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Plane Geometry Informal Statements: (1) The sum of the three angles of a triangle is equal to one straight angle. (2) Two straight lines can intersect at only one point. (3) A line can be divided into two equal parts by only one point. (4) A given angle can be divided into two equal parts by only one line. (5) All straight angles are equal. (6) All right angles are equal. (7) Only one perpendicular can be drawn to a line at a point in the line. (8) Complements of equal angles or of the same angles are equal. (9) Supplements of equal angles or of the same angle are equal. (10) The sum of all the angles about a point on one side of a straight line through the point is a straight angle. (11) The sum of all the angles about a point is equal to 360 degrees. (12) If the sum of two adjacent angles equals a straight angle, their exterior sides form one straight line. (13) When two straight lines intersect, the vertical angles formed are equal. (14) The sum of two sides of a

Plane Geometry Postulates in Geometry: (1) Through two given points one and only one straight line can be drawn. (2) A straight line is the shortest line that can be drawn two points. (3) A straight line may be extended indefinitely or it may be limited at any point. (4) A circle or part of one may be drawn about any point as the center and with any given radius. (5) A geometric figure may be freely moved in space without any change in form or size. (6 - Parallel Postulate) Through a given point one and only one straight line can be drawn parallel to a given line. (6.1 - Parallel Postulate Corollary) If two straight lines are parallel to a third straight line, they are parallel to each other.

Plane Geometry Principles in Geometry (1) A straight line is the shortest line between two points. (2) Only one straight line can be drawn between two points. (3) Two straight lines intersect at only one point. (4) Quantities (Lines) equal to the same quantity (line) are equal to each other. (5) The whole is equal to the sum of all its parts. (6) The whole is greater than any of its parts. (7) If equals are added to equals, then the sums are equal. (8) If equals are subtracted from equals then the remainders are equal. (9) Doubles of equals are equal. (10) Halves of equals are equal. (11) If the first of three quantities is greater than the second, and the second is greater than the third, then the first is greater than the third. (12) All straight lines are equal.

Plane Geometry Basic Axioms in Mathematics (1) Quantities that are equal to the same quantity or to equal quantities are equal to each other. (2 - Addition Axiom) If equals are added to equals, the sums are equal. (3 - Subtraction Axiom) If equals are subtracted from equals, the remainders are equal. (4 - Multiplication Axiom) Doubles of equals are equal. In general, if equals are multiplied by equals, the products are equal. (5 - Divison Axiom) Halves of equals are equal. In general, if equals are divided by equals, the quotients are equal. The divisor must not be zero. (6) The whole is equal to the sum of all its parts. (7) The whole is greater than any of its parts. (8 - Substitution Axiom) A quantity may be substituted for its equal in any process. (9) If the first of three quantities is greater than the second quantity, and the second is greater than the third, then the first is greater than the third. (10) If equals are added to, or subtracted from, unequal, the re

Plane Geometry Table of Metric Square Measures 100 square millimeters = 1 square centimer = 0.154 sq in 100 square centimeters = 1 square decimeter = 10.8 sq ft 100 square decimeters = 1 square meter = 1.19 sq yd 100 square meters = 1 square decameter = 119.6 sq yd 100 square decameters = 1 square hectometer = 2.5 A. 100 square hectometers = 1 square kilometer = 0.4 sq mi

Plane Geometry Table of Metric Linear Measures 10 millimeters = 1 centimeter = 0.4 inches 10 centimeters = 1 decimeter = 4.0 inches 10 decimeters = 1 meter = 39.37 inches 10 meters = 1 decameter = 10.9 yards 10 decameters = 1 hectometer = 109.0 yards 10 hectometers = 1 kilometer = 0.6 miles

Plane Geometry Table of Square Measures: 144 square inches = 1 square foot 9 square feet = 1 square yard 30.25 square yards = 1 square rod 160 square rods = 1 acre

Plane Geometry Table of Linear Measures: 12 inches = 1 foot 3 feet = 1 yard 5.5 yards = 1 rod 320 rods = 1 mile

Plane Geometry Chapter 7: Solution of Right Triangles by Ratios Terms to Know: Section 1-2: Definitions of Trigonometric Ratios Sine = side opposite/hypotenuse Cosine = sie adjacent/hypotenuse Tangent = side opposite/side adjacent Section 3-4: Interpolation Interpolation is the general method used for determining values intermediate to those found in a given table.

