| A | B |
|---|
| ReflexiveProperty | For every number a, a = a. |
| SymmetricProperty | For all numbers a & b, if a = b, then b = a.(ex. the segment GH = segment HG) |
| TransitiveProperty | For all numbers a, b & c, if a = b & b = c, then a = c. (A bit like the law of syllogism) |
| Add/Subtract Property | For all numbers a, b, & c, if a = b, then a + c = b + c and a - c = b - c.(ex. 1 ft = 12 inches, 1 ft + 3 inches = 12 in ches+ 3 inches) |
| Mult/Division Property | For all numbers a, b, and c, if a = b, then a * c = b * c, and if c not equal to zero, a ÷ c = b ÷ c.(ex. 1 m = 1000 mm, 1 m * 5 = 1000 mm * 5, 5 m = 5000 mm) |
| SubstitutionProperty | For all numbers a & b, if a = b, then a may be replaced by b in any equation or expression. |
| DistributiveProperty | For all numbers a, b, & c, a(b + c) = ab + ac. |
| THEOREM 2-1 Segment Properties | Congruence of segments is reflexive, symmetric, and transitive. |
| Theorem 2-2 Supplement Theorem | If two angles form a linear pair,then they are supplementary angles. |
| Theorem 2-3 Angle Properties | Congruence of angles is reflexive, symmetric, and transitive. |
| Theorem 2-4 supplementary congruent | Angles supplementary to the same angle or to congruent angles are congruent. |
| Theorem 2-5complementary congruent | Angles complementary to the same angle or to congruent angles are congruent. |
| Theorem 2-6 right congruent | All right angles are congruent. |
| Theorem 2-7 vertical angles | Vertical angles are congruent. |
| Theorem 2-8 perpendicular lines form | Perpendicular lines intersect to form four right angles.. |
| Postulate 3-1 Corresponding Angles | If two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent., |
| Theorem 3-1 Alternate Interior | If two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent, |
| Theorem 3-2 Consecutive Interior Angle | If two parallel lines are cut by a transversal, then each pair of consecutive interior angles is supplementary |
| Theorem 3-3 Alternate Exterior Angle | If two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent, |
| Theorem 3-4 Perpendicular Transversal | In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other., |
| Postulate 3-5 Euclidean Parallel Postulate | In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other. |
| Theorem 3-5 transversal alt int angles | If there is a line and a point not on the line, then there exists exactly one line though the point that is parallel to the given line., |
| Theorem 3-5 transversal alt int angles | If two lines in a plane are cut by a transversal so that a pair of alternate interior angles is congruent, then the two lines are parallel., |
| Theorem 3-6 | If two lines in a plane are cut by a transversal so that a pair of consecutive interior angles is supplementary, then the lines are parallel., |
| Theorem 3-8 | In a plane, if two lines are perpendicular to the same line, then they are parallel., |
| Theorem 3-7 | to be added |
| Postulate 3-2 | Two nonvertical lines have the same slope if and only if they are parallel., |
| Postulate 3-3 | Two nonvertical lines are perpendicular if and only if the product of their slopes is -1., |
| Postulate 3-4 | If two lines in a plane are cut by a transversal so that corresponding angles are congruent, then the lines are parallel., |
| Theorem 4-2 Third Angle Theorem | If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent., |
| Theorem 4-1 Angle Sum Theorem | The sum of the measures of the angles of a triangle is 180., |
| Theorem 4-3 Exterior Angle Theorem | The measure of an exterior angle of a trianlge is equal to, |
| Corollary 4-1 | The acute angles of a right triangle are complementary., |
| Postulate 4-1 SSS | (Side - Side - Side) - If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent., |
| Postulate 4-2 SAS | Side - Included Angle - Side) - 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 4-3 ASA | (Angle - Included Side - Angle) - 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. |
| Postulate 4-3 AAS | (Angle - Angle - Side) - If two angles and a NON-INCLUDED side of one triangle are congruent to the corresponding two angles and side of a second triangle, the two triangles are congruent. |
| Theorem 4-6 Isosceles Triangle Theorem (ITT) | If two sides of a triangle are congruent, then the angles opposite those sides are congruent., |
