Term
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Definition
| a reflection is an isometry |
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Term
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Definition
| a rotation is an isometry |
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Term
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Definition
| if two lines, L and M, intersect at point O, then a reflection in L followed by a reflection in M is a rotation about point O. The angle of rotation is 2x degrees, where x degrees is the measure of the acute or right angle between L and M. |
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Term
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Definition
| a translation is an isometry |
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Term
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Definition
| if lines L and M are parallel, then a reflection in line L followed by a reflection in line M is a translation. if P'' is the image of P after two reflections, then line PP'' is perpendicular to L and PP''= 2d, where d is the distance between L and M |
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Term
| postulate 20, Angle-Angle (AA) Similarity Postulate |
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Definition
| if two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar |
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Term
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Definition
| if two polygons are similar, then the ratio of their perimeters is equal to the ratio of their corresponding sides |
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Term
| theorem 8.2, Side-Side-Side (SSS) Similarity Theorem |
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Definition
| if corresponding sides of two triangles are proportional, then the two triangles are similar |
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Term
| theorem 8.3, Side-Angle-Side (SAS) Similarity Theorem |
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Definition
| if an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar |
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Term
| theorem 8.4, Triangle Proportionality Theorem |
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Definition
| if a line parallel to one side of a triangle intersects to other two sides, then it divides the two sides proportionally |
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Term
| theorem 8.5, Triangle Proportionality Converse |
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Definition
| if a line divides two sides of a triangle proportionally, then it is parallel to the third side |
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Term
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Definition
| if three parallel lines intersect two transversals, then they divide the transversals proportionally |
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Term
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Definition
| if a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides |
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Term
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Definition
| if the altitude is drawn of a right triangle, then the two triangles formed are similar to the original triangle and to each other |
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Term
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Definition
| in a right triangle, the length of the altitude from the right angle to the hypotenuse is the geometric mean of the lengths of the two segments of the hypotenuse |
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Term
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Definition
| in a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. each leg of the right triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg |
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Term
| theorem 9.4, The Pythagorean Theorem |
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Definition
| in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs |
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Term
| theorem 9.5, Converse of the Pythagorean Theorem |
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Definition
| if the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the two shorter sides, then the triangle is a right triangle |
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Term
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Definition
| if the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the two shorter sides, then the triangle is acute |
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Term
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Definition
| if the square of the length of the longest side of a triangle is greater than the sum of the the squares of the lengths of the two shorter sides, then the triangle is obtuse |
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Term
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Definition
| in a 45-45-90 degree triangle, the hypotenuse is the square root of two times as long as each leg. that is, the side-length ratios of leg:leg:hyp are 1:1: the square root of 2 |
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Term
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Definition
| in a 30-60-90 degree triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is the square root of 3 as long as the shorter leg. That is the side-length ratio of short leg: long leg: hyp are 1: the square root of 3: 2 |
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Term
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Definition
| if a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency |
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Term
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Definition
| in a plane, if a line is perpendicular to a radius of a circle at its endpoint on a circle, then the line is tangent to the circle |
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Term
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Definition
| if two segments from the same exterior point are tangent to a circle, then they are congruent |
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Term
| postulate 21, Arc Addition Postulate |
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Definition
| the measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs |
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Term
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Definition
| in the same circle, or in congruent circles, two arcs are congruent if and only if their central angles are congruent |
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Term
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Definition
| in the same circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent |
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Term
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Definition
| if a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc |
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Term
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Definition
| if chord AB is a perpendicular bisector of another chord, then AB is a diameter |
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Term
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Definition
| in the same circle or in congruent circles, two chords are congruent if and only if they are equidistant from the center |
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Term
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Definition
| if and angle is inscribed in a circle, then its measure is half the measure of its intercepted arc |
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Term
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Definition
| if two inscribed angles of a circle intercept the same arc, then the angles are congruent |
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Term
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Definition
| an angle that is inscribed in a circle is a right angle if and only if its corresponding arc is a semicircle |
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Term
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Definition
| if a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is half the measure of its intercepted arc |
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Term
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Definition
