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Linear Pair Theorem

If two angles form a linear pair, then they are supplementary.

Congruent Supplements Theorem

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

RIght Angle Supplements Theorem

All right angles are congruent.

Congruent Complements Theorem

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

Common Segments Theorem

Given collinear points A, B, C, and D arranged as shown, if segment AB is congruent to segment CD, then segment AC is congruent to segment BD.

Vertical Angles Theorem

VAT Vertical 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.

Same-Side Interior Angles Theorem

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

Converse of the Alternate Interior Angles Theorem

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

Converse of the Alternate Exterior Angles Theorem

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

Converse of the Same-Side Interior Angles Theorem

If two coplanar lines are cut by a transversal so that a pair of same-side interior angles are supplementary, then the two lines are parallel.

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.

Parallel Lines Theorem

two nonvertical lines are parallel if and only if they have the same slope.

Perpendicular Lines Theorem

two nonvertical lines are perpendicular if and only if the product of their slopes is -1.

Triangle Sum Theorem

The sum of the angle measures of a triangle is 180°.

Exterior Angle Theorem

The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles.

corollary 1: Third Angles Theorem

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

Angle-Angle-Side (AAS) Congruence Theorem

If two angles and a nonincluded side of one triangle are congruent to the corresponding parts of another triangle then the triangles are congruent.

Hypotenuse-Leg (HL) Congruence Theorem

If the hypotenuse and a leg of a right triangle are congruent to the corresponding parts of another right triangles then the triangles are congruent.

Isosceles Triangle Theorem (ITT)

If two sides of a triangle are congruent, the the angles opposite to those sides are congruent.

Converse of the Isosceles Triangle Theorem (converse of ITT)

If two angles of a triangle are congruent, the the sides opposite those angles are congruent.

Perpendicular Bisector Theorem

If a point is on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment.

Converse of the Perpendicular Bisector Theorem

If a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment.

Angle Bisector Theorem

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

Converse of the Angle Bisector Theorem

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

Circumcenter Theorem

The perpendicular bisectors of the sides of a triangle intersect in a point that is equidistant from the three verities of the triangle.

Incenter Theorem

The bisector of the angles of a triangle intersect in a point that is equidistant from the three sides of the triangle.

Centroid Theorem

The medians of a triangle intersect in a point that is located 2/3 of the distance from each vertex to the midpoint of the opposite side.

Triangle Midsegment Theorem

The segment that joins the midpoints of two sides of a triangle is parallel to the third side, and its length is half the length of that side.

Triangle Inequality Theorem

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

Converse of the Pythagorean Theorem

If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle.

Pythagorean Inequalities Theorem

In ΔABC, c is the length of the longest side. If c² > a² + b², then ΔABC is an obtuse triangle. If c² < a² + b², then ΔABC is an acute triangle.

45°-45°-90° Triangle Theorem

In a 45°-45°-90° triangle, both legs are congruent, and the length of the hypotenuse is the length of a leg times √2.

30°-60°-90° Triangle Theorem

In a 30°-60°-90° triangle, the length of the hypotenuse is 2 times the length of the shorter leg, and the length of the longer leg is the length of the shorter leg times √3.

Polygon Angle Sum Theorem

The sum of the interior angle measures of a convex polygon with n sides is ( n - 2 )180°

Polygon Exterior Angle Sum Theorem

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

Triangle Midsegment Theorem-

opposite sides of a parallelogram are congruent.

Perpendicular Bisector Theorem-

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

Triangle Inequality Theorem-

If a parallelogram is a rectangle, then its diagonals are congruent.

The Hinge Theorem (SAS Inequality Theorem)-

If a parallelogram is a rhombus, then its diagonals are perpendicular.

Converse Theorem

If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus.

Trapezoid Midsegment Theorem

The midsegment (or median) of a trapezoid is parallel to each base, and its length is one half the sum of the lengths of the bases.

Side-Side-Side (SSS) Similarity Theorem

If the three sides of one triangle are proportional to the three corresponding sides of another triangle, then the triangles are similar.

Side-Angle-Angle (SAA) Similarity Theorem

If two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar.

Triangle Proportionality Theorem

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

Triangle Angle Bisector Theorem

An angle bisector of a triangle divides the opposite side into two segments whose lengths are proportional to the lengths of the other two sides.

Proportional Perimeters and Areas Theorem

If the similarity ratio of two similar figures is a / b, then the ratio of their perimeters is a / b, and the ratio of their areas is a² / b² or ( a / b )².

Pythagorean Theorem

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

The Law of Sines

For any ΔABC with side lengths a, b, and c, sin A / a = sin B / b = sin C / c.

The Law of Cosines

For any ΔABC with sides a, b, and c, a² = b² + c² - 2b cos A, b² = a² + c² - 2ac cos B, and c² = a² + b² - 2ab cos C.

Inscribed Angle Theorem

The measure of an inscribed angle is half the measure of its intercepted arc.

Chord-Chord Product Theorem

If two chords intersect in the interior of a circle, then the products of the lengths of the segments of the chords are equal.

Secant-Secant Product Theorem

If two secants intersect in the exterior of a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment.

Secant-Tangent Product Theorem

If a secant and a tangent intersect in the exterior of a circle, then the product of the lengths of the secant segment and its external segment equals the length of the tangent segment squared.

Equation of a Circle

the equation of a circle with center ( h, k ) and radius r is ( x - h )² + ( y - k )² = r².

intersection of two lines and point

If two lines intersect, then they intersect in exactly one point.

Line and point

Through a line and a point not in the line there is exactly one plain.

Intersection of two lines and plane.

If two lines intersect, then exactly one plane contains the lines

mid point theorem

if M is the midpoint of ab, then AM=1/2AB and MB=1/2AB

Angle Bisector theorem

If (ray) →BX is the bisector of ∠ABC, then m∠ABX =1/2m∠ABC and m∠XBC=1/2m∠ABC.