| Theorem 4-7 Converse of the ITT | If two angles of a triangle are congruent, then the sides opposite those angles are congruent. |
| Corollary 4-3 | A triangle is equilateral if and only if it is equiangular. |
| Corollary 4-4 | Each angle of an equilateral triangle measures 60 degrees. |
| Theorem 5-5 LL (Leg - Leg) | If the legs of one right triangle are congruent to the corresponding legs of another right triangle, then the triangles are congruent., |
| Theorem 5-6 HA (Hypotenuse - Angle) | If the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and corresponding acute angle of another right triangle, then the two triangles are congruent., |
| Theorem 5-7 LA (Leg - Angle) | If the leg and an acute angle of one right triangle are congruent to the corresponding leg and acute angle of another right triangle, then the triangles are congruent., |
| Postulate 5-1 HL (Hypotenuse -Leg) | If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent. |
| Theorem 5-8 Exterior Angle Inequality Theorem | If an angle is an exterior angle of a triangle, then its measure is greater than the measure of either of its corresponding remote interior angles., |
| Theorem 5-9 | If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite |
| Theorem 5-10 | If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle, |
| Theorem 5-11 | The perpendicular segment from a point to a line is the shortest segment from the point to the line., |
| Theorem 5-12 | The perpendicular segment from a point to a plane is the shortest segment from the point to the plane. |
| Theorem 5-12 Triangle Inequality Theorem | The sum of the lengths of any two sides of a triangle is greater than the length of the third side. |
| Theorem 6-5 | If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram |
| Theorem 6-6 | If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram |
| Theorem 6-7 | If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. |
| Theorem 6-8 | If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. |
| Parallelogram Property Summary | 1. Both pairs of opposite sides are parallel.2. Both pairs of opposite sides are congruent. 3. Both pairs of opposite angles are congruent. 4.Diagonals bisect each other. 5.A pair of opposite sides is both parallel and congruent. |
| Theorem 6-9 | If a parallelogram is a rectangle, then its diagonals are congruent |
| Theorem 6-10 (Converse of Thm 6-9) | If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. |
| Rectangle Property Summary | 1. Opposite sides are congruent and parallel. (All parallelograms) 2. Opposite angles are congruent. (All parallelograms) 3. Consecutive angles are supplementary. (All parallelograms) 4. Diagonals bisect each other. (All parallelograms) 5.Diagonals are congruent. (Rectangle) 6.All four angles are right angles. (Rectangle) |
| Theorem 6-11 | The diagonals of a rhombs are perpendicular |
| Theorem 6-12 (Converse of Thm 6-11) | If the diagonals of a parallelogram are perpendicular, |
| Theorem 6-13 | Each diagonal of a rhombus bisects a pair of opposite angles. |
| Trapezoids | Definition #1: A trapezoid is a quadrilateral with exactly one pair of parallel sides.Definition #2:A trapezoid is a quadrilateral with at least one pair of parallel sides. |
| Definition of Similar Polygons | Similar Polgyons (Scale Factor) Two polygons are Similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are proportional |
| Postulate7-1 AA (Angle - Angle) Similarity | f two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. |
| Theorem 7-1 SSS (Side - Side - Side) Similarity | If the measures of the corresponding sides of two triangles are proportional, then the triangles are similar. |
| Theorem 7-2 SAS (Side - Included Angle - Side) Similarity | If the measures of two sides of a triangle are proportional to the measures of two corresponding sides of another triangle and the included angles are congruent, then the triangles are similar. |
| Theorem 7-4 Triangle Proportionality | If a line is parallel to one side of a triangle and intersects the other two sides in two distinct points, then it separates these sides into segments of proportional lengths. |
| Theorem 7-5 Converse of the Triangle Proportionality | If a line intersects two sides of a triangle and separates the sides into corresponding segments of proportional lengths, then the line is parallel to the third side. |
