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Theorems

Page history last edited by Erika Rosales 12 years, 2 months ago

Geometry Theorems

 

 

 

Theorem 2.1  Congruence of Segments

 

 Segment congruence is reflexive, symmetric, and transitive.

                                                         ___                      ___

     Reflexive     For any segment AB, AB is congruent to AB

                           ___                       ___        ___                       ___

     Symmetric   If AB is congruent to CD, then CD is congruent to AB.

                           ___                      ___       ___                      ___         ___                      ___

     Transitive    If AB is congruent to CD and CD is congruent to EF, then AB is congruent to EF.

 

Theorem 2.2  Congruent Angles

 

Angle congruence is reflexive, symmetric, and transitive.

 

     Reflexive     For any angle A, angle A is congruent to angle A.

 

     Symmetric   If angle A is congruent to angle B, then angle B is congruent to angle A.

 

     Transitive     If angle A is congruent to angle B and angle B is congruent to angle C, then angle A is

                         congruent to angle C.

 

Theorem 2.3  Right Angle Congruence Theorem

 

 All right angles are congruent. 

 

 

Theorem 2.4 Congruent Supplements Theorem

 

If two angles are supplementary to the same angle(or to congruent angles), then they are congrutent.

 

 

Theorem 2.5  Congruent Complements Theorem

 

If two angles are complementary to the same angle (or to congruent angles), then they are congruent.

 

 

Theorem 2.6 Vertical Angles Congruence Theorem

 

Vertical angles are congruent.

 

Theorem 3.1  Alternate Interior Angles Theorem

 

If two parallel lines are cut by a transversal, then the pairs of alternate interior angls are congruent.

 

Theorem 3.2  Alternate Exterior Angles Theorem

 

If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.

 

Theorem 3.3 Consecutive Interior Angles Theorem

 

If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary.

 

Theorem 3.4 Alternate Interior Angles Converse

 

If two lines are cut by a transversal so the alternate interior angles are congruent, then the lines are parallel. 

 

Theorem 3.5 Alternate Exterior Angles Converse

If two lines are cut by a transversal so the alternate exterior angles are congruent, then the lines are parallel.

 

Theorem 3.6 Consecutive Interior Angles Converse

 

If two lines are cut by a transversal so the consecutive interior angles are suplementary, then the lines are parallel.

 

Theorem 3.7 Transitive Property of Parallel Lines

 

If tow lines are parallel to the same line, then they are parallel to each other.

 

Theorem 3.8

 

If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular.

 

Theorem 3.9

 

If two lines are perpendicular, then they intersect to form four right angles.

 

Theorem 3.10

 

If two sides of two adjacent acute angles are perpendicular, then the angles are complementary.

 

Theorem 3.11 Perpendicular Transversal Theorem

 

If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.

 

Theorem 3.12 Lines Perpendicular to a Transversal Theorem

 

In a plane, if two lines ar perpendicular to the same line, then they are parallel to each other.

 

 Thereom 4.1 Triangle Sum Theorem

 

  The sum of the measures of the interior angles of a triangle is 180 degrees.

 

Theorem 4.2 Exterior Angle Theorem

 

The measure of an exterior angle of a triangle is egual to the sum of the measures of the two nonadjacent   interior angles.

 

 

Corollary to the Triangle Sum Theorem 

 

The acute angles of a right triangle are complementary.

 

Theorem 4.3 Third Angles Theorem

 

If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.

 

Theorem 4.4 Third Angles Theorem

 

Reflexive Property of Congruent Triangles

 

For any triangle ABC, Formula

 

Symmetric Property of Congruent Triangles

 

 

If Formulathen,Formula

 

 

Transitive Property of Congruent Triangles

 

 

If FormulaandFormula, then Formula

 

Theorem 4.5: Hypotenuse Leg(HL) Congruence Theorem

 

if the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the 2 triangles are congruent.

 

 Theorem 4.6: Angle-Angle- Side (AAS) Congruence Theorem

 

If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent. 

 

Theorem  5.1: Midsegment Theorem

 

The segment connecting the midpoints of twp sides of a triangle is parrallel to the third side and is half as long as that side.

