one triangle are congru- ent, respectively, to two sides of a second triangle, and the included angle of the first triangle has a greater measure than the included angle of the second triangle, then the third side of the first triangle is longer than the third side of the second triangle.
Theorem 5.17 SSS Inequality Theorem:If two sides of one triangle are congruent, respectively, to two sides of a second triangle, and the length of the third side of the first triangle is greater than the length of the third side of the second triangle, then the angle opposite the third side of the first triangle has a greater measure than the angle opposite the third side of the second triangle.
Theorem 6.1The sum of the measures of the interior angles of a convex polygon with n sides is (n - 2)180.
Theorem 6.1- Corollary 1The sum of the measures of the interior angles of a convex quadrilateral is 360.
Theorem 6.1- Corollary 2The measure of an angle of a regular polygon with n sides is(n - 2)180/n.
Theorem 6.2The sum of the measures of the exterior angles, one at each vertex, of any convex polygon is 360.
Theorem 6.2 - CorollaryThe measure of an exterior angle of a regular polygon with n sides is 360/n.
Theorem 6.3A diagonal of a parallelogram forms two congruent triangles.
Theorem 6.3 - Corollary 1Opposite sides of a parallelogram are congruent.
Theorem 6.3 - Corollary 2Opposite angles of a parallelogram are congruent.
Theorem 6.4Consecutive angles of a parallelogram are supplementary.
Theorem 6.5The diagonals of a parallelogram bisect each other.
Theorem 6.6If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
Theorem 6.7If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
Theorem 6.8If two sides of a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram.
Theorem 6.9If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
Theorem 6.10 Midsegment Theorem:The segment joining the midpoints of two sides of a triangle is parallel to the third side, and its length is half the length of the third side.
Theorem 6.11If two fines are parallel, then all points of each line are equidistant from the other line.
Theorem 6.12If three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.
Theorem 6.12 - CorollaryIf any number of parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.
Theorem 6.13If a segment is parallel to one side of a triangle and contains the midpoint of a second side, then this segment bisects the third side.
Theorem 7.1 WAS for Congruent QuadrilateralsTwo quadrilaterals are congruent if any three sides and the included angles of one are congruent, respectively, to three sides and the included angles of the other.
Theorem 7.2 ASASA for Congruent Quadrilaterals:Two quadrilaterals are congruent if any three angles and the included sides of one are congruent, respectively, to three angles and the included sides of the other.
Theorem 7.3The diagonals of a rhombus are perpendicular.
Theorem 7.4The diagonals of a rectangle are congruent.
Theorem 7.5Each diagonal of a rhombus bisects two angles of the rhombus.
Theorem 7.6A parallelogram with one right angle is a rectangle.
Theorem 7.7A parallelogram with two adjacent, congruent sides is a rhombus.
Theorem 7.8A parallelogram with perpendicular diagonals is a rhombus.
Theorem 7.9A parallelogram with congruent diagonals is a rectangle.
Theorem 7.10A parallelogram with a diagonal that bisects opposite angles is a rhombus.
Theorem 7.11A quadrilateral with four congruent sides is a rhombus.
Theorem 7.12All altitudes of a trapezoid are congruent.
Theorem 7.13The median of a trapezoid is parallel to its bases. Its length is one-half the sum of the lengths of the two bases.
Theorem 7.14The base angles of an isosceles trapezoid are congruent.
Theorem 7.15If the base angles of a trapezoid are congruent, then the trapezoid is isosceles.
Theorem 7.16The diagonals of an isosceles trapezoid are congruent.
Theorem 7.17If the diagonals of a trapezoid are congruent, then the trapezoid is isosceles.
Theorem 8.1In a proportion, the product of the extremes equals the product of the means.
Theorem 8.1 - CorollaryIf the product of the extremes equals the product of the means, then a proportion exists.
Theorem 8.2If a/b = c/d, then: b/d = d/c; a/c = b/d; a+b/b = c+d/d; a-b/b = c-d/d; a/b = a+c/b+d.
Theorem 8.3Congruent triangles are similar.
Theorem 8.4 Transitive Property of Triangle Similarity:If Triangle ABC – Triangle DEF and Triangle DEF – Triangle GHI, then Triangle ABC – Triangle GHI.
