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Level 122

Theorems


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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.