Theorems & Postulates Math Fundamentals Reference Sheet Page 1

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1 Math Fundamentals Reference Sheet Page Triangle In a triangle, the length of the hypotenuse is two times the length of the shorter leg, and the length of the longer leg is the length of the shorter leg times the square root of three Triangle In a triangle, both legs are congruent, and the length of the hypotenuse is the length of a leg times the square root of two. Angle, Angle, Side (AAS) If two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle, then the triangles are congruent. Addition Property of Equality If the same number is added to equal numbers, then their sums are equal. Alternate Exterior Angles If two parallel lines are intersected by a transversal, then their alternate exterior angles are equal in measure. Alternate Interior Angles If two parallel lines are intersected by a transversal, then their alternate interior angles are equal in measure. Angle Addition Postulate For any angle, the measure of the whole is equal to the sum of the measures of its nonoverlapping parts. Angle Bisector If a segment, ray, line or plane is an angle bisector, then it divides an angle so that each part of the angle is equal to one half of the whole angle.

2 Math Fundamentals Reference Sheet Page 2 Angles Outside the Circle If a tangent and a secant, two tangents, or two secants intersect outside a circle, then the measure of the angle formed is one-half the difference of the measures of the intercepted arcs. Arc Addition Postulate The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. Area Addition Postulate The area of a region is equal to the sum of the areas of its nonoverlapping parts. Angle, Side, Angle (ASA) If two angles and an included side of one triangle are congruent to two corresponding angles and an included side of another triangle, then the triangles are congruent. Centriod The centriod of a triangle is located 2/3 of the distance from each vertex to the midpoint of the opposite side. Combining Like Terms If we add or subtract expressions on the same side of the equal sign, then we have combined like terms. Common Segments If two collinear segments adjacent to a common segment are congruent, then the overlapping segments formed are congruent. Congruent Complements If two angles are complements of the same angle, then they are congruent.

3 Math Fundamentals Reference Sheet Page 3 Congruent Corresponding Chords In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. Congruent Supplements If two angles are supplements of the same angle, then they are congruent. Converse of Alternate Exterior Angle If alternate exterior angles are congruent, then the lines that form them will be parallel to one another. Converse of Alternate Interior Angle If alternate interior angles are congruent, then the lines that form them will be parallel to one another. Converse of Corresponding Angles Postulate If two lines are intersected by a transversal and the corresponding angles are equal in measure, then the lines are parallel. Converse of Isosceles Triangle If two angles of a triangle are equal in measure, then the sides opposite those angles are equal in measure. Converse of Same Side Interior Angle If same side interior angles are supplementary, then the lines that form them will be parallel to one another.

4 Math Fundamentals Reference Sheet Page 4 Converse of the Angle Bisector 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. Converse of the Perpendicular Bisector If a point is the same distance from both the endpoints of a segment, then it lies on the perpendicular bisector of the segment Converse of the Perpendicular Chord Bisector If one chord of a circle is a perpendicular bisector of another chord, then the first chord is a diameter. Converse of the Triangle Proportionality If a line divides two sides of a triangle proportionally, then it is parallel to the third side. Corresponding Angles Postulate If two parallel lines are intersected by a transversal, then the corresponding angles are equal in measure. CPCTC If two triangles are congruent, then all other corresponding parts (sides & angles) of the congruent triangles are congruent. Distributive Property If an expression is being multiplied by a value and that value is multiplied out to all terms in the expression (or parentheses), then the value has been distributed.

5 Math Fundamentals Reference Sheet Page 5 Division Property of equality If equal numbers are divided by the same number, then the quotients are equal. Equation of a Circle The equal of a circle with center (h, k) and radius r is (x h) 2 + (y k) 2 = r 2. Exterior Angle If we have an exterior angle of a triangle, then its measure will equal the sum of its two remote interior angles. External Tangent Congruence If two segments from the same exterior point are tangent to a circle, then they are congruent. Geometric Mean (Altitude) In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the geometric mean of the lengths of the two segments of the hypotenuse. Geometric Mean (Leg) In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg. Geometric Mean The nth root of a sum of n numbers is that set s geometric mean. Hypotenuse Leg Congruence (HL) 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.