Postulate 18 AA Similarity Postulate:AA Similarity Postulate: If two angles of a triangle are congruent to two angles of another triangle, then the two triangles are similar.
Theorem 8.5Triangle Proportionality Theorem: If a line is parallel to one side of a triangle and intersects the other two sides, then it divides the two sides proportionally.
Theorem 8.6If a line divides two sides of a triangle proportionally, then the line is parallel to the third side of the triangle.
Theorem 8.7 SAS Similarity Theorem:If an angle of one triangle is congruent to an angle of another triangle and the corresponding sides that include these angles are proportional, then the triangles are similar.
Theorem 8.8 SSS Similarity Theorem:If all three pairs of corresponding sides of two triangles are proportional, then the two triangles are similar.
Theorem 8.9Corresponding altitudes of similar triangles are proportional to corresponding sides.
Theorem 8.10Corresponding medians of similar triangles are proportional to corresponding sides.
Theorem 8.11The bisector of an angle of a triangle divides the opposite side of the triangle into segments proportional to the other two sides.
Theorem 9.1In a right triangle, the altitude to the hypotenuse forms two similar right triangles, each of which is also similar to the original triangle.
Theorem 9.1 - Corollary 1In a right triangle, the square of the length of the altitude to the hypotenuse equals the product of the lengths of the segments formed on the hypotenuse.
Theorem 9.1 - Corollary 2If the altitude is drawn to the hypotenuse of a right triangle, then the square of the length of either leg equals the product of the lengths of the hypotenuse and the segment of the hypotenuse adjacent to that leg.
Theorem 9.2 Pythagorean Theorem:In any right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.
Theorem 9.3 Converse of the Pythagorean Theorem:If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle.
Theorem 9.4If the square of the longest side of a triangle is greater (less) than the sum of the squares of the lengths of the other two sides, then the triangle is obtuse (acute).
Theorem 9.5In a 45-45-90 triangle, the hypotenuse is N/2- times as long as a leg.
Theorem 9.6In a 30-60-90 triangle, the hypotenuse is twice as long as the leg opposite the 30 angle. The leg opposite the 60 angle is Ö3 times as long as the leg opposite the 30 angle.
Theorem 10.1 Distance Formula:The distance d between P1(x1,y1) and P2(x2,y2) is given by the formula d = Ö (X2 - X1)^2 + (Y2 - Y1)^2
Theorem 10.2 Midpoint Formula:Given P1(xl,yl) and P2(X2,Y2), the coordinates (x ... y .. ) of M, the midpoint of PQ, are (x1+x2)/2, (y1+y2)/2.
Theorem 10.3All segments of a non-vertical line have equal slopes.
Theorem 10.4An equation of a line with slope m containing the point P1(xl,yl) is y - y1 = m(x - x1).
Theorem 10.6If two non-vertical lines have the same slope, then they are parallel.
Theorem 10.5If a line has slope m and y-intercept b, then an equation of the line is y = mx + b.
Theorem 10.7If two non-vertical lines are parallel, then they have equal slopes.
Theorem 10.8If the product of the slopes of two non-vertical perpendicular lines is -1, then the lines are perpendicular.
Theorem 10.9The product of the slopes of two non-vertical perpendicular lines is - 1.
Theorem 11.1If a line or segment contains the center of a circle and is perpen- dicular to a chord, then it bisects the chord.
Theorem 11.2In the same circle or in congruent circles, congruent chords are equidistant from the center(s).
Theorem 11.3In the same circle or in congruent circles, chords that are equidistant from the center(s) are congruent
Theorem 11.4In the same circle or congruent circles, if two chords are unequally distant from the center(s), then the chord nearer its corresponding center is the longer chord.
Theorem 11.5In the same circle or congruent circles, if two chords are unequal in length, then the longer chord is nearer the center of its circle.
Theorem 11.6If a line is perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle.
Theorem 11.7If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency.
Theorem 11.8Two segments drawn tangent to a circle from an exterior point are congruent.
Theorem 11.8 - CorollaryThe angle between two tangents to a circle from an exterior point is bisected by the segment joining its vertex and the center of the circle.