| Theorem 7-4 Triangle Midpoint Proportionality | A segment whose endpoints are the midpoints of two sides of a triangle is parallel to the third side of the triangle, and its length is one-half the length of the third side. |
| Corollary 7-1 | If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally. |
| Corollary 7-2 | If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. |
| Theorem 7-7 Proportional Perimeters | If two triangles are similar, then the perimeters are proportional to the measures of corresponding sides. |
| Theorem 7-8 Proportional Altitudes | If two triangles are similar, then the measures of the corresponding altitudes are proportional to the measures of the corresponding sides. |
| Theorem 7-9 Proportional Angle Bisectors | If two triangles are similar, then the measures of the corresponding angle bisectors are proportional to the measures of the corresponding sides. |
| Theorem 7-10 Proportional Medians | If two triangles are similar, then the measures of the corresponding medians are proportional to the measures of the corresponding sides. |
| Theorem 7-11 Angle Bisector Theorem | An angle bisector in a triangle separates the opposite side into segments that have the same ratio as the other two sides. |
| Theorem 7-7 Proportional Perimeters | If two triangles are similar, then the perimeters are proportional to the measures of corresponding sides. |
| Theorem 7-8 Proportional Altitudes | If two triangles are similar, then the measures of the corresponding altitudes are proportional to the measures of the corresponding sides. |
| Theorem 7-9 Proportional Angle Bisectors | If two triangles are similar, then the measures of the corresponding angle bisectors are proportional to the measures of the corresponding sides. |
| Theorem 7-10 Proportional Medians | If two triangles are similar, then the measures of the corresponding medians are proportional to the measures of the corresponding sides. |
| Theorem 7-11 Angle Bisector Theorem | An angle bisector in a triangle separates the opposite side into segments that have the same ratio as the other two sides. |
| The Geometric Mean in Special Similar Triangles | Consider right triangle ABC with altitude CD drawn from the right angle C to the hypotenuse AB. This create three triangles that are all similar to each other. In the diagrams below we show the triangles relationships and how we can use them to calculate the geometric mean. |
| Theorem 8-4 The Pythagorean Theorem | In a right triangle, the sum of the squares of the measures of the legs equals the square of the measure of the hypotenuse. |
| Theorem 8-5 The Converse of the Pythagorean Theorem | If the sum of the squares of the measures of two sides of a triangle equals the square of the measure of the longest side, then the triangle is a right triangle. |
| Theorem 8-6 | In a 45-45-90 triangle, the hypotenuse is sqrt(2) times as long as a leg. (In this particular triangle the legs are of equal length.) |
| Theorem 8-7 | In a 30-60-90 triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is sqrt(3) times as long as the shorter leg. |
| Theorem 9-1 - | In n a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. |
| Theorem 9-2 - | In a circle, if a diameter is perpendicular to a chord, then it bisects the chord and its arc. |
| Theorem 9-3 | In a circle or in congruent circles, two chords are congruent if and only if they are equidistant from the center. |
| Theorem 9-4 | If an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of its intercepted arc |
| Theorem 9-5 | Inscribed Angles on Same Arc =If two inscribed angles of a circle or congruent circles intercept congruent arcs or the same arc, then the angles are congruent. |
| Theorem 9-7 | Cyclic QuadrilateralIf a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. |
| Theorem 9-8 | If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.The Converse of 9-8 then states: In a plane, if a line is perpendicular to a radius of a circle at the endpoint on the circle, then the line is a tangent of the circle. |
| Theorem 9-10 | External PointIf two segments from the same exterior point are tangent to the circle, then they are congruent. |
| A SECANT | is a line that intersects a circle in exactly two points |
