Postulate 1-7 If two points lie in the same plane, then the line containing them lies in the plane.

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1 Geometry Postulates and Theorems Unit 1: Geometry Basics Postulate 1-1 Through any two points, there exists exactly one line. Postulate 1-2 A line contains at least two points. Postulate 1-3 Two lines intersect at exactly one point. Postulate 1-4 Through any three non-collinear points, there exists exactly one plane. Postulate 1-5 A plane contains at least three non-collinear points. Postulate 1-6 If two planes intersect, their intersection is a line. Postulate 1-7 If two points lie in the same plane, then the line containing them lies in the plane. Postulate 1-8 The Ruler Postulate The points on a line can be paired one-to-one with the real numbers. Postulate 1-9 Segment Addition Postulate If B is between A and C, then AB + BC = AC. Also, if AB + BC = AC, then B is between A and C. Postulate 1-10 Any segment has exactly one midpoint. Postulate 1-11 Protractor Postulate The sides of an angle can be paired one-to-one with the real numbers from 0 to 180. The points on one side can always be paired with 0, and the points on the other side can always be paired with a number between 0 and 180. Postulate 1-12 Angle Addition Postulate If B is in the interior of ADC, then m ADB m BDC m ADC. Postulate 1-13 Every angle except a straight angle has exactly one bisector. Unit 2: Geometric Reasoning Theorem 2-1 All right angles are congruent.

2 Theorem 2-2 Linear Pair Theorem If two angles form a linear pair, then they are supplementary. Theorem 2-3 Congruent Complements Theorem If two angles are complementary to the same angle or to congruent angles, then they are congruent. Theorem 2-4 Congruent Supplements Theorem If two angles are supplementary to the same angle or to congruent angles, then they are congruent. Theorem 2-5 Vertical Angles Theorem Vertical angles are congruent. Theorem 2-6 Congruence of angles is reflexive, symmetric, and transitive. Theorem 2-7 Congruence of segments is reflexive, symmetric, and transitive. Unit 3: Parallel and Perpendicular Lines Theorem 3-1 If two parallel planes are intersected by a third plane, then the lines of intersection are parallel. Postulate 3-2 Parallel Postulate Given a line and a point P that is not on the line, there is exactly one line through point P that is parallel to. Postulate 3-3 Corresponding Angles Postulate If two parallel lines are intersected by a transversal, then the corresponding angles are congruent. Theorem 3-4 Alternate Interior Angles Theorem If two parallel lines are intersected by a transversal, then the alternate interior angles are congruent. Theorem 3-5 Alternate Exterior Angles Theorem If two parallel lines are intersected by a transversal, then the alternate exterior angles are congruent. Theorem 3-6 Consecutive Interior Angles Theorem If two parallel lines are intersected by a transversal, then the consecutive interior angles are supplementary.

3 Postulate 3-7 Converse of the Corresponding Angles Postulate If two lines are intersected by a transversal so that the corresponding angles are congruent, then the lines are parallel. Theorem 3-8 Converse of the Alternate Interior Angles Theorem If two lines are intersected by a transversal so that the alternate interior angles are congruent, then the lines are parallel. Theorem 3-9 Converse of the Alternate Exterior Angles Theorem If two lines are intersected by a transversal so that the alternate exterior angles are congruent, then the lines are parallel. Theorem 3-10 Converse of the Consecutive Interior Angles Theorem If two lines are intersected by a transversal so that the consecutive interior angles are supplementary, then the lines are parallel. Theorem 3-11 If two different lines are parallel to a third line, then they are parallel to each other. Theorem 3-12 If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. Theorem 3-13 If two lines are perpendicular, then they intersect to form four right angles. Postulate 3-14 Perpendicular Postulate Given a line and a point P that is not on the line, there is exactly one line through point P that is perpendicular to. Theorem 3-15 If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other. Theorem 3-16 If two lines are perpendicular to the same line, then they are parallel to each other. Postulate 3-17 Slopes of Parallel Lines If two lines have the same slope, then the lines are parallel. Postulate 3-18 Slopes of Perpendicular Lines If two lines have slopes that are opposite reciprocals, then the lines are perpendicular. Unit 4: Congruent Triangles Theorem 4-1 Triangle Sum Theorem The sum of the angle measures in a triangle is 180.