Postulate 19If P is a point on A-PB, then mA_B + mFB_ = mX-P-B.
Theorem 11.9In the same circle or in congruent circles: 1. If chords are congruent, then their corresponding arcs and central angles are congruent; 2. If arcs are congruent, then their corresponding chords and central angles are congruent; 3. If central angles are congruent, then their corresponding arcs and chords are congruent.
Theorem 11.10 Inscribed Angle Theorem:The measure of an inscribed angle is one-half of the degree measure of its intercepted arc.
Theorem 11.10 - Corollary 1If two inscribed angles intercept the same arc or congruent arcs, then the angles are congruent.
Theorem 11.10 - Corollary 2An angle inscribed in a semicircle is a right angle.
Theorem 11.10 - Corollary 3If two arcs of a circle are included between parallel chords or secants, then the arcs are congruent.
Theorem 11.10 - Corollary 4The opposite angles of an inscribed quadrilateral are supplementary.
Theorem 11.11The measure of an angle formed by two secants or chords intersecting in the interior of a circle is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
Theorem 11.12The measure of an angle formed by two secants intersecting in the exterior of the circle is one-half the difference of the measures of the intercepted arcs.
Theorem 11.13If a tangent and a secant (or a chord) intersect at the point of tangency on a circle, then the measure of the angle formed is one-half the measure of its intercepted arc.
Theorem 11.14The measure of an angle formed either by (1) a tangent and secant intersecting at a point exterior to a circle, or (2) two tangents intersecting at a point exterior to a circle equals one-half the difference of the measures of the intercepted arcs.
Theorem 11.15If two chords of a circle intersect, then the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord.
Theorem 11.16If a tangent and a secant intersect in the exterior of a circle, then the square of the length of the tangent segment equals the product of the lengths of the secant segment and the external secant segment.
Theorem 11.16 - CorollaryIf two secants intersect in the exterior of a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment.
Theorem 11.17The equation of a circle with the coordinates of the center (b,k) and a radius of length r is (x - h)' + (y - k)l = r.
Postulate 20Congruent polygons have equal areas.
Postulate 21The area of a rectangle is the product of the lengths of a base and a corresponding altitude (Area of rectangle = bb).
Theorem 12.1The area of a square is the square of the length s of a side (A = S2).
Postulate 22 Area Addition Postulate:If a region is the union of two or more nonoverlapping regions, then its area is the sum of the areas of these nonoverlapping regions.
Theorem 12.2The area of a parallelogram is the product of the lengths of a base and a corresponding altitude (Area of parallelogram = bb).
Theorem 12.3The area of a triangle is one-half the product of the lengths of a base and a corresponding altitude (Area of triangle = lbh).
Theorem 12.4The area of a kite is one-half the product of the lengths of the diagonals (Area of kite = 1djd2).
Theorem 12.4 - CorollaryThe area of a rhombus is one-half the product of the lengths of the two diagonals.
Theorem 12.5If s is the length of a side of an equilateral triangle, then the area is s^2Ö ¾
Theorem 12.6 Heron's Formula.If a, b, and c are the lengths of the sides of a triangle and s is the serniperimeter, such that s = 1/2(a + b + c), then Area(triangle) = Ö s(s- a)(s - b)(s - c).
Theorem 12.7The area of a trapezoid is one-half the product of the sum of the lengths of the upper and lower bases and the length of an altitude.
Theorem 12.8A circle can be circumscribed about any regular polygon.
Theorem 12.9The area of a regular polygon is one-half the product of the apothem and the perimeter [Area (n-gon) = 1/2ap)].
Theorem 12.10The area of a regular polygon is n[sin(18-0)] [cos(180 ]r 2, orns2 180 , where n is the number of sides, s is the length of a 4 tan (T) side, and r is the length of a radius.
Theorem 12.11The ratio of the perimeters of two similar polygons is the same as the ratio of the lengths of any two corresponding sides.
Theorem 12.12The ratio of the areas of two similar triangles is the square of the ratio of the lengths of any two corresponding sides.
Theorem 12.13The ratio of the areas of two similar polygons is the square of the ratio of the lengths of any two corresponding sides.
