Geometric Rules Quick Reference PDF

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Geometric Rules Quick Reference - MathBitsNotebook (Geo - CCSS Math)

Geometric Rules Quick Reference [Junior Level] MathBitsNotebook.com Topical Outline | JrMath Outline | MathBits' Teacher Resources Terms of Use Contact Person: Donna Roberts

This is a partial listing of the more popular rules (theorems, postulates, and properties) that you will be using in your study of Geometry. First a few words that refer to types of geometric "rules": • A theorem is a statement (rule) that has been proven true using facts, operations and other rules that are known to be true. These are usually the "big" rules of geometry. A short theorem referring to a "lesser" rule is called a lemma. • A corollary is a follow-up to an existing proven theorem. Corollaries are off-shoots of a theorem that require little or no further proof. • A postulate (or axiom) is a statement (rule) that is taken to be true without proof. Euclid derived many of the rules for geometry starting with a series of definitions and only five postulates. • A property is a quality or characteristic belonging to something. For example, the real numbers have the associative, commutative and distributive properties.

Your textbook (and your teacher) may want you to remember these "rules" with slightly different wording. Be sure to follow the directions from your teacher.

Real Number Properties: Reflexive Property

A quantity is equal to itself. a = a

Symmetric Property

If a = b, then b = a.

Transitive Property

If a = b and b = c, then a = c.

Geometric Rules Quick Reference - MathBitsNotebook (Geo - CCSS Math)

Addition Postulate

If equal quantities are added to equal quantities, the sums are equal.

Subtraction Postulate

If equal quantities are subtracted from equal quantities, the differences are equal.

Multiplication Postulate

If equal quantities are multiplied by equal quantities, the products are equal.

Division Postulate

If equal quantities are divided by equal nonzero quantities, the quotients are equal.

Substitution Postulate

A quantity may be substituted for its equal in any expression.

Segments: Ruler Postulate

Points on a line can be paired with the real numbers.

Segment Addition Postulate

The whole is equal to the sum of its parts. When B lies between A and C on a segment, AB + BC = AC

Midpoint of Segment

The midpoint of a segment is a point on the segment forming two congruent segments (equal segments).

Bisector of Segment

The bisector of a segment is a line, a ray, or segment which cuts the given segment into two congruent segments (equal segments).

Euclid's Postulate 1

A straight line segment can be drawn joining any two points.

Euclid's Postulate 3

Any straight line segment can be extended indefinitely in a straight line.

Angles: Angle Addition Postulate

The whole is equal to the sum of its parts. m∠ABD + m∠DBC = m∠ABC

Right Angles

All right angles are congruent (equal in measure). (They all have a measure of 90º.)

(Euclid's Postulate 4)

Straight Angles

All straight angles are congruent (equal in measure). (They all have a measure of 180º.)

Geometric Rules Quick Reference - MathBitsNotebook (Geo - CCSS Math)

Vertical Angles

Vertical angles are congruent (equal in measure). m∠1 = m∠2 m∠ 3 = m∠ 4

Triangle Sum

The sum of the measures of the interior angles of a triangle is 180º.

Exterior Angle

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.

Base Angle Theorem (Isosceles Triangle)

Base Angle Converse (Isosceles Triangle)

If two sides of a triangle are congruent, the angles opposite these sides are congruent (equal in measure). If two angles of a triangle are congruent, the sides opposite these angles are congruent (equal in length).

Angles forming a straight line

Angles around a point

Complementary Angles

Two angles the sum of whose measures is 90º.

Supplementary Angles

Two angles the sum of whose measures is 180º.

Triangles: Pythagorean Theorem

c2 = a2 + b2 In a right triangle, the square of the hypotenuse equals the sum of the square of the lengths of the legs.

Sum of Two Sides

The sum of the lengths of any two sides of a triangle must be greater than the third side. In a triangle, the longest side is across from the largest

Geometric Rules Quick Reference - MathBitsNotebook (Geo - CCSS Math)

Longest Side

angle.

Largest Angle

In a triangle, the largest angle is across from the longest side

Side-Side-Side (SSS) Congruence

If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

Side-Angle-Side (SAS) Congruence

If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

Angle-Side-Angle (ASA) Congruence

If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

Angle-Angle-Side (AAS) Congruence

If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

Hypotenuse-Leg (HL) Congruence (right triangle)

If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the two right triangles are congruent.

CPCTC

Corresponding parts of congruent triangles are congruent.

Angle-Angle (AA) Similarity

If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.

Sides of Similar Δs

Corresponding sides of similar triangles are in proportion.

Parallels: Construction

Through a point not on a line, one and only one parallel to that line can be drawn.

Construction

From a given point on (or not on) a line, one and only one perpendicular can be drawn to the line.

Corresponding Angles

If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.

Alternate Interior Angles

If two parallel lines are cut by a transversal, then the alternate interior angles are congruent.

Alternate Exterior

If two parallel lines are cut by a transversal, then the

Geometric Rules Quick Reference - MathBitsNotebook (Geo - CCSS Math)

Angles

alternate exterior angles are congruent.

Interiors on Same Side

If two parallel lines are cut by a transversal, the interior angles on the same side of the transversal are supplementary.

Quadrilaterals: Quadrilateral

Parallelograms

• a figure with exactly four sides • the sum of the interior angles is 360º About Sides

• opposite sides are parallel • opposite sides are congruent

About Angles

• opposite angles are congruent • consecutive angles are supplementary

About Diagonals

• diagonals bisect each other • diagonals form two congruent triangles

Rectangle

• is a parallelogram • has 4 right angles • diagonals are congruent

Rhombus

• is a parallelogram • has 4 congruent sides • diagonals bisect the angles • diagonals are perpendicular

Square

• has all the properties of a parallelogram, a rectangle, and a rhombus

Trapezoid

• has at least one pair of parallel sides

Isosceles Trapezoid

• has at least one pair of parallel sides • legs congruent • base angles congruent • diagonals are congruent • opposite angles supplementary

Kite

• figure with four sides • two distinct pairs of adjacent sides congruent • diagonals perpendicular • one pair opposite angles congruent • one diagonal creates 2 isosceles triangles • one diagonal creates 2 congruent triangles

Geometric Rules Quick Reference - MathBitsNotebook (Geo - CCSS Math)

• one diagonal bisects the angles • one diagonal bisects the other

Area (A), Volume (V), Surface Area (SA): Rectangle

Arectangle = l × w = b • h l= length; w = width; b = base; h = height

Parallelogram

Aparallelogram = b • h

Triangle

AΔ = ½ • b• h

Trapezoid

Atrapezoid = ½ h (b1 + b2)

Regular Polygon

Aregular polygon = ½ • a • p a = apothem; p = perimeter

C = 2πr = πd

Circle (circumference)

r = radius; d = diameter

Circle (area)

Acircle = πr2

Rectangular Solid

SA formula assumes a "closed box" with all 6 sides.

Cube [special case of rectangular solid]

SA formula assumes a "closed box" with all 6 sides. s = side

Geometric Rules Quick Reference - MathBitsNotebook (Geo - CCSS Math)

Cylinder

SA formula assumes a "closed container" with a top and a bottom.

Cone SA formula assumes a "closed container", with a bottom. s = slant height

Sphere

Right Prism

Vright prism = B • h; SA = 2B + p • h B = area of the base; h = height; p = perimeter of base

Pyramid [assuming all of the faces (not the base) are the same]

B = area of the base; h = height; p = perimeter of base; s = slant height

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