Volume 1 Postulates, Properties, Theorems, Corollaries,

Volume 1 Postulates, Propert ies , Theorems, Corol lar ies ,

and More …

Table Of Contents PPTC Volume 1

(p01) Ruler Postulate 01

(p02) Segment Addition Postulate 02

(p03) Midpoint Formula

(p04) Distance Formula

(p05) Protractor Postulate 03

(p06) Angle Addition Postulate 04

(p07) Angle Classifications

(p08) Interpreting a Diagram

(p09) Conditional Statement (CCIC)

(p10) Laws of Logic

(p11) Point, Line, and Plane Postulate 05








(p19) Line Perpendicular to a Plane

(p20) Properties of Equality

(p21) Reflective Property of Equality

(p22) Symmetric Property of Equality

(p23) Transitive Property of Equality

(p24) Congruence of Segments Theorem 2.1

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(p25) Congruence of Angles Theorem 2.2


(p27) Right Angle Congruence Theorem 2.3

(p28) Congruent Supplements Theorem 2.4

(p29) Congruent Complements Theorem 2.5


(p31) Vertical Angles Congruence Theorem 2.6

(p32) Parallel Postulate 13

(p33) Perpendicular Postulate 14

(p34) Angles Formed By Transversals

(p35) Corresponding Angles Postulate 15

(p36) Alternate Interior Angles Theorem 3.1

(p37) Alternate Exterior Angles Theorem 3.2

(p38) Consecutive Interior Angles Theorem 3.3

(p39) Corresponding Angles Converse Postulate 16 (see postulate 15)

(p40) Alternate Interior Angles Converse Theorem 3.4 (see theorem 3.1)

(p41) Alternate Exterior Angles Converse Theorem 3.5 (see theorem 3.2)

(p42) Consecutive Interior Angles Converse Theorem 3.6 (see theorem 3.3)

(p43) Transitive Property of Parallel Lines Theorem 3.7

(p44) The Slope Formula

(p45) Slopes of Parallel Lines Postulate 17

(p46) Slopes of Perpendicular Lines Postulate 18

(p47) Theorem 3.8

(p48) Theorem 3.9

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(p49) Theorem 3.10

(p50) Perpendicular Transversal Theorem 3.11

(p51) Lines Perpendicular to a Transversal Theorem 3.12

(p52) Distance from a Point to a Line

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(p68) Vocabulary chapter 0





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Postulate 1 Ruler Postulate The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is the coordinate of the point. The distance between points A and B, written as AB, is the absolute value of the difference of the coordinates of A and B.

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Postulate 2 Segment Addition Postulate If (B) is between (A) and (C), Then (AB) + (BC) = (AC). Or If (AB) + (BC) = (AC), Then (B) is between (A) and (C).

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Formula The Midpoint Formula The coordinates of the midpoint of a segment are the averages of the x-coordinates and of the y-coordinates of the endpoints.

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Formula The Distance Formula If A(x1, y1) and B(x2, y2) are points in a coordinate plane, then the distance between A and B is

Postulate 3 Protractor Postulate Consider line (OB) and a point (A) on one side of line (OB). The rays of the form ray (OA) can be matched one to one with the real numbers from 0 to 180. The measure of angle (AOB) is equal to the absolute value of the difference between the real numbers for ray (OA) and ray (OB).

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Postulate 4 Angle Addition Postulate If (P) is in the interior of angle (RST), Then the measure of angle (RST) is equal to The sum of the measures of angle (RSP) and angle (PST). m∠RST = m∠RSP + m∠PST

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Classifications Angle Classifications Acute Right Obtuse Straight Reflex

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Concept Summary Interpreting A Diagram

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Classification Conditional Statements (ccic) Conditional Statement: p → q (If) hypothesis (then) conclusion Converse: q → p (If) conclusion (then) hypothesis Inverse: ~p → ~ q (If) not hypothesis (then) not conclusion Contrapositive: ~ q → ~ p (If) not conclusion (then) not hypothesis Biconditional: p ↔ q (If) if and only if The conditional statement and the contrapositive