Theorem 12.14The ratio of the circumference to the length of a diameter is the same for all circles.
Theorem 12.14 - CorollaryThe circumference of a circle with radius of length r is 2rP.
Theorem 12.15The area of a circle with radius of length r is Pr^2
Theorem 12.16The area of a sector of a circle is one-half the product of the length s of the arc and the length r of its radius (A = 1/2rs).
Theorem 13.1The locus of points in a plane equidistant from two given points is the perpendicular bisector of the segment having the two points as endpoints.
Theorem 13.2In a plane, the locus of points equidistant from the sides of an angle is the bisector of the angle.
Theorem 13.3The perpendicular bisectors of the sides of a triangle are concur- rent at a point equidistant from the vertices of the triangle.
Theorem 13.4The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides of the triangle.
Theorem 13.5The lines containing the altitudes of a triangle are concurrent.
Theorem 13.6Two medians of a triangle intersect at a point two-thirds of the distance from each vertex to the midpoint of the opposite side.
Theorem 13.7The medians of a triangle are concurrent at a point that is two-thirds the distance from each vertex to the midpoint of the opposite side.
Theorem 14.1The lateral area of a right prism is the product of the perimeter of a base and the length of an altitude (L = ph).
Theorem 14.2The lateral area of a right cylinder is the product of the circumference of a base and the length of an altitude. The lateral area of a right cylinder with radius r and altitude of length h is 2Prh.
Theorem 14.3The total area of a right cylinder with radius of length r and altitude of length h is 2Pr^2 + 2Prh, or 2Pr(r + b).
Theorem 14.4If the ratio of the lengths of corresponding edges of two similar polyhedra is then the ratio of the lateral areas and of the total areas is (a/b)^2.
Postulate 23For any rectangular solid, the volume V = lwh, where 1, w, and b are the lengths of three edges with a common vertex.
Theorem 14.5The volume of a cube with edges of length S is S3.
Postulate 24If a solid is the union of two or more nonoverlapping solids, then its volume is the sum of the volumes of these nonoverlapping parts.
Postulate 25 Cavalieri's Principle:If two solids have equal heights, and if the cross sections formed by any plane parallel to the bases of both solids have equal areas, then the volumes of the solids are equal.
Theorem 14.6For any prism or cylinder, the volume is the product of the area of a base and the length of an altitude (V = Bh, where B = area of base and h = altitude).
Theorem 14.6 - CorollaryThe volume of a cylinder is Bh, or Pr^2h.
Theorem 14.7The lateral area of a regular pyramid is one-half the product of the perimeter of the base and the slant height (L = 1/2pl).
Theorem 14.8The lateral area L of a right cone is Prl. The total area (A) is, Prl + Pr^2 = Pr(l + r).
Theorem 14.9The volume of a pyramid is one-third the volume of a prism with the same base and altitude as the pyramid. The volume of a cone is one-third the volume of a cylinder with the same base and altitude as the cone (V = 1/3Bh).
Theorem 14.9 - Corollary 1The volume of a pyramid or cone is one-third the product of the area of its base and the length of its altitude (V = 1/3Bh).
Theorem 14.9 - Corollary 2The volume of a cone with a base radius of length r and an altitude of length h is 1/3P r^2h.
Theorem 14.10The volume of a sphere with radius of length r is 4/3Pr^3.
Theorem 14.11The area of a sphere is 4Pr^2.
Theorem 14.12The distance between points P(x1, y1, z1) and Q(x2, y2, z2) is given by the formula d = Ö(x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2.
Theorem 15.1A reflection is an isometry.
Theorem 15.2A translation is an isometry.
Theorem 15.3The resultant image determined by two successive reflections about parallel lines is a translation.
Theorem 15.4A rotation is an isometry.
Theorem 15.5The measure of the angle of rotation formed by two successive reflections (or by the product of two reflections) is twice the measure of the non-obtuse angle between the two lines of symmetry.
Theorem 15.6The transformation defined by adding a constant to the coordinates of each point is a translation.
Theorem 15.7The transformation defined by multiplying each coordinate of each point by a constant is a dilation.


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