Theorems & Postulates

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linear pair theorem
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if two angles form a linear pair, then they are supplementary
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congruent supplements theorem
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if two angles are supplementary to the same angle (or to congruent angles) then they are congruent
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right angle congruence theorem
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all right angles are congruent
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congruent complements theorem
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if two angles are complementary to the same angle (or to congruent angles) then the two angles are congruent
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common segments theorem
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Given collinear points A, B, C and D arranged as shown if line AB is congruent to line CD then line AC is congruent to line BD a__,__b_______c__,__d
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vertical angles theorem
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vertical angles are congruent
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corresponding angles postulate
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if two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent
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alternate interior angles theorem
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if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent
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alternate exterior angles theorem
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If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent
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same side interior angles theorem
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if a transversal intersects 2 parallel lines, then the same side interior angles are supplementary
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converse of the corresponding angles postulate
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if two coplanar lines are cut by a transversal so that a pair of corresponding angles are congruent, then the two lines are parallel
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parallel postulate
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Through a point not on a line, there is one and only one line parallel to the given line.
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converse of the alternate interior angles theorem
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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
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converse of the alternate exterior angles theorem
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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.
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converse of the same side interior angles theorem
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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.
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perpendicular transversal theorem
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in a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other
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parallel lines theorem
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in a coordinate plane, two non-vertical lines are parallel if and only if they have the same slope. any two vertical lines are parallel
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perpendicular lines theorem
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In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1. Vertical and horizontal lines are perpendicular.
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definition of a linear pair
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A linear pair is two adjacent angles whose exterior sides form a straight line.
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corollary
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a theorem whose proof follows directly from another theorem.
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exterior angle theorem
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the measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles
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third angles theorem
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if two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are also congruent
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definition of corresponding angles
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two angles that are formed by two lines and a transversal and occupy corresponding positions
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side-side-side congruence postulate
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If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
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side-angle-side congruence postulate
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If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
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angle-side-angle congruence postulate
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If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.
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angle-angle-side congruence theorem
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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.
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hypotenuse-leg congruence theorem
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If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.
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vertex angle
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the angle formed by the legs.
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isosceles triangle theorem
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If two sides of a triangle are congruent,then the angles opposite the sides are congruent.
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converse of isosceles triangle theorem
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if two angles of a triangle are congruent, then the sides opposite those angles are congruent
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equilateral triangle corollary
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if a triangle is equilateral, then it is equiangular
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equiangular triangle corollary
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if a triangle is equiangular, then it is equilateral
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ruler postulate
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the points on a line can be put into a one-to-one correspondence with the real numbers
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definition of a line segment
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A part of a line with two endpoints.
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angle addition postulate
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if point B lies in the interior of <aoc then m<aob+m<boc=m<AOC

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