# Geometry Definitions, Postulates, and Theorems

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Order NowAlgebra Postulates Name Addition Prop. Of equality Subtraction Prop. Of equality Multiplication Prop. Of equality Division Prop. Of equality Reflexive Prop. Of equality Symmetric Property of Equality Substitution Prop. Of equality Transitive Property of Equality Distributive Property

Definition If the same number is added to equal numbers, then the sums are equal If the same number is subtracted from equal numbers, then the differences are equal If equal numbers are multiplied by the same number, then the products are equal If equal numbers are divided by the same number, then the quotients are equal A number is equal to itself If a = b then b = a If values are equal, then one value may be substituted for the other. If a = b and b = c then a = c a(b + c) = ab + ac

Visual Clue

Congruence Postulates Name Definition Reflexive Property of Congruence A ≅ A Symmetric Property of If A ≅ B, then B ≅ A Congruence Transitive Property of Congruence If A ≅ B and B ≅ C then A≅C Page 2 of 11

Visual Clue

Definitions, Postulates and Theorems

Angle Postulates And Theorems Name Definition Angle Addition For any angle, the measure of the whole is postulate equal to the sum of the measures of its nonoverlapping parts Linear Pair Theorem If two angles form a linear pair, then they are supplementary. Congruent If two angles are supplements of the same supplements theorem angle, then they are congruent. Congruent If two angles are complements of the same complements angle, then they are congruent. theorem Right Angle All right angles are congruent. Congruence Theorem Vertical Angles Vertical angles are equal in measure Theorem Theorem If two congruent angles are supplementary, then each is a right angle. Angle Bisector If a point is on the bisector of an angle, then Theorem it is equidistant from the sides of the angle. Converse of the If a point in the

interior of an angle is Angle Bisector equidistant from the sides of the angle, then Theorem it is on the bisector of the angle. Visual Clue

Lines Postulates And Theorems Name Definition Visual Clue Segment Addition For any segment, the measure of the whole postulate is equal to the sum of the measures of its non-overlapping parts Postulate Through any two points there is exactly one line Postulate Common Segments Theorem If two lines intersect, then they intersect at exactly one point. Given collinear points A,B,C and D arranged as shown, if A B ≅ C D then AC ≅ BC If two parallel lines are intersected by a transversal, then the corresponding angles are equal in measure If two lines are intersected by a transversal and corresponding angles are equal in measure, then the lines are parallel Page 3 of 11

Corresponding Angles Postulate Converse of Corresponding Angles Postulate

Definitions, Postulates and Theorems

Lines Postulates And Theorems Name Definition Postulate Through a point not on a given line, there is one and only one line parallel to the given line Alternate Interior If two parallel lines are intersected by a Angles Theorem transversal, then alternate interior angles are equal in measure Alternate Exterior If two parallel lines are intersected by a Angles Theorem transversal, then alternate exterior angles are equal in measure Same-side Interior If two parallel lines are intersected by a Angles Theorem transversal, then same-side interior angles are supplementary. Converse of Alternate If two lines are intersected by a transversal Interior Angles and alternate interior angles are equal in Theorem measure, then the lines are parallel Converse of Alternate If two lines are intersected by a transversal Exterior Angles and alternate exterior angles are equal in Theorem measure, then the lines are parallel Converse of Same-side If two lines are intersected by a transversal Interior Angles and same-side interior angles are Theorem supplementary, then the lines are parallel Theorem If two intersecting lines form a linear pair of congruent angles, then the lines are perpendicular Theorem If two lines are perpendicular to the same transversal, then they are parallel Perpendicular Transversal Theorem

Perpendicular Bisector Theorem Converse of the Perpendicular Bisector Theorem Parallel Lines Theorem Perpendicular Lines Theorem Two-Transversals Proportionality Corollary If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other one. If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment If a point is the same distance from both the endpoints of a segment, then it lies on the perpendicular bisector of the segment In a coordinate plane, two nonvertical lines are parallel IFF they have the same slope. In a coordinate plane, two nonvertical lines are perpendicular IFF the product of their slopes is -1. If three or more parallel lines intersect two transversals, then they divide the transversals proportionally. Page 4 of 11