6 Math Fundamentals Reference Sheet Page 6 Inscribed Angle The measure of an inscribed angle is half the measure of its intercepted arc. Inscribed Angles of a Circle If two inscribed angles of a circle intercept the same arc (or congruent arcs), then the angles are congruent. Inscribed Quadrilateral If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. Inscribed Right Angle An inscribed angle of a triangle intercepts a diameter or semicircle if and only if the angle is a right angle. Isosceles triangle theorem If two sides of a triangle are equal in measure, then the angles opposite those sides are equal in measure. Cosine (cos) For an acute angle of a right triangle the ration of the side adjacent to the angle to the measure of the hypotenuse. (adj/hyp) Law of Cosines For any triangle ABC with sides a, b, and c, the equation c 2 = a 2 + b 2 2abcosC is the law of cosines. Sine (sin) For an acute angle of a right triangle, the ratio of the side opposite the angle to the measure of the hypotenuse. (opp/hyp)

7 Math Fundamentals Reference Sheet Page 7 Law of Sines For any triangle ABC with side lengths a, b, and c, the equation sina a = sinb b = sinc c is the law of sines. Tangent (tan) For an acute angle of a right triangle, the ratio of the side opposite to the angle to the measure of the side adjacent (opp/adj) Linear Pair If two angles form a linear pair, then they are supplementary. Midpoint If a point is a midpoint, then the point divides a segment so that each part of the segment is equal to one half of the whole segment. Multiplication Property of Equality If equal numbers are multiplied by the same number, then the products are equal. Parallel Lines In a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope. Partition Postulate The whole is equal to the sum of its parts. Perpendicular Chord Bisector If a diameter (or radius) of a circle is perpendicular to a chord, then it bisects the chord and its arc. Perpendicular Lines In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1.

8 Math Fundamentals Reference Sheet Page 8 Perpendicular Transversal If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other one. Polygon Exterior Angle Sum The sum of the exterior angle measures, one angle at each vertex, of a convex polygon is 360. Polygon Interior Angles Sum The sum of the measures of the interior angles of a convex polygon with n sides is (n-2)x180. Pythagorean theorem In any right triangle, the square of the length of the hypotenuse is equal to the sum of the square of the lengths of the legs. a 2 + b 2 = c 2 Reflexive Property Something is congruent to itself. Right Angle Congruence All right angles are congruent. Right Triangle Similarity If an altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and each other. Same-Side Interior Angles If two parallel lines are intersected by a transversal, then same-side interior angles are supplementary.

9 Math Fundamentals Reference Sheet Page 9 Side, Angle, Side (SAS) If two sides and an included angle of one triangle are congruent to two corresponding sides and an included angle of another triangle, then the triangles are congruent. Segment Addition Postulate For any segment, the measure of the whole is equal to the sum of the measures of its non-overlapping parts. Segments of Chords If two chords intersect in a circle, then the products of the lengths of the chord segments are equal. Segments of Secants & Tangents If a tangent and a secant intersect in the exterior of a circle, then the square of the measure of the tangent is equal to the product of the measures of the secant and its external secant segment. Segments of Secants If two secants intersect in the exterior of a circle, then the product of the measures of one secant segment and its external secant segment is equal to the product of the measures of the other secant and its external secant segment. Side, Side, Side (SSS) If all three sides of one triangle are congruent to all three corresponding sides of another triangle, then the triangles are congruent. Substitution Property of Equality If values are equal, then one value may be substituted for the other. Subtraction Property of Equality If the same number is subtracted from equal numbers, then the differences are equal. Symmetric Property of Congruence If something is equal to another thing, then the second thing is also equal to the first thing. If A = B, then B = A.

10 Math Fundamentals Reference Sheet Page 10 Tangent Line to a Circle In a plane, a line is tangent to a circle if and only if it is perpendicular to a radius drawn to the point of tangency. Third Angles If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are also congruent. Three Point Postulate Through any three noncollinear points there is exactly one plane containing them. Transitive Property If one expression is equal/congruent to a second expression, and that second expression is equal/congruent to a third expression, then the first and third expressions are also equal/congruent. Transitive Property of Parallel Lines If two lines are parallel to the same line, then they are parallel to each other. Trapezoid Midsegment The midsegment of a trapezoid is parallel to each base, and its length is one half the sum of the lengths of the bases. Triangle Angle Bisector An angle bisector of a triangle divides the opposite sides into two segments whose lengths are proportional to the lengths of the other two sides. Triangle Inequality The sum of any two side lengths of a triangle is greater than the third side length.

11 Math Fundamentals Reference Sheet Page 11 Triangle Larger Angle If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle. Triangle Longer Side If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side. Triangle Midsegment A midsegment of a triangle is parallel to a side of triangle, and its length is half the length of that side. Triangle Proportionality If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. Triangle Sum The sum of the measure of the interior angles of a triangle is 180. Two Point Postulate Through any two points there is exactly one line. Vertical Angle If two angles are vertical angles, then their measures are congruent to one another.