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Level 123

Theorems 2


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Line Intersection Theorem
Two different lines intersect in at most one point.
Vertical angles theorem
If two angles are vertical angles, then they are congruent.
Parallel Lines and Slopes Theorem
Two non-vertical lines are parallel if and only if they have the same slope.
Transitivity of Parallelism Theorem
If line AB is parallel to line DC and line DC is parallel to line FG, the line AB is parallel to line FG.
Parallel Property Theorem
Under a size change Sk, the line through any two preimage points is parallel to the line through their images.
Collinearity Is Preserved Theorem
Under Sk, the images of collinear points are collinear.
Angle Measure Is Preserved Theorem
Under Sk, an angle and its image have the same measure.
Two Perpendiculars Theorem
If two coplanar lines L and M are each perpendicular to the same line, then they are parallel to each other.
Perpendicular to Parallels Theorem
In a plane, if a line is perpendicular to 1 of 2 parallel lines, then it is also perpendicular to the other
Perpendicular Lines and Slopes Theorem
Two non-vertical lines are perpendicular if and only if the product of their slopes if -1.
Figure Transformation Theorem
If a figure is determined by certain points, then its transformation image is the corresponding figure determined by the transformation images of those points.
Two-Reflection Theorem for Translations
If m || n, the translation rn ༠ rm has magnitude two times the distance between m and n in the direction from m perpendicular to m.
A-B-C-D Theorem
Every isometry preserves Angle measure, Betweenness, Collinearity (lines), and Distance (lengths of segments).
Theorem (Equivalence Properties of Congruence)
For any figures F,G and H:
Segment Congruence Theorem
Two segments are congruent if and only if they have the same length.
Angle Congruence Theorem
Two angles are congruent if and only if they have the same measure.
Corresponding Parts in Congruent Figures (CPCF) Theorem
If two figures are congruent, then any pair of corresponding parts are congruent.
Parallel Lines Theorem
If two parallel lines are cut by a transversal:
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.
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 so that the two pairs of same-side interior angles are supplementary, then the lines are parallel.
Perpendicular Bisector Theorem
if the perpendicular bisector goes through a vertex of a triangle the legs will be congruent
Unique Circle Theorem
There is exactly one circle (a unique circle) through three given non-collinear points.
Uniqueness of Parallels Theorem
Through a point not on a line, there is exactly one line parallel to the given line.
Triangle-Sum Theorem
The sum of the measures of the angles of any triangle is 180 degrees.
Quadrilateral-Sum Theorem
The sum of the measures of the angles of a convex quadrilateral is 360.
Polygon-Sum Theorem
The sum of the measures of the angles of a convex n-gon is (n-2) x 180.
Exterior Angle Theorem for Triangles
In a triangle, the measure of an exterior angle of a triangle is equal to the sum of the measures of the interior angles at the other two vertices of the triangle.
Polygon Exterior Angle Theorem
The sum of the measures of the exterior angles of any convex n-gon is 360.
Flip-Flop Theorem
1.) If F and G are points and rl(F) = G, then rl(G) = F
Segment Symmetry Theorem
Every segment has exactly two symmetry lines:
Side-Switching Theorem
If one side of an angle is reflected over the line containing the angle bisector, its image is the other side of the angle
Angle Symmetry Theorem
The line containing the bisector of an angle is a symmetry line of the angle
Circle Symmetry Theorem
A circle is reflection-symmetric to any line through its center.
Symmetric Figures Theorem
If a figure is symmetric, then any pair of corresponding parts is congruent.
Isosceles Triangle Symmetry Theorem
The line containing the bisector of the vertex angle of an isosceles triangle is a symmetry line for the triangle.
Isosceles Triangle Coincidence Theorem
In an isosceles triangle, the bisector of the vertex angle, the perpendicular bisector of the base, and the median to the base determine the same line.
Isosceles Triangle Base Angles Theorem
If two sides of a triangle are congruent, then the angles opposite these sides are congruent.
Unequal Sides Theorem
If two sides of a triangle are not congruent, then the angles opposite them are not congruent, and the larger angle is opposite the larger side.
Converse of the Isosceles Triangle Bases Angles Theorem
If two angles of a triangle are congruent, then the sides opposite the angles are congruent.
Unequal Angles Theorem
If two angles of a triangle are not congruent, the sides opposite them are not congruent, and the longer side is opposite the larger angle.
Inscribed Angle Theorem
The measure of an angle inscribed in a circle is equal to one-half the measure of the intercepted arc.
Thales' Theorem
If an inscribed angle intercepts a semicircle than the angle is a right angle.
Kite Symmetry Theorem
The line containing the ends of a kite is a symmetry line for the kite.
Kite Diagonal Theorem
The symmetry line of a kite is the perpendicular bisector of the other diagonal and bisects the two angles at the ends of the kite.
Rhombus Diagonal Theorem
Each diagonal of a rhombus is a symmetry line of the rhombus and the perpendicular bisector of the other diagonal.
Trapezoid Angle Theorem
In a trapezoid, consecutive angles between a pair of parallel sides are supplementary.
Isosceles Trapezoid Symmetry Theorem
the perpendicular bisectors of one base of an isosceles trapezoid is the perpendicular bisectors of the other base and a symmetry line for the trapezoid
Isosceles Trapezoid Theorem
In an isosceles trapezoid, the non-base sides are congruent.
Rectangle Symmetry Theorem
The perpendicular bisectors of the sides of the rectangle are symmetry lines for the rectangle.
Theorem
A statement that is proved by reasoning deductively from already accepted statements.
Center of a Regular Polygon Theorem
For any regular polygon, there is a unique point (its center) that is equidistant from its vertices.
Regular Polygon Rotation Symmetry Theorem
A regular n-gon has n-fold rotation symmetry.
Regular Polygon Reflection Symmetry Theorem
Every regular polygon has reflection symmetry about:
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.
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.
All right angles are congruent.
Theorem 2-6-3 RIght Angle Supplements Theorem
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
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
In a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel.
Perpendicular Lines Theorem
In a coordinate plane, 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.
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 angles and nonincluded side 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 hypotenuse and leg of another right triangle, then the triangles are congruent.
Isosceles Triangle Theorem
If two sides of a triangle are congruent, the the angles opposite the sides are congruent.
Converse of the Isosceles Triangle Theorem
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 it 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 it is on the perpendicular bisector of the segment.
Angle Bisector Theorem
If a point is on the bisector of an angle, then it is equidistant from the sides of the angle.
Converse of the Angle Bisector Theorem
If a point in the interior of an angle is equidistant from the sides of the angle, then it is 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 located 2/3 of the distance from each vertex to the midpoint of the opposite side.
Triangle Midsegment Theorem
A midsegment of a triangle is parallel to a side of a triangle, and its length is half the length of that side.
Triangle Inequality Theorem
The sum of any two side lengths of a triange is greater than the third side length.
Hinge Theorem
If two sides of one triangle are congruent to two sides of another triangle and the included angles are not congruent, then longer third side is across from the larger included angle.
Converse of the Hinge Theorem
If two sides of one triangle are congruent to two sides of another triangle and the third sides are not congruent, then the larger included angle is across from the longer third side.
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.