Visual Clue

Definitions, Postulates and Theorems

Triangle Postulates And Theorems Name Definition Visual Clue Angle-Angle If two angles of one triangle are equal in measure (AA) to two angles of another triangle, then the two Similarity triangles are similar Postulate Side-side-side If the three sides of one triangle are proportional to (SSS) the three corresponding sides of another triangle, Similarity then the triangles are similar. Theorem Side-angleIf two sides of one triangle are proportional to two side SAS) sides of another triangle and their included angles Similarity are congruent, then the triangles are similar. Theorem Third Angles If two angles of one triangle are congruent to two Theorem angles of another triangle, then the third pair of angles are congruent Side-AngleIf two sides and the included angle of one triangle Side are equal in measure to the corresponding sides Congruence and angle of another triangle, then the triangles are Postulate congruent. SAS Side-side-side If three sides of one triangle are equal in measure Congruence to the corresponding sides of another triangle, then Postulate the triangles are congruent SSS Angle-sideIf two angles and the included side of one triangle angle are congruent to two angles and the included side Congruence of another triangle, then the triangles are Postulate congruent. ASA Triangle Sum The sum of the measure of the angles of a triangle Theorem is 180o

Corollary The acute angles of a right triangle are complementary. Exterior angle An exterior angle of a triangle is equal in measure theorem to the sum of the measures of its two remote interior angles. Triangle If a line parallel to a side of a triangle intersects the Proportionality other two sides, then it divides those sides Theorem proportionally. Converse of If a line divides two sides of a triangle Triangle proportionally, then it is parallel to the third side. Proportionality Theorem

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Definitions, Postulates and Theorems

Triangle Postulates And Theorems Name Definition Triangle Angle An angle bisector of a triangle divides the opposite Bisector sides into two segments whose lengths are Theorem proportional to the lengths of the other two sides. Angle-angleside Congruence Theorem AAS HypotenuseLeg Congruence Theorem HL Isosceles triangle theorem Converse of Isosceles triangle theorem Corollary Corollary Corollary Theorem Pythagorean theorem Geometric Means Corollary a Geometric Means Corollary b Circumcenter Theorem Incenter Theorem If two angles and a non-included side of one triangle are equal in measure to the corresponding angles and side of another triangle, then the triangles are congruent. 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. If two sides of a triangle are equal in measure, then the angles opposite those sides are equal in measure If two angles of a triangle are equal in measure, then the sides opposite those angles are equal in measure If a triangle is equilateral, then it is equiangular The measure of each angle of an equiangular triangle is 60o If a triangle is equiangular, then it is also equilateral If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. 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. The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the two segments of the hypotenuse. The length of a leg of a right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse adjacent to

that leg. The circumcenter of a triangle is equidistant from the vertices of the triangle. The incenter of a triangle is equidistant from the sides of the triangle. Visual Clue

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Definitions, Postulates and Theorems

Triangle Postulates And Theorems Name Definition Centriod The centriod of a triangle is located 2/3 of the Theorem distance from each vertex to the midpoint of the opposite side. Triangle A midsegment of a triangle is parallel to a side of Midsegment triangle, and its length is half the length of that Theorem side. Theorem If two sides of a triangle are not congruent, then the larger angle is opposite the longer side. Theorem Triangle Inequality Theorem Hinge Theorem Converse of Hinge Theorem Converse of the Pythagorean Theorem Pythagorean Inequalities Theorem 45˚-45˚-90˚ Triangle Theorem 30˚-60˚-90˚ Triangle Theorem Law of Sines If two angles of a triangle are not congruent, then the longer side is opposite the larger angle. The sum of any two side lengths of a triangle is greater than the third side length. If two sides of one triangle are congruent to two sides of another triangle and the third sides are not congruent, then the longer third side is across from the larger included angle. 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. 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. In ∆ABC, c is the length of the longest side. If c² > a² + b², then ∆ABC is an obtuse triangle. If c² < a² + b², then ∆ABC is acute. In a 45˚-45˚-90˚ triangle, both legs are congruent, and the length of the hypotenuse is the length of a length times the square root of 2. 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 the square root of 3. For any triangle ABC with side lengths a, b, and c, sin A sin B sin C = = a b c For any triangle, ABC with sides a, b, and c, a 2 = b 2 + c 2 − 2bc cos A, b 2 = a 2 + c 2 − 2ac cos B, c 2 = a 2 + b 2 − 2ac cos C Page 7 of 11

Visual Clue

Law of Cosines

Definitions, Postulates and Theorems

Plane Postulates And Theorems Name Definition Postulate Through any three noncollinear points there is exactly one plane containing them. Visual Clue