4 Theorem 4-2 Exterior Angle Theorem The measure of an exterior angle in a triangle is the sum of its remote interior angle measures. Theorem 4-3 The acute angles of a right triangle are complementary. Theorem 4-4 The measure of each angle of an equiangular triangle is 60. Theorem 4-5 Third Angle Theorem If two angles in a triangle are congruent to two angles in another triangle, then the third angles are also congruent. Postulate 4-6 Side-Side-Side Congruence Postulate (SSS) If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. Postulate 4-7 Side-Angle-Side Congruence (SAS) If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. Postulate 4-8 Angle-Side-Angle Congruence Postulate (ASA) If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. Theorem 4-9 Angle-Angle-Side Congruence Theorem (AAS) If two angles and a non-included side of one triangle are congruent to two angles and a corresponding non-included side of another triangle, then the triangles are congruent. Theorem 4-10 Hypotenuse-Leg Congruence Theorem (HL) If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. Theorem 4-11 Isosceles Triangle Theorem The base angles of an isosceles triangle are congruent. Theorem 4-12 Equilateral Triangle Theorem If a triangle is equilateral, then it is equiangular. Theorem 4-13 Converse of the Isosceles Triangle Theorem If a triangle has two congruent angles, then the triangle is isosceles and the congruent sides are opposite the congruent angles. Theorem 4-14 Converse of the Equilateral Triangle Theorem If a triangle is equiangular, then it is equilateral.

5 Unit 5: Triangles Theorem 5-1 Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Theorem 5-2 Converse of the Perpendicular Bisector Theorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. Theorem 5-3 Circumcenter Theorem The circumcenter of a triangle is equidistant from the vertices of the triangle. Theorem 5-4 Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. Theorem 5-5 Converse of the Angle Bisector Theorem If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle. Theorem 5-6 Incenter Theorem The incenter of the triangle is equidistant from the sides of the triangle. Theorem 5-7 Centroid Theorem The medians of a triangle intersect at a point that is located 2 3 the distance from each vertex to the midpoint of the opposite side. Theorem 5-8 Orthocenter Theorem The lines containing the altitudes of a triangle are concurrent. Theorem 5-9 Triangle Midsegment Theorem A midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side. Theorem 5-10 Exterior Angle Inequality Theorem An exterior angle of a triangle is greater than either of the nonadjacent interior angles. Theorem 5-11 If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side.

6 Theorem 5-12 If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle. Theorem 5-13 Triangle Inequality Theorem The sum of any two side lengths of a triangle is greater than the third side length. Theorem 5-14 Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second. Theorem 5-15 Converse of the Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle and the third side of the first is longer than the third side of the second, then the included angle of the first is larger than the included angle of the second. Unit 6: Similarity Postulate 6-1 Angle-Angle (AA) Similarity If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Theorem 6-2 Side-Side-Side (SSS) Similarity Theorem If the three sides of one triangle are proportional to the three corresponding sides of another triangle, then the triangles are similar. Theorem 6-3 Side-Angle-Side (SAS) Similarity Theorem If two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar. Theorem 6-4 If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. Theorem 6-5 Geometric Mean Altitude Theorem The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the two segments of the hypotenuse. Theorem 6-6 Geometric Mean Leg Theorem The length of a leg of a right triangle is the geometric mean of the length of the hypotenuse and the length of the segment of the hypotenuse adjacent to that leg. Theorem 6-7 If two triangles are similar, the lengths of the corresponding altitudes are proportional to the lengths of the corresponding sides.

7 Theorem 6-8 If two triangles are similar, the lengths of the corresponding medians are proportional to the lengths of the corresponding sides. Theorem 6-9 If two triangles are similar, the lengths of the corresponding angle bisectors are proportional to the lengths of the corresponding sides. Theorem 6-10 Triangle Proportionality Theorem If a line parallel to the side of the triangle intersects the other two sides, then it divides those sides proportionally. Theorem 6-11 Converse of the Triangle Proportionality Theorem If a line divides two sides of a triangle proportionally, then it is parallel to the third side of the triangle. Theorem 6-12 If three or more parallel lines intersect two transversals, then they divide the transversals proportionally. Theorem 6-13 Triangle Angle Bisector Theorem An angle bisector of a triangle divides the opposite side into two segments whose lengths are proportional to the lengths of the other two sides. Theorem 6-14 Proportional Perimeters and Areas Theorem If the similarity ratio of two similar figures is a b, then the ratio of their perimeters is a, and the b ratio of their areas is 2 a 2 b, or 2 a b. Unit 7: Right Triangles and Trigonometry Theorem 7-1 The Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. a 2 + b 2 = c 2 Theorem 7-2 Converse of the Pythagorean Theorem 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 other two sides, then the triangle is a right triangle. Theorem 7-3 If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, then the triangle is an obtuse triangle.

