Basic theorems of Euclidean geometry

Set 1: Isosceles triangles

    Definitions: A triangle is isosceles if two of its sides have the same length. Two lines M and N are perpendicular if they intersect at a point P and for some points Q on M and R on N, with Q, R distinct from P, a(QPR) = 90. The perpendicular bisector of a line segment AB is the line perpendicular to AB through the midpoint of AB.

    Straight angle: If the points A, B, C are distinct, then B is between A and C if and only if a(ABC) = 180.

    Isosceles triangle: If the points A, B, C are noncollinear, then AB = AC if and only if a(CBA) = a(BCA).

    Perpendicular bisector: The perpendicular bisector of a line segment AB is the set of all points which are equidistant from A and B.

    Existence of perpendicular: For every line L and every point A there exists a line through A perpendicular to L.

Set 2: Parallels

    Definitions: Suppose that L and M are distinct lines, and T is a transversal. Let A be the point of intersection of L and T, B the point of intersection of M and T, C a point of T so that B is between A and C, A' a point of L, and B' a point of M such that B' is on the same side of T as A', and B'' a point of M so that B is between B'' and B'. Then A'AB and B'BC are corresponding angles, A'AB and B''BA are alternate interior angles, and A'AB and ABB' are same side interior angles.

    Existence of parallel: Given a line L and a point P not on L, there exists a line M through P parallel to L.

    Corresponding angles: Suppose that L, M, T are distinct lines. Then L and M are parallel if and only if corresponding angles of intersection of L and T, M and T are equal.

    Alternate interior angles: Suppose that L, M, T are distinct lines. Then L and M are parallel if and only if alternate interior angles of intersection of L and T, M and T are equal.

    Same side interior angles: Suppose that L, M, T are distinct lines. Then L and M are parallel if and only if same side interior angles of intersection of L and T, M and T are supplementary.

    Triangle angle sum: The sum of the interior angles of a triangle is 180 degrees. More precisely, if A, B, C are distinct points, then a(ACB) + a(CBA) + a(BAC) = 180.

Set 3: Congruence

    Angle-Side-Angle (ASA): If A, B, C are distinct points, and A', B', C' are distinct points, with AB = A'B', a(BAC) = a(B'A'C') and a(CBA) = a(C'B'A'), then the triangles ABC and A'B'C are congruent.

    Side-Side-Side (SSS): If A, B, C are distinct points, and A', B', C' are distinct points, with AB = A'B', AC = A'C', and BC = B'C', then triangles ABC and A'B'C' are congruent.

Set 4: Area

    Definitions: A quadrilateral is a four-sided polygon. A parallelogram is a quadrilateral whose opposite sides are parallel. A rectangle is a quadrilateral whose adjacent sides are perpendicular.

    Area of parallelogram: The area of a parallelogram is its base times its height. More precisely, if the parallelogram is ABCD, the area is bh, where b = AB, and h = CE, where E is the intersection of the line L through AB with the perpendicular to L through the point C. (Note: There two possible bases b and b', with corresponding heights h and h'. A corollary of this theorem is that bh = b'h'.)

    Area of triangle: The area of a triangle is 1/2 its base times its height. (Note: There are three possible bases b, b', b'', with corresponding heights h, h', h''. A corollary of this theorem is that 1/2 bh = 1/2 b'h' = 1/2 b''h''.)

 Set 5: Similarity

    Definitions: Two triangles are similar if there is a bijection (one-to-one correspondence) between the vertices, and a real number k > 0, so the ratio of the lengths of corresponding sides is k and corresponding angles have the same angle measure. In other words, we can label the vertices ABC and A'B'C' so that kAB = A'B', kAC = A'C', kBC = B'C', a(C,A,B) = a(C',A',B'), a(A,B,C) = a(A',B',C'), a(B,C,A) = a(B',C',A').

    The real number k is called the similarity ratio (of A'B'C' to ABC).

    Side-Angle-Side: If A, B, C are distinct points, and A', B', C' are distinct points, and k > 0 is a real number, with kAB = A'B', kAC = A'C', and a(B,A,C) = a(B',A',C'), then the triangles ABC and A'B'C are similar, with similarity ratio k.

    Angle-Angle: If A, B, C are distinct points, and A', B', C' are distinct points, with a(B,A,C) = a(B',A',C') and a(C,B,A) = a(C',B',A'), then the triangles ABC and A'B'C are similar.

    Side-Side-Side: If A, B, C are distinct points, and A', B', C' are distinct points, and k > 0 is a real number, with kAB = A'B', kAC = A'C', and kBC = B'C', then triangles ABC and A'B'C' are similar, with similarity ratio k.

    Pythagorean Theorem: If ABC is a right triangle, with right angle ACB, then AC² + BC² = AB².

C. McCrory 9/18/04