Plane Geometry Chapter 6: Regular Polygons and Circles Terms to Know: Section 7-15: Measurement of the Area of a Circle The area of a circle is the area of the surface enclosed by the circle. A sector of a circle is the figure formed by two radii and the arc intercepted by them. In the same circle or in different circles similar sectors, similar arcs, and similar segments are vectors, arcs, and segments that correspond to equal central angles.

Plane Geometry Chapter 6: Regular Polygons and Circles Terms to Know: Section 2-6: Circles and Polygons The center of a regular polygon is the common center of the inscribed and circumscribed circles. The radius of a regular polygon is the radius of the circumscribed circle. The angle at the center of a regular polygon is the angle between two radii drawn to the extremities of any side. The apothem of a regular polygon is the radius of the inscribed circle.

Plane Geometry Chapter 6: Regular Polygons and Circles Terms to Know: Section 1: Proving Polygons Regular A regular polygon is a polygon which is both equiangular and equilateral.

Plane Geometry Chapter 5: Surface Measurement Terms to Know: Section 15-17: Transforming a Plane Figure To transform a plane figure means to construct another figure of different form but of equal area.

Plane Geometry Chapter 5: Surface Measurement Terms to Know: Section 1: Surface Measurement The area of any plane number that indicates how many square units of a given kind it contains. Two planes figures are equal if they have the same area, irrespective of their shape. Equal figures are not necessarily congruent. A constant is a quantity which remains the same throughout a given problem or discussion. A variable is a quantity which continually changes throughout a given problem or discussion. Section 2-14: Area of a Rectangle The number of surface units in a rectangle is equal to the product of the number of linear units in the base by the number of linear units of the same kind in the altitude.

Plane Geometry Chapter 4: Proportions and Similar Polygons Terms to Know: Section 15-18: Applications The resultant is represented in direction and value by the diagonal of a parallelogram. Section 19-24: Numerical and Literal Fractions In representing a ratio expressed as a numerical fraction, any two lines may be used which have this ratio. In representing a ratio expressed as a literal fraction, such as a:b, where the numerical value of the ratio is not given, consider the two terms a and b as any two given lines. Section 25: Projection The projection of a point upon a line is the foot of the perpendicular from the point to the line. The projection of one line upon another is the locus of the projection of all its points. Section 26: Numerical Properties of Triangles In any triangle, the square of the side opposite an acute angle is equal to the sum of the squares of the other two sides diminished by twice the product of one of those sides and the proje

Plane Geometry Chapter 4: Proportions and Similar Polygons Terms to Know: Section 11: Pythagorean Theorem A constant is a quantity which remains the same throughout a given problem or discussion. A Pythagorean triplet is a group of three positive non-zero integers that satisfy the Pythagorean theorem, such as 3, 4, 5, and 12, 35, 37. Section 12-14: Proportional Segments in Circles The perimeter of a polygon is the sum of the sides of the polygon. The 45-45-90 triangle and the 30-60-90 triangle has many common applications in the study of mathematics.

Plane Geometry Chapter 4: Proportions and Similar Polygons Terms to Know: Section 2-6: Similar Polygons Two polygons are similar if their corresponding angles are equal and their corresponding sides are proportional. Two equal angles in two similar polygons (one angle in each polygon) are called corresponding angles. Two sides in the two similar polygons (one side in each polygon) which join the vertices of two pairs of corresponding angles are called corresponding sides. In the case of similar triangles, corresponding angles will always lie opposite a pair of equal. The ratio of two corresponding sides of two similar polygons is called the ratio of similitude of the polygons. Section 7-10: Proving Lines Proportional C.S.S.T.P stands for corresponding sides of similar triangles are proportional. To prove that two angles are equally based on similar triangles, use corresponding angles of similar triangles are equal. (C.A.S.T.E.) C.A.S.T.E is another way to express th

Plane Geometry Chapter 4: Proportions and Similar Polygons Terms to Know: Section 1: Proportional Quantities Ratios are common fractions, and common fraction may be regarded as a ratio. The antecedent is the numerator or first term in a ratio. The consequent is the denominator or second term in the ratio. The value of a ratio remains the same if both of its terms are multiplied or divided by the same number. Proportion A proportion is an expression of equality between two ratios. The extremes of a proportion are the first and fourth terms. The means of a proportion are the second and third terms. The fourth proportional is to three given quantities is the fourth term of the proportion whose first three terms are the three quantities taken in order. In any proportion in which the two means are equal, either means is said to be the mean proportional between the first and last terms.