Postulate If two points lie in a plane, then the line containing those points lies in the plane Postulate If two points lie in a plane, then the line containing those points lies in the plane Polygon Postulates And Theorems Name Definition Visual Clue Polygon Angle The sum of the interior angle measures of a Sum Theorem convex polygon with n sides. Polygon Exterior The sum of the exterior angle measures, one Angle Sum angle at each vertex, of a convex polygon is Theorem 360˚. Theorem If a quadrilateral is a parallelogram, then its opposite sides are congruent. Theorem If a quadrilateral is a parallelogram, then its opposite angles are congruent. Theorem If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. Theorem If a quadrilateral is a parallelogram, then its diagonals bisect each other. Theorem If one pair of opposite sides of a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram. Theorem If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem If both pairs of opposite angles are congruent, then the quadrilateral is a parallelogram. Theorem If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. Theorem If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Theorem Theorem Theorem

If a quadrilateral is a rectangle, then it is a parallelogram. If a parallelogram is a rectangle, then its diagonals are congruent. If a quadrilateral is a rhombus, then it is a parallelogram.

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Definitions, Postulates and Theorems

Polygon Postulates And Theorems Name Definition Theorem If a parallelogram is a rhombus then its diagonals are perpendicular. Visual Clue

Theorem Theorem Theorem Theorem

If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle. If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. If one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus. If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus. If a quadrilateral is a kite then its diagonals are perpendicular. If a quadrilateral is a kite then exactly one pair of opposite angles are congruent. If a quadrilateral is an isosceles trapezoid, then each pair of base angles are congruent. If a trapezoid has one pair of congruent base angles, then the trapezoid is isosceles. A trapezoid is isosceles if and only if its diagonals are congruent. The midsegment of a trapezoid is parallel to each base, and its length is one half the sum of the lengths of the bases.

Theorem

Theorem

Theorem Theorem Theorem Theorem Theorem Trapezoid Midsegment Theorem

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Definitions, Postulates and Theorems

Polygon Postulates And Theorems Name Definition Visual Clue Proportional a If the similarity ratio of two similar figures is , Perimeters and b Areas Theorem a then the ratio of their perimeter is and the b a2 ⎛a⎞ ratio of their areas is 2 or ⎜ ⎟ b ⎝b⎠ The area of a region is equal to the sum of the areas of its nonoverlapping parts. 2

Area Addition Postulate

Circle Postulates And Theorems Name Definition Visual Clue Theorem If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. Theorem If a line is perpendicular to a radius of a circle at a point on the circle, then the line is tangent to the circle. Theorem If two segments are tangent to a circle from the same external point then the segments are congruent. Arc Addition The measure of an arc formed by two adjacent arcs Postulate is the sum of the measures of the two arcs. Theorem In a circle or congruent circles: congruent central angles have congruent chords, congruent chords have congruent arcs and congruent acrs have congruent central angles. Theorem In a circle, if a radius (or diameter) is perpendicular to a chord, then it bisects the chord and its arc.

Theorem Inscribed Angle Theorem Corollary Theorem Theorem

In a circle, the perpendicular bisector of a chord is a radius (or diameter). The measure of an inscribed angle is half the measure of its intercepted arc. If inscribed angles of a circle intercept the same arc or are subtended by the same chord or arc, then the angles are congruent An inscribed angle subtends a semicircle IFF the angle is a right angle If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

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Definitions, Postulates and Theorems

Circle Postulates And Theorems Name Definition Visual Clue Theorem If a tangent and a secant (or chord) intersect on a circle at the point of tangency, then the measure of the angle formed is half the measure of its

intercepted arc. Theorem If two secants or chords intersect in the interior of a circle, then the measure of each angle formed is half the sum of the measures of the intercepted arcs. Theorem If a tangent and a secant, two tangents or two secants intersect in the exterior of a circle, then the measure of the angle formed is half the difference of the measure of its intercepted arc. Chord-Chord If two chords intersect in the interior of a circle, Product then the products of the lengths of the segments of Theorem the chords are equal. SecantIf two secants intersect in the exterior of a circle, Secant then the product of the lengths of one secant Product segment and its external segment equals the product Theorem of the lengths of the other secant segment and its external segment. SecantIf a secant and a tangent intersect in the exterior of a Tangent circle, then the product of the lengths of the secant Product segment and its external segment equals the length Theorem of the tangent segment squared. Equation of a The equal of a circle with center (h, k) and radius r Circle is (x – h)2 + (y – k)2 = r2