Geometry Theorems
Theorem 2.1 Congruence of Segments
Segment congruence is reflexive, symmetric, and transitive.
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Reflexive For any segment AB, AB is congruent to AB
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Symmetric If AB is congruent to CD, then CD is congruent to AB.
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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,
Symmetric Property of Congruent Triangles
If then,
Transitive Property of Congruent Triangles
If and, then
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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