Plane Geometry Chapter 3: The Circle Terms to Know: Section 20-26: Excenter An excenter is the intersection of the bisectors of two exterior angles of the triangle. Section 27-29: Loci A locus is the location of all the points, and only those points, that satisfy a given condition or conditions. Section 30: Simple Loci - Determining a Locus (1) Prove that every point on the assumed locus satisfies the given condition. (2) Prove that every point that satisfies the given condition is on this assumed locus. Section 31: Fundamental Loci Theorems The Distance of a Point from a Circle means the shortest distance from the point to the circle. Section 32-38: Intersecting Loci A locus is a path traced by a point moving under some restriction.

Plane Geometry Chapter 3: The Circle Terms to Know: Section 19f: Two Arcs Are Unequal (Requirements) Arcs of unequal central angles or of unequal chords.

Plane Geometry Chapter 3: The Circle Terms to Know: Section 19e: Two Angles Are Unequal (Requirements) Central angles whose chords or arcs are unequal. Angles with vertices on the circle intercepting unequal arcs.

Plane Geometry Chapter 3: The Circle Terms to Know: Section 19d: Two Lines Are Unequal (Requirements) Chords of unequal central angles or of unequal arcs. Chords unequally distant from the center.

Plane Geometry Chapter 3: The Circle Terms to Know: Section 19c: Two Arcs Are Equal (Requirements) Arcs of equal central angles or of equal chords. Arcs intercepted by parallel lines. Arcs intercepted by equal angles whose vertices are a circle.

Plane Geometry Chapter 3: The Circle Terms to Know: Section 19b: Two Angles Are Equal (Requirements) Central angles whose chords or arcs are equal. Angles with vertices on the circle intercepting the same or equal arcs.

Plane Geometry Chapter 3: The Circle Terms to Know: Section 19a: Two Lines Are Equal (Requirements) Chords of equal central angles or of equal arcs. Chords the same distance from the center. Tangents from the same external point.

Plane Geometry Chapter 3: The Circle Terms to Know: Section 12: Concentric Circles Concentric circles are circles which have the same center. Section 13-18: Measurements of Angles in a Circle The numerical measure of any magnitude is the number which tells how many times his magnitude contains the given unit. A rational number can be expressed as the ratio of two integers. An irrational number cannot be expressed as the ratio of two integers and is a nonrepeating decimal. Magnitudes Two magnitudes of the same kind which have a common unit of measure are said to be commensurable. Two magnitudes of the same kind which have no common unit of measure are said to be incommensurable. Angles A central angle is measured by its intercepted arc. A numerical measure of angle = numerical measure of its arc. A secant is a straight line that cuts the circle in two points. The length of a secant from an external point to a circle means the length of the line from the ext

Plane Geometry Chapter 3: The Circle Terms to Know: Section 8: Tangents A tangent to a circle is a straight line which, however far it may be extended, has one and only one point in common with the circle. The one and only point is the point of contact or point of tangency. A polygon is circumscribed about a circle when its sides are tangent to the circle. The circle is the inscribed in the polygon. Section 9: Equal Tangents By the length of a tangent from a point to a circle is meant the length of time from that point to the point of tangency. The line joining the points of tangency of two tangents drawn from an external point to a circle is called the chord of tangency of the two tangents. Section 10-11: Common Tangents and Common Chords Two circles are tangent to each other if both are tangent to the same line at the same point. Two circles are tangent internally or externally, according to the tangent circles that lies wholly within or outside the other.

Plane Geometry Chapter 3: The Circle Terms to Know: Section 7: Chords Not Equidistant from the Centers of Circles A polygon is inscribed in a circle when all its vertices lie on the circle. The circle is then circumscribed about the polygon.

