Big Ideas Math Geometry Theorems/Postulates

Right Angles Congruence Theorem

All right angles are congruent

Congruent Supplements Theorem

If two angles are supplementary to the same angle(or to congurent angles), Then they are congruent

Congruent Complements Theorem

If two angles are complementary to the same angle(or to congurent angles), Then they are congruent

Vertical Angles Congruence Theorem

Vertical Angles are congruent

Corresponding Angles Theorem

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

Alternate Interior Angles Theorem

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

Alternate Exterior Angles Theorem

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

Consecutive Interior Angles Theorem

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

Corresponding Angles Converse

If two lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel

Alternate Interior Angles Converse

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

Alternate Exterior Angles Converse

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

Consecutive Interior Angles Converse

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

Transitive Property of Parallel Lines

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

Linear Pair Perpendicular Theorem

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

Perpendicular Transversal Theorem

In a plane, if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other line

Lines Perpendicular to a Transversal Theorem

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

Slopes of Parallel Lines

Two nonvertical lines are parallel iff they have the same slope. Any two vertical lines are parallel

Slopes of Perpendicular Lines

Two nonvertical lines are perpendicular iff the product of their slopes is -1. Horizontal lines are perpendicular to vertical lines

Composition Theorem

The composition of two(or more) rigid motions is a rigid motion

Reflections in Parallel Lines Theorem

If lines k and m are parallel then a reflection in line k followed by a reflection in line m is the same as a translation. if A'' is the image of A, then

1. AA'' is perpendicular to k and m,

2 AA'' = 2d where d is the distance between k and m

Reflections in Intersecting Lines Theorem

Relection in two lines that intersect is the same as a rotation around point p. The angle of rotation is 2x where x is the measure of the acutre or right angle formed by the two lines

Triangle Sum Theorem

The sum of the measure of the interior angles of a triangle is 180 degrees

Exterior Angle Theorem

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior anlges

Corollary to the Triangle Sum Theorem

The acute angles of right triangle are complementary

Properties of Triangle Congruence

Reflexive, Symmetric, Transitive

Third Angles Theorem

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

Side-Angle-Side (SAS) Congruence Theorem

If two sides and the included angle of one triangle are congruent to two sides and the included angles of a second triangle, then the two triangles are congruent

Base Angles Theorem

If two sides of a triangle are congruent, then the angles opposite them are congruent

Converse to the Base Angles Theorem

If two angles of a triangle are congruent, then the sides opposite them are congruent

Corollary to the Base Angles Theorem

If a triangle is equilateral, then it is equiangular

Corollary to the Converse of the Base Angles Theorem

If a triangle is equiangular, then it is equilateral

Side-Side-Side(SSS) Congruence Theorem

If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent

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 two triangles are congruent

Angle-Side-Angle(ASA) Congruence Theorem

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

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

Perpendicular Bisector Theorem

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

Converse of the Perpendicular Bisector Theorem

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

Angle Bisector Theorem

If a point lies on the bisector of an angle, then it is equidistant form the two sides of the angle

Converse of the Angle Bisector Theorem

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

Circumcenter Theorem

The circumcenter of a triangle is equidistant from the vertices of the triangle

Incenter Theorem

The incenter of a triangle is equidistant from the sides of the triangle

Centroid Theorem

The centroid of a triangle is two-thirds of the distance from each vertex to the midpoint to the opposite side

Triangle Midsegment Theorem

The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long as that side

Triangle Longer Side Theorem

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

Triangle Larger Angle Theorem

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

Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle is greater than the length of the third side

Hinge Theorem

If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the second, then the third side of the first is longer than the third side of the second

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

Polygon Interior Angles Theorem

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

Corollary to the Polygon Interior Angles Theorem

The sum of the measures of the interior angles of a quadrilateral is 360

Polygon Exterior Angles Theorem

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

Parallelogram Opposite Sides Theorem

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

Parallelogram Opposite Angles Theorem

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

Parallelogram Consecutive Angles Theorem

If a quadrilateral is a parallelogram, then its consecutive angles are supplementary

Parallelogram Diagonals Theorem

If a quadrilateral is a parallelogram, then its diagonals bisect each other

Parallelogram Opposite Sides Converse

If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram

Parallelogram Opposite Angles Converse

If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram

Opposite Sides Parallel and Congruent Theorem

If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram

Parallelogram Diagonals Converse

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

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 anlges

Square Corollary

A quadrilateral is a square if and only if it has a rhombus and a rectangle

Rhombus Diagonals Theorem

A parallelogram is a rhombus if and only if its diagonals are perpendicular

Rhombus Opposite Angles Theorem

A parallelogram if and only if each diagonal bisects a pair of opposite angles

Rectangle Diagonals Theorem

A parallelogram is a rectangle if and only if its diagonals are congruent

Isosceles Trapezoid Base Angles Theorem

If a trapezoid is isosceles, then each pair of base angles is congruent

Isosceles Trapezoid Base Angles Converse

If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid

Isosceles Trapezoid Diagonals Theorem

A trapezoid is isosceles if and only if its diagonals are congruent

Trapezoid Midsegement 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

Kite Diagonals Theorem

If a quadrilateral is a kite, then its diagonals are perpendicuar

Kite Opposite Angles Theorem

If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent

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

Areas of Similar Polygons

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

Angle-Angle(AA) Similarity Theorem

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

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

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

Converse of the Triangle Proportionality Theorem

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

Three Parallel Lines Theorem

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

Triangle Angle Bisector Theorem

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

Pythagorean Theorem

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

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 trianlge

Pythagorean Inequalities Theorem

For any triangle ABC, where c is the length of the longest side,

if c^2>a^2+b^2 then acute if c^2

45-45-90 Triangle Theorem

The legs are the same length, Hypotenuse is root 2 times as long

30-60-90 Triangle Theorem

Hypotenuse is x2 length of short leg

Long leg is xroot 3 length of short leg

Right Triangle Similarity Theorem

If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle to each other

Geometric Mean Altitude Theorem

Remember the Equation

Geometric Mean Leg Theorem

Remember the Equation

Law of Sines

sin a/a = sin b/b =sin c/c

a/sin a = b/sin b =c/sin c

Law of Cosines

a^2 =b^2+c^2-2bc cos A

b^2 =a^2+c^2-2ac cos b

c^2 =b^2+a^2-2ba cos c

Tangent Line to Circle Theorem

In a plane, a line is tangent to a circle iff theline is perpendicular to a radius of the cirlce at its endpoint on the circle

External Tangent Congruence Theorem

Tangent segments from a common external point are congruent

Congruent Circles Theorem

Two circles are congruent circles iff they have the same radius

Congruent Central Angles Theorem

In the same circle, or in congruent circles, two minor arcs are congruent iff their corresponding central angles are congruent

Similar Circles Theorem

All circles are similar

Congruent Corresponding Chords Theorem

In the same circle, or in congruent circles, tow minor arcs are congruent iff their corresponding chords are congruent

Perpendicular Chord Bisector Theorem

If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc

Perpendicular Chord Bisector Converse

If one chord of a circle is a perpendicular bisector of another chord, then the first chord is a diameter

Equidistant Chords Theorem

In the same circle, or in congruent circles, two chords are congruent iff they are equidistant from the center

Measure of an Inscribed Angle Theorem

The measure of an inscribed anlge is one-half the measure of its intercepted arc

Inscribed Angles of a Circle Theorem

If two inscribed Angles of a circle intercept the same arc, then the angles are congruent