8 Theorem 7-4 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 other two sides, then the triangle is an acute triangle. Theorem Triangle Theorem In a triangle, the hypotenuse is 2 times as long as a leg. Theorem Triangle Theorem In a triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is 3 times as long as the shorter leg. Theorem 7-7: Law of Sines For any triangle ABC, with side lengths a, b, and c, sin A sinb sinc a b c Theorem 7-8 Law of Cosines For any triangle ABC, with side lengths a, b, and c, a 2 = b 2 + c 2 2bc cos A b 2 = a 2 + c 2 2ac cos B c 2 = a 2 + b 2 2ab cos C Unit 8: Polygons Theorem 8-1 Angle Sum Theorem The sum of the measures of the interior angles of a convex polygon with n sides is (n 2)180. Corollary 1 to Theorem 8-1 The measure of an angle of a regular polygon with n sides is n n Corollary 2 to Theorem 8-1 The sum of the interior angles of a quadrilateral is 360. Theorem 8-2 The sum of the measures of the exterior angles of a convex polygon, one at each vertex, is 360. Theorem 8-3 If a quadrilateral is a parallelogram, then its opposite sides are congruent. Theorem 8-4 If a quadrilateral is a parallelogram, then its opposite angles are congruent. Theorem 8-5 If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.

9 Theorem 8-6 If a quadrilateral is a parallelogram, then its diagonals bisect each other. Theorem 8-7 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem 8-8 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem 8-9 If one pair of opposite sides of a quadrilateral is congruent and parallel, then the quadrilateral is a parallelogram. Theorem 8-10 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Theorem 8-11 A parallelogram is a rhombus if and only if its diagonals are perpendicular. Theorem 8-12 A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles. Theorem 8-13 A parallelogram is a rectangle if and only if its diagonals are congruent. Rhombus Corollary A quadrilateral is a rhombus if and only if it has four congruent sides. Rectangle Corollary A quadrilateral is a rectangle if and only if it has four right angles. Square Corollary A quadrilateral is a square if and only if it is a rhombus and a rectangle. Theorem 8-14 If a trapezoid is isosceles, then each pair of base angles is congruent. Theorem 8-15 If a trapezoid has one pair of congruent base angles, then it is an isosceles trapezoid. Theorem 8-16 A trapezoid is isosceles if and only if its diagonals are congruent. Theorem 8-17 Trapezoid Midsegment Theorem The midsegment of a trapezoid is parallel to each base, and its length is one half the sum of the lengths of the bases. Theorem 8-18 If a quadrilateral is a kite, then its diagonals are perpendicular. Theorem 8-19 If a quadrilateral is a kite, then exactly one pair of opposite angles is congruent.

10 Unit 9: Transformations Theorem 9-1 Converse of the Segment Addition Postulate Given distinct points A, B, and C, if AC + CB = AB, then A, B, and C are collinear and C is between A and B. Unit 10: Perimeter and Area Theorem 10-1 Area Congruence Postulate The area of a region is equal to the sum of the areas of its non-overlapping parts. Postulate 10-2 Area Addition Postulate If two polygons are congruent, then they have the same area. Theorem 10-3 Pick s Theorem If a lattice polygon has b boundary points and i interior points, then its area is A = i + b 2 1. Theorem 10-4 Changing Dimensions Proportionally If the dimensions of a figure are multiplied by a scale factor k to produce a figure that is similar to the original, then the perimeter is multiplied by k and the area is multiplied by k 2. Unit 11: Surface Area and Volume Theorem 11-1 Euler s Theorem For all convex polyhedra, there is a relationship between the number of vertices, edges, and faces. Unit 12: Circles Theorem 12-1 In a plane, a line is tangent to a circle if and only if it is perpendicular to the radius drawn to the point of tangency. Theorem 12-2 If two segments from the same exterior point are tangent to a circle, then they are congruent. Postulate 12-3 Arc Addition Postulate The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

11 Theorem 12-4 Two minor arcs of the same circle, or congruent circles, are congruent if and only if their central angles are congruent. Theorem 12-5 In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. Theorem 12-6 In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center. Theorem 12-7 If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. Theorem 12-8 If one chord is a perpendicular bisector of another chord, then the first chord is a diameter of the circle. Theorem 12-9 Measure of an Inscribed Angle Theorem If an angle is inscribed in a circle, then the measure of the angle equals half the measure of its intercepted arc. Theorem If two inscribed angles of a circle intercept the same arc or congruent arcs, then the angles are congruent. Theorem An inscribed angle of a triangle intercepts a semicircle if and only if the angle is a right angle. Theorem If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. Theorem If a tangent and a secant intersect at the point of tangency, then the measure of each angle formed is half the measure of its intercepted arc. Theorem If two secants intersect in the interior of a circle, then the measure of each angle formed is half the sum of the measures of the arcs intercepted by the angle and its vertical angle. Theorem If two secants, two tangents, or a secant and a tangent 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. Theorem If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. Theorem If two secant segments share an endpoint outside a circle, then the product of the lengths of one secant segment and its external secant segment equals the product of the lengths of the other secant segment and its external secant segment.

12 Theorem If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the lengths of the secant segment and its external segment equals the square of the length of the tangent segment. Copyright 2012 The American Education Corporation. A+, A+LS, and A+nyWhere Learning System are either trademarks or registered trademarks of The American Education Corporation