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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 exterior angle measures, one angle at each vertex, of a convex polygon is 360°.

Trapezoid Midsegment 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.

midpoint theorem

if M is the midpoint of AB, then AM is congruent to MB

Alternate Interior Angles Theorem

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

consecutive interior angles theorem

if two parallel lines are cut by a transversal, then each pair of consecutive interior angles is supplementary

Alternate Exterior Angles Theorem

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

Perpendicular Transversal Theorem

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

Triangle Sum Theorem

angles add up to 180°

Third Angle Theorem

If two angles of a triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent.

Exterior angle theorem

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

right angles congruence theorem

all right angles are congruent

Congruent supplements theorem

if two angles are supplementary to the same angle (or to congruent angles), then they are congruent

Congruent Complements Theorem

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

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

lines perpendicular to a transversal theorem

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

corollary to the triangle sum theorem

the acute angles of a right triangle are complementary

hypotenuse-leg 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-angle-side 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

Base Angles Theorem

If 2 sides of a triangle are congruent then the angles opposite are congruent (used with isosceles)

converse of base angles theorem

if two angles of a triangle are congruent, then the side opposite them are congruent

corollary to the base angles theorem

if a triangle is equilateral, then it is equiangular

corollary to the converse of a base angles theorem

if a triangle is equiangular, then it is equilateral

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

Perpendicular Bisector Theorem

if the perpendicular bisector goes through a vertex of a triangle the legs will be congruent

Converse of the Perpendicular Bisector Theorem

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

concurrency of perpendicular bisectors of a triangle

the perpendicular bisectors of a triangle intersect at the point that is equidistant from the vertices of the triangle

angle bisector theorem

if a point is on the bisector of an angle, then it is equidistant from the other two sides of the angle

Converse of the Angle Bisector Theorem

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

concurrency of angle bisectors of a triangle

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

concurrency of medians of a triangle

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

concurrency of altitudes of a triangle

the lines containing the altitudes of a triangle are concurrent

Triangle Inequality Theorem

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

Properties of Segment Congruence

Segment congruence is reflexive, symmetric, and transitive.

Properties of Angle Congruence

Angle congruence is reflexive, symmetric, and transitive.

Vertical Angles Congruence Theorem

Vertical angles are congruent.

Corollary

The acute angles of a right triangle are complementary.

Third Angles Theorem

If 2 angles of one triangle are congruent to 2 angles of another then the third angles are also congruent

Properties of Triangles Congruence

Triangle congruence is reflexive, symmetric, and transitive.

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-Angle-Side (AAS) Congruence Theorem

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

Converse of the Base Angles Theorem

If 2 angles of a triangle are congruent then the sides opposite are congruent (used with isosceles)

Concurrency of Perpendicular Bisectors Theorem

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

Side-Side-Side (SSS) 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.

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

Converse of the Pythagorean Theorem

If a^2 + b^2 = c^2, then the triangle is a right triangle, where a and b are the lengths of the legs and c is the length of the hypotenuse of the triangle.