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Geometry Theorems
These are all the theorems we've learned this year... Hope this helps.
105
Mathematics
9th Grade
06/02/2009
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Term
Theorem 2.1 Properties of Segment Congruence
 Definition
Segment congruence is reflexive, symmetric, and transitive.
Term
Theorem 2.2 Properties of Angle Congruence
 Definition
Angle congruence is reflexive, symmetric, and transitive.
Term
Theorem 2.3 Right Angle Congruence Theorem
 Definition
All right angles are congruent.
Term
Theorem 2.4 Congruent Supplements Theorem
 Definition
If two angles are supplementary to the same angle (or to congruent angles) then they are congruent.
Term
Theorem 2.5 Congruent Complements Theorem
 Definition
If two angles are complementary to the same angle (or to congruent angles) then they are congruent.
Term
Theorem 2.6 Vertical Angles Theorem
 Definition
Vertical angles are congruent.
Term
Theorem 3.1
 Definition
If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular.
Term
Theorem 3.2
 Definition
If two sides of two adjacent acute angles are perpendicular, then the angles are complementary.
Term
Theorem 3.3
 Definition
If two lines are perpendicular, then they intersect to form four right angles.
Term
Theorem 3.4 Alternate Interior Angles
 Definition
If two parallel lines are cut by a transversal, then the alternate interior angles are congruent.
Term
Theorem 3.5 Consecutive Interior Angles
 Definition
If two parallel lines are cut by a transversal, then the consecutive interior angles are supplementary.
Term
Theorem 3.6 Alternate Exterior Angles
 Definition
If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent.
Term
Theorem 3.7 Perpendicular Transversal
 Definition
If a transversal is perpendicular to one of two parallel lines, then it it perpendicular to the other.
Term
Theorem 3.8 Alternate Interior Angles Converse
 Definition
If two lines are cut by a transversal so that the alternate interior angles are congruent, then the lines are parallel.
Term
Theorem 3.9 Consecutive Interior Angles Converse
 Definition
If two lines are cut by a transversal so that the consecutive lines are supplementary, then the lines are parallel.
Term
Theorem 3.11
 Definition
If two lines are parallel to the same line, then they are parallel to each other.
Term
Theorem 3.12
 Definition
In a plane, if two lines are perpendicular to the same line, then they are parallel to each other.
Term
Theorem 4.1 Triangle Sum Theorem
 Definition
The sum of the measures of the interior angles of a triangle is 180 degrees.
Term
Corollary to Theorem 4.1
 Definition
The acute angles of a right triangle are complementary.
Term
Theorem 4.2 Exterior Angle Theorem
 Definition
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.
Term
Theorem 4.3 Third Angles Theorem
 Definition
If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.
Term
Theorem 4.4 Reflexive Property of Congruent Triangles
 Definition
Every triangle is congruent to itself.
Term
Theorem 4.4 Symmetric Property of Congruent Triangles
 Definition
If triangle ABC is congruent to triangle DEF, then triangle DEF is congruent to triangle ABC.
Term
Theorem 4.4 Transitive Property of Congruent Triangles
 Definition
If triangle ABC is congruent to DEF and DEF is congruent to triangle JKL, then triangle ABC is congruent to triangle JKL.
Term
Theorem 4.5 AAS Congruence Theorem
 Definition
If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of a second triangle, then the two triangles are congruent.
Term
Theorem 4.6 Base Angles Theorem
 Definition
If two sides of a triangle are congruent, then the angles opposite them are congruent.
Term
Corollary to Theorem 4.6
 Definition
If a triangle is equiangular, then it is equilateral.
Term
Theorem 4.7 Converse of the Base Angles Theorem
 Definition
If two angles of a triangle are congruent, then the sides opposite them are congruent.
Term
Theorem 4.8 HL Congruence Theorem
 Definition
If the hpotenuse and a leg of a right triangle are congruent to the hypotenuse and leg of a second right triangle, then the triangles are congruent.