| a quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary |
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Term
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Definition
| if two chords intersect in the interior of a circle, then the measure of each angle is half the sum of then measures of the arcs intercepted by the angle an its vertical angle |
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Term
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Definition
| if a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is half the difference of the measures of the intercepted arcs |
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Term
| postulate 22, Area of a Square Postulate |
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Definition
| the area of a square is the square of the length of its side, or A=s^2 |
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Term
| postulate 23, Area Congruence Postulate |
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Definition
| if two polygons are congruent, then they have the same area |
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Term
| postulate 24, Area Addition Postulate |
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Definition
| the area of a region is the sum of the areas of all its nonoverlapping parts |
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Term
| theorem 11.1, Area of a Rectangle Theorem |
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Definition
| the area of a rectangle is the product of its base and height, or A= bh |
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Term
| theorem 11.2, Area of a Parallelogram Theorem |
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Definition
| the area of a parallelogram is the product of a base and its corresponding height, or A= bh |
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Term
| theorem 11.3, Area of a Triangle Theorem |
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Definition
| the area of a triangle is half the product of a base and its corresponding height, or A= 1/2bh |
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Term
| theorem 11.4, Area of a Trapezoid Theorem |
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Definition
| the area of a trapezoid is half the product of the height and the sum of the bases, or A= 1/2(b+b)h |
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Term
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Definition
| if the diagonals of a quadrilateral are perpendicular, then the area of the quadrilateral is half the product of the lengths of the diagonals, or A= 1/2 d(d) |
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Term
| theorem 11.6, Area of an Equilateral Triangle Theorem |
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Definition
| the area of an equilateral triangle is one-fourth the square of the length of the side times the square root of 3, or A= 1/4s^2(square root of 3) |
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Term
| theorem 11.7, Area of a Regular Polygon Theorem |
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Definition
| the area of a regular polygon is half the product of the apothem, a, and the perimeter, or A= 1/2aP |
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Term
| theorem 11.8, Circumference of a Circle Theorem |
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Definition
| for all circles, the ratio of the circumference, C, to the diameter, d, is the same. this ratio C/d is denoted by the number 3.14 or pi. thus, the circumference or a circle is C=(pi)d or C=2r(pi) |
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Term
| theorem 11.10, Area of a Sector Theorem |
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Definition
| the ratio of the area, A, of a sector to the area of its circle is equal to the ratio of the measure of the intercepted arc to 360 degrees. |
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Term
| theorem 11.11, Areas of Similar Polygons Theorem |
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Definition
| if two polygons are similar with corresponding sides in the ratio of a:b, then the ratio of the areas is a^2:b^2 |
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Term
| theorem 12.1, Euler's Theorem |
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Definition
| the number of faces (F), vertices (V), and edges (E) of a polyhedron is related by F+V= E+2 |
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Term
| theorem 12.2, Surface Area of a Right Prism |
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Definition
the surface area, S, of a right prism is:
S= 2B + Ph
where B is the area of a base, P is the perimeter of a base, and h is the height |
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Term
| theorem 12.3, Surface Area of a Right Cylinder |
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Definition
| the surface area, S, of a right cylinder is S= 2B+ Ch= 2r^2(pi)+ 2rh(pi) where B is the area of a base, C is the circumference of a base, r is the radius of a base, and h is the height |
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Term
| theorem 12.4, Surface Area of a Regular Pyramid |
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Definition
| the surface area, S, of a regular pyramid is S= B+ 1/2Pl, where B is the area of the base, P is the perimeter of the base, and l is the slant height |
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Term
| theorem 12.5, Surface Area of a Right Cone |
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Definition
| the surface area, S, of a right cone is S= (pi)r^2+ (pi)rl, where r is the radius of the base and l is the slant height of the cone |
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Term
| postulate 25, Volume of Cube Postulate |
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Definition
| the volume of a cube is the cube of the length of its side, or V=s^2 |
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Term
| postulate 26, Volume Congruence Postulate |
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Definition
| if two polyhedron are congruent, then they have the same volume |
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Term
| postulate 27, Volume Addition Postulate |
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Definition
| The volume of a solid is the sum of the volumes of all its nonoverlapping parts |
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Term
| theorem 12.6, Cavalieri's Principle |
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Definition
| if two solids have the same height and the same cross-sectional area at every level, then they have the same volume |
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Term
| theorem 12.7, Volume of a Prism |
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Definition
| the volume, V, of a prism is V=Bh, where B is the area of a base and h is the height |
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Term
| theorem 12.8, Volume of a Cylinder |
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Definition
| the volume, V, of a cylinder is V= Bh= (pi)r^2h, where B is the area of a base, h is the height, and r is the radius of a base |
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Term
| theorem 12.9, Volume of a Pyramid |
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Definition
| the volume, V, of a pyramid is given by V= 1/3Bh, where B is the area of the base and h is the height |
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Term
| theorem 12.10, Volume of a Cone |
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Definition
| the volume, V, of a cone is given by V= 1/3Bh= 1/3(pi)r^2h, where B is the area of the base, h is the height, and r is the radius of the base |
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Term
| theorem 12.11, Surface Area of a Sphere |
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Definition
| the surface area, S, of a sphere of radius r is S=4(pi)r^2 |
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Term
| theorem 12.12, Volume of a Sphere |
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Definition
| the volume, V, of a sphere of radius r is V= 4/3(pi)r^3 |
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Term
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Definition
| if two solids are similar with a scale factor of a:b, then corresponding areas have a ratio of a^2:b^2 and corresponding volumes have a ratio of a^3:b^3 |
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