Plane Geometry Chapter 3: The Circle Terms to Know: Section 1: Arcs Congruent arcs, in the same circle or in congruent circles, are arcs that can be made to coincide throughout. Congruent arc are referred as equal arcs. A minor arc is an arc less than a semicircle. A major arc is an arc greater than a semicircle. A central angle is an angle formed by two radii. An angle is said to intercept the arc cut off between its sides, and the arc is said to subtend or determine the angle. Section 2-6: Equal Arcs A chord is said to subtend or determine an arc.

Plane Geometry Chapter 2: Rectilinear Plane Figures Terms to Know: Section 28-38: Analytic and Synthetic Methods of Proof The analytic method of proof traces out a path from the unknown statement made in the conclusion to some known statement, by a series of successive substitutions. The synthetic method consists continually putting together known statements to obtain new statements until eventually the unknown statement made in the conclusion is obtained.

Plane Geometry Chapter 2: Rectilinear Plane Figures Terms to Know: Section 27: Ways a Line May Be Drawn Joining two points. Bisecting a given angle. Through a point perpendicular to a given line. Through a point parallel to a given line. Extending a given line. Marking off the shorter of two lines on the longer. Making with another line an angle equal to a given angle.

Plane Geometry Chapter 2: Rectilinear Plane Figures Terms to Know: Section 26h: Quadrilateral is a Parallelogram If the opposite sides are parallel. If the opposite sides are equal. If two sides are equal and parallel. If the diagonals bisect each other. If the opposite angles are equal or the adjacent angles are supplementary.

Plane Geometry Chapter 2: Rectilinear Plane Figures Terms to Know: Section 26g: Two Lines are Unequal If they lie opposite unequal angles in the same triangle. If when drawn from a point in a perpendicular to a given line they cut off the given line unequal distances from the foot of the perpendicular. By the use of axioms.

Plane Geometry Chapter 2: Rectilinear Plane Figures Terms to Know: Section 26f: Two Angles are Unequal If they lie opposite unequal sides in the same triangle. If one is an exterior angle and the other is an opposite interior angle of the same triangle. By the use of axioms.

Plane Geometry Chapter 2: Rectilinear Plane Figures Terms to Know: Section 26e: Two Lines are Parallel If two alternate interior angles or two corresponding angles are equal when the lines are cut by the same transversal. If they are parallel or perpendicular to a third line. If they are opposite sides of a parallelogram. If the sum of the interior angles on the same side of a common transversal equals a straight angle.

Plane Geometry Chapter 2: Rectilinear Plane Figures Terms to Know: Section 26d: Two Lines are Equal If they are corresponding sides of congruent triangles (C.P.C.T.E.) If they are sides opposite equal angles in the same triangle. If they are opposite sides of a parallelogram. If they are segments between parallels which cut equal segments from another transversal. By the use of axioms. If they are subtended by equal central angles or arcs in the same or congruent circles.

Plane Geometry Chapter 2: Rectilinear Plane Figures Terms to Know: Section 26c: Two Angles are Equal If they are straight angles, right angles, or vertical angles. If they are complements or supplements of the same angle or of equal angles. If they are corresponding angles of congruent triangles (C.P.C.T.E) If they are corresponding angles of similar triangles (C.A.S.T.E) If they are opposite equal sides in the same triangle. If they are alternated interior or corresponding angles of parallel lines made by the same transversal. If they are angles whose sides are parallel or perpendicular respectively. By the use of axioms. If they are subtended by equal arcs or chords in the same or congruent circles.

Plane Geometry Chapter 2: Rectilinear Plane Figures Terms to Know: Section 26b: Two Right Triangles are Congruent If they have two legs equal respectively. (L.L.) If they have one leg and either acute angle equal respectively. (L.A.) If they have the hypotenuse and either leg equal respectively. (H.L.) If they have the hypotenuse and either acute angle equal respectively. (H.A.)

Plane Geometry Chapter 2: Rectilinear Plane Figures Terms to Know: Section 26a: Two Triangles are Congruent If they have two sides and the included angle equal respectively (S.A.S) If they have three sides equal respectively (S.S.S) If they have one side and any two angles equal respectively (S.A.A)