 

Theorem 5.2: Perpendicular Bisector Theorem

 

 In a plane, if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

 

Theorem 5.3: Converse of Perpendicular Bisector Theorem

 

 In a plane, if a point os eguidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

 

Theorem 5.4: Concurrecy of Perpendicular Bisectors of a Triangle

 

The perpedicular bisectors of a triangle intersect at a point that is equidistant from the vertices of a triangle.

 

Theorem 5.5: Angle Bisector Theorem 

 

If a point is on the bisector of an angle, then it is equidistant from the two soides of the angle.

 

Theorem 5.6: Converse to the Angle Bisector Theorem

 

If a point is on the interior of an angle and is equidistant from the sides of the angle,  then it lies on the bisector of the the angle. 

 

Theorem 5.7: Concurrency of Angle Bisectors of a Triangle

 

The angle bisectors of a triangel intersect at a point that is equidistant from the sides of the triangle.

 

Theorem 5.8: Concurrency of Medians of a Triangle

 

The medians of atriangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of  the opposite side.

 

Theorem 5.9: Concurrecy of Alltitudes of a Triangle

 

The lines containing the alttitudes of a triangle are concurrent.

 

Theorem 5.10

 

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.

 

Theorem 5.11

 

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 5.12: 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 5.13: 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, than the third side of the first is longer than the third side of the second.

 

Theorem 5.14 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.

 

Theorem 6.1 Perimeters of Similar Polygons

 

 If two polygons are similar,then the ratio of their perimeters is equal to the ratios of their corresponding side lengths.

 

Theorem 6.2 Side-Side-Side (SSS) Similarity Theorem

 

 If the corresponding side lengths of two triangles are proportional, then the triangles are similar.

 

Theorem 6.3 Side-Angle-Side (SAS) Similarity Theorem

If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.

 

Theorem 6.4 Triangle Proportionality Theorem

If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.

 

Theorem 6.5 Converse of the Triangle Proportionality Theorem

 

If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

 

Theorem 6.6

 

If three parallel lines intersect two transversals, then they divide the transversals proportionally.

 

Theorem 6.7

 

If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides.

 

Theorem 7.1 Pythagorean Theorem

 

In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

 

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 less than the sum of the squares of the lengths of the other two sides, then the triangle is an acute triangle.

 

Theorem 7.4

 

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.

 

Theorem 7.5

 

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 7.6 Geometric Mean (Altitude) Theorem

 

In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. 

The length of the altitude is the geometric mean of the lengths of the two segments.

 

Theorem 7.7 Geometric Mean (Leg) Theorem

 

In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments.

The lenght of each leg of the right triangle is the geometric mean of the lenghts of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.

 

Theorem 7.8 45°-45°-90° Triangle Theorem

 

In a 45°-45°-90° triangle, the hypotenuse is √2 times as long as each leg.

hypotenuse = leg × √2                                                                                    

 

Theorem 7.9 30°-60°-90° Triangle Theorem

 

In a 30°-60°-90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is√3 times as long as the shorter leg.

 

hypotenuse = 2 × shorter leg                                                    longer leg = shorter leg × √3 

 

 

Theorem 8.1 Polygon Interior Angle Theorem

 

The sum of the measures of the interior angles of a convex n-gon is (n-2) times 180 degrees. 

 

Collary to Theorem 8.1

 

The sums of the measures of the interior angles of a quadrilateral is 360 degrees.

 

Theorem 8.2

 

The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360 degrees.

 

Theorem 8.3

 

If a quadrilateral is a parallelogram, then its opposite sides are congruent

 

Theorem 8.4

 

If a quadrilateral si a parallelogra, then its opposite anglesare congruent.

 

Theorem 8.5

 

If a quadrilateral is a parellelogram, then its consecutive angles are supplementary.

 

Theorem 8.6

 

If a quadrateral is a parallelogram, then its diagons 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 are congruent and parallel, then the quadrilateral is a parallelogram.

Theorem 8.10

If the diagnals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

 

Comments (1)

David Cox said

at 10:28 am on Oct 30, 2008

Nice Job, Vanessa.

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