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Level 124

Theorems 3


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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.