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AABC apply assume Axiom bounded Chapter choose circle claim classes closed collection complete composition compute congruent conjugate consider construction containing coordinate corresponding define Definition denote described desired determine directed angle measure distance distinct equal equivalent established Euclidean example Exercise exists fact fixed point formula four frieze groups geometry given gives glide reflection Hence Hint implies important interior intersect invariant isometry lattice length line segment maps means obtain opposite orbit original pair parallel parallelogram particular patterns perpendicular plane point group polygonal regions positive possible preserves proof properties Proposition Prove radius real number reflection result rotation shown in Figure side similarity split square Structure subgroup Suppose symmetry group Theorem transformations translation triangle union unique verify vertices wallpaper
Page xiii - That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
Page 58 - The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Page 48 - The SAS Postulate: Given a correspondence between two triangles (or between a triangle and itself). If two sides and the included angle of the first triangle are congruent to the corresponding parts of the second triangle, then the correspondence is a congruence.
Page 4 - the best way to explain it is to do it." (And, as you might like to try the thing yourself some winter day, I will tell you how the Dodo managed it.) First it marked out a race-course, in a sort of circle, ("the exact shape doesn't matter," it said,) and then all the party were placed along the course, here and there. There was no "One, two...
Page 61 - If two sides of one triangle are equal to two sides of a second triangle and the included angle of the first is greater than the included angle of the second, then the third side of the first triangle is greater than the third side of the second.
Page 294 - A median of a triangle is a line which joins a vertex of the triangle to the midpoint of the opposite side. Let ABC be any triangle, and let a, b and c be position vectors, of A, B and C relative to some fixed origin.
Page 114 - The radius of a circle is a line segment from the center of the circle to a point on the circle.
Page 48 - If two sides of a triangle are congruent, then the angles opposite these sides are congruent.
Page 68 - The sum of the measures of the interior angles of any triangle is 180°.
Page 120 - Suppose a correspondence between two triangles is such that two sides and the included angle of the first triangle are congruent to the corresponding parts of the second triangle.