Term
Theorem 5.1 Perpendicular Bisector Theorem
 Definition
If a point is on a perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
Term
Theorem 5.2 Converse of the Perpendicular Bisector Theorem
 Definition
If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.
Term
Theorem 5.3 Angle Bisector Theorem
 Definition
If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle.
Term
Theorem 5.4 Converse of the Angle Bisector Theorem
 Definition
If a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle.
Term
Theorem 5.5 Concurrency of Perpendicular Bisectors of a Triangle
 Definition
The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle.
Term
Theorem 5.6 Concurrency of Angle Bisectors of a Triangle
 Definition
The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle.
Term
Theorem 5.7 Concurrency of Medians of a Triangle
 Definition
The medians of a triangle intersect at a poin that is two thirds of the distance from each vertex to the midpoint of the opposite side.
Term
Theorem 5.8 Concurrency of Altitudes of a Triangle
 Definition
The lines containing the altitudes of a triangle are concurrent.
Term
Theorem 5.9 Midsegment Theorem
 Definition
The segment connecting the midpoints of two sides of a triangle is parallel to the third side and half as long.
Term
Theorem 5.10
 Definition
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.
Term
Theorem 5.11
 Definition
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.
Term
Theorem 5.12 Exterior Angle Inequality
 Definition
The measure of an exterior angle of a triangle is greater than the measure of either of the two nonadjacent interior angles.
Term
Theorem 5.13 Triangle Inequality
 Definition
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Term
Theorem 5.14 Hinge Theorem
 Definition
If two sides of one triangle are congruent to two sides of another tirangle, an 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.
Term
Theorem 5.15 Converse of the Hinge Theorem
 Definition
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.
Term
Theorem 6.1 Interior Angles of a Quadrilateral
 Definition
The sum of the measures of the interior angles of a quadrilateral is 360 degrees.
Term
Theorem 6.2
 Definition
If a quadrilateral is a parallelogram, then its opposite sides are congruent.
Term
Theorem 6.3
 Definition
If a quadrilateral is a parallelogram, then its opposite angles are congruent.
Term
Theorem 6.4
 Definition
If a quadrilateral is a parallogram, then its consecutive angles are supplementary.
Term
Theorem 6.5
 Definition
If a quadrilateral is a parallelogram, then its diagonals bisect each other.
Term
Theorem 6.6
 Definition
If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
Term
Theorem 6.7
 Definition
If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
Term
Theorem 6.8
 Definition
If an angle of a quadrilateral is supplementary to both of it's consecutive angles, then the quadrilateral is a parallelogram.
Term
Theorem 6.9
 Definition
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallogram.
Term
Theorem 6.10
 Definition
If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram.
Term
Rhombus Corollary
 Definition
A quadrilateral is a rhombus if and only if it has four congruent sides.
Term
Rectangle Corollary
 Definition
A quadrilateral is a rectangle if and only if it has four right angles.
Term
Square Corollary
 Definition
A quadrilateral is a square if and only if it is a rhombus and a rectangle.
Term
Theorem 6.11
 Definition
A parallelogram is a rhombus if and only if its diagonals are perpendicular.
Term
Theorem 6.12
 Definition
A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles.
Term
Theorem 6.13
 Definition
A parallelogram is a rectangle if and only if its diagonals are congruent.
Term
Theorem 6.14
 Definition
If a trapezoid is isoseles, then each pair of base angles is congruent.
Term
Theorem 6.15
 Definition
If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid.
Term
Theorem 6.16
 Definition
A trapezoid is isosceles if and only if its diagonals are congruent.
Term
Theorem 6.17 Midsegment Theorem for Trapezoids
 Definition
The midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of its bases.
Term
Theorem 6.18
 Definition
If a quadrilateral is a kite, then its diagonals are perpendicular.
Term
Theorem 6.19
 Definition
If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.
