Theorems
Points, Lines, Planes, and Angles1-1 If 2 lines intersect, then they intersect in exactly 1 point.
1-2 Through a line and a point not in the line there is exactly 1 plane.
1-3 If 2 lines intersect, then exactly 1 plane contains the lines.
Deductive Reasoning2-1 Midpoint Theorem
2-2 Angle Bisector Theorem
2-3 Vertical Angles are congruent.
2-4 If 2 lines are perpendicular, then they form congruent adjacent angles.
2-5 If 2 lines form congruent adjacent angles, then the lines are perpendicular.
2-6 If 2 exterior sides of 2 adjacent acute angles are perpendicular,then the angles are complementary.
2-7 If 2 angles are supplements of congruent angles(or of the same angle), then the 2 angles are congruent.
2-8 If 2 angles are complements of congruent angles(or of the same angle), then the 2 angles are congruent.
Parallel Lines and Planes3-1 If 2 // planes are cut by a 3rd plane, then the lines of the intersection are //.
3-2 If 2 // lines are cut by a transversal, then alternate interior angles are congruent.
3-3 If 2 // lines are cut by a transversal, then same-side interior angles are supplementry.
3-4 If a transversal is perpendicular to 1 of 2 // lines, then it is perpendicular to the other 1 also.
3-5 If 2 lines are cut by a transversal and alternate interior angles are congruent, then the lines are //.
3-6 If 2 lines are cut by a transversal and same-side interior angles are supplementary, then the lines are //.
3-7 In a plane 2 lines are perpendicular to the same line are //.
3-8 Through a point outside a line, there is exactly 1 line // to the given line.
3-9 Through a point outside a line, there is exactly 1 lineperpendicular to the given line.
3-10 2 lines are // to a 3rd line are // to each other.
3-11 The sum of the measures of the angles of a triangle is 180.
Corollary 1 If 2 angles of 1 triangle are congruent to 2 angles of another triangle, then the 3rd angles are congruent.
Corollary 2 Each angle of an equiangular triangle has measure 60.
Corollary 3 In a triangle, there can be at most 1 right angle or obtuse angle.
Corollary 4 The acute angles of a right triangle are complementary.
3-12 The measure of an exterior angle of a triangle qwuals the sum of the measures of the 2 remote interior angles.
3-13 The sum of the measures of the anglesof a convex polygon with n sides is 180(n-2).
3-14 The sum of the measures of the exterior angles of any convex polygon, 1 angle at each vertex, is 360.
Congruent Triangles4-1 (The Isosceles Triangle Theorem) If 2 sides of a triangle are congruent, then the angles opposite those sides are congruent.
Corollary 1 An equilateral triangle is also equiangular.
Corollary 2 An equilateral triangle has 3 60-degree angles.
Corollary 3 The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint.
4-2 If 2 angles are congruent, then the sides opposite those angles are congruent.
4-3 (AAS Theorem) If 2 angles and a non-included side of 1 triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent.
4-4 (HL Theorem) If the hypotenuse and a leg of 1 right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent.
4-5 If a point lies on the perpendicular busector of a segment, then the point is equidistant from the endpoints of the segment.
4-6 If a point is equidistant from the endpoints of the segment, then the poin lies on the perpendicular bisector of the segment.
4-7 If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle.
4-8 If a point is equidistant from the sides of an angle, then the point lies on the bisector of an angle.
Quadrilaterals5-1 Opposite sides of a //ogram are congruent.
5-2 Opposite angles of a //ogram are congruent.
5-3 Diagonals of a //ogram bisect each other.
5-4 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a //ogram.
5-5 If a pair of opposite sides of a quadrilateral are both congruent and //, then the quadrilateral is a //ogram.
5-6 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a //ogram .
5-7 If the diagonals of a quadrilatreal bisect each other, then the quadrilateral is a //ogram.
5-8 If 2 lines are //, then all points on 1 line are equidistant from the other line.
5-9 If 3 // lines cut off congruent segments on 1 transversal, then they cut off congruent segments on every transversal.
5-10 A line that contains the midpoint of 1 side of a triangle and is / to another side passes throguhthe midpoint of the 3rd side.
5-11 The segment that joins the midpoints of 2 sides of a triangle
(1) is // to the 3rd side.
(2) is 1/2 as long as the 3rd side.
5-12 The diagonals of a rectangle are congruent.
5-13 The diagonals of a rhombus are congruent.
5-14 Each diagonals of a rhombus bisect each other.
5-15 The midpoint of the hypoenuse of a right triangle is equidistant from the 3 vertices.
5-16 If an angle of a //ogram is a right angle, then the//ogram us a rectangle.
5-17 If 2 consecutive sides of a //ogram are congruent, then rthe //ogram is a rhombus.
5-18 Base angles of an isosceles trapezoid are congruent.
5-19 The median of a trapezoid.
(1) is // to the bases.
(2) has a length equal to the average of the base lengths.
Inequalities in Geometry6-1 (The Exterior Angle Inequality Theorem) The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle.
6-2 If 1 side of a triangle is longer that a 2nd side, then the angle opposite the 1st side is larger than the angle opposite the 2nd side.
6-3 If 1 angle of a triangle is larger than a 2nd angle, the the side opposite the 1st angle is longer than the side opposite the 2nd angle.
Corollary 1 The perpendicular segment from a point to a line is the shortest segment from the point to the line.
Corollary 2 The perpendicular segment from a point to a plane is the shortest segment from the point to the plane.
6-4 (The Triangle Inequality) The sum of the lengths of any 2 sides of a triangle is greater than the length of the 3rd side.
6-5 (SAS Inequality Theorem) If 2 sides of 1 triangle are congruent to the 2 sides of another triangle, but the included angle of the 1st triangle is larger than the included angle of the 2nd, then the 3rd side of the 1st triangle is longer than the 3rd side of the 2nd triangle.
6-6 (SSS Inequality Theorem) If 2 sides of 1 triangle are congruent to the 2 sides of another triangle, but the 3rd side of the 1st triangle is longer than the 3rd side of the 2nd, then the included angle of the 1st triangle is larger than the include angle of the 2nd.
Similar Polygons7-1 (SAS Similarity Theorem) If an angle of 1 triangle is congruent to an angle of another triangle and the sides including those angles are in proportion, then the triangles are similar.
7-2 (SSS Similarity Theorem) If the sides of 2 triangles ar in proportion, then the triangles are similar.
7-3 (SAS Similarity Theorem) If a line // to 1 side of a triangle intersects the other 2 sides, then it divides those sides propoertionally.
Corollary If 2 // lines intersect 2 transversals, then they divide the transversals proportionally.
7-4 (Triangle Angle-Bisector Theorem) If a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the other 2 sides.
Right Triangles8-1 If the altitude is drawn to the hypotenuse of a right triangle, then the 2 triangles formed are similar to the original triangle and to each other.
Corollary 1 When the altitude is drawn to the hypotenuse of a right triangle, the length of the altitude is geometric mean between the segments of the hypotenuse.
Corollary 2 When the altitude is drawn to the hypotenuse of a right triangle, each leg is geometric mean between the hypotenuse and the segmennt of the hypotenuse that is adjacent to that leg.
8-2 (Pythagorean Theorem) In a right triangle, the square of the hypotenuse is equal to teh sum of the squares of the legs.
8-3 If the square of 1 side of a triangle is equal to the sum of the squares of the other 1 sides, then the triangle is a right triangle.
8-4 Tf the square of the longest side of a triangle is less than the sum of the squares of the other 2 sides, then the triangle is an acute triangle.
8-5 Tf the square of the longest side of a triangle is greater than the sum of the squares of the other 2 sides, then the triangle is an obtuse triangle.
8-6 In a 45-45-90 triangle, the hypotenuse is square root of 2 times as long as a leg.
8-7 In a 30-60-90 triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is square root of 3 times as long as the shorter leg.