Term
Theorem 6.20 Area of a Rectangle
 Definition
The area of a rectangle is the product of its base and height. A=bh
Term
Theorem 6.21 Area of a Parallelogram
 Definition
The area of a parallelogram is the product of a base and its corresponding height. A=bh
Term
Theorem 6.22 Area of a Triangle
 Definition
The area of a triangle is one half the product of the base and its corresponding height. A=1/2bh
Term
Theorem 6.22 Area of a Triangle
 Definition
The area of a triangle is one half the product of a base and its corresponding height. A=1/2bh
Term
Theorem 6.23 Area of a Trapezoid
 Definition
The area of a trapezoid is one half the product of the height and the sum of the bases. A=1/2(b1+b2)
Term
Theorem 6.24 Area of a Kite
 Definition
The area of a kite is one half the product of the lengths of its diagonals. A=1/2d1d2
Term
Theorem 8.1
 Definition
If two polygons are similar, then the ratio of thir perimeters is equal to the ratios of their corresponding side lengths.
Term
Theorem 8.2 SSS Similarity Theorem
 Definition
If the lengths of the corresponding sides of two triangles are proportional, then the triangles are similar.
Term
Theorem 8.3 SAS Similarity Theorem
 Definition
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.
Term
Theorem 8.4 Triangle Proportionality Theorem
 Definition
If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.
Term
Theorem 8.5 Converse of the Triangle Proportionality Theorem
 Definition
If a line divides two sides of a triangle proportionally, then it is parallel to the third side.
Term
Theorem 8.6
 Definition
If three parallel lines intersect two transversals, then they divide the transversals proportionally.
Term
Theorem 8.7
 Definition
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.
Term
Theorem 9.1
 Definition
If an 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.
Term
Theorem 9.2
 Definition
In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into ttwo segments. The length of the altitude is the geometric mean of the lengths of the two segments.
Term
Theorem 9.3
 Definition
In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. Each leg of the right triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.
Term
Theorem 9.4 Pythagorean Theorem
 Definition
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.
Term
Theorem 9.5 Converse to the Pythagorean Theorem
 Definition
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.
Term
Theorem 9.6
 Definition
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 acute.
Term
Theorem 9.7
 Definition
If the square of the length of the longest side of a triangle is greeater than the sum of the squares of the length of the other two sides, then the triangle is obtuse.
Term
Theorem 9.8 45-45-90 Triangle Theorem
 Definition
In a 45-45-90 triangle, the hypotenuse is the square root of 2 times as long as each leg.
Term
Theorem 9.9 30-60-90 Triangle Theorem
 Definition
In a 30-60-90 triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is the square root of 3 times as long as the shorter leg.
Term
Theorem 10.1
 Definition
If a line is tangent to a circle, then it is perpendicular to the raduis drawn to the point of tangency.
Term
Theorem 10.2
 Definition
In a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle.
Term
Theorem 10.3
 Definition
If two segments fromt eh same exterior point are tangent to a circle, then they are congruent.
Term
Theorem 10.4
 Definition
In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.
Term
Theorem 10.5
 Definition
If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.
Term
Theorem 10.6
 Definition
If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.
Term
Theorem 10.7
 Definition
In the same circle or in congruent circles, two chords are congruent if and only if they are equidistant from the center.
Term
Theorem 10.8
 Definition
If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.
Term
Theorem 10.9
 Definition
If two inscribed angles of a circle intercept the same arc, the the angles are congruent.
Term
Theorem 10.10
 Definition
If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle.
Term
Theorem 10.11
 Definition
A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.
Term
Theorem 10.12
 Definition
If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc.
Term
Theorem 10.13
 Definition
If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
Term
Theorem 10.14
 Definition
If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.
Term
Theorem 10.15
 Definition
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.
Term
Theorem 10.16
 Definition
If two secant segments share the same endpoint ouside a circle, then the product of the length of one secant segment and the length of its external segment equals the product of the length of the other secand segment and the length of its external segment.
Term
Theorem 10.17
 Definition
If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the length of the secant segment and the length of its external segment equals the square of the length of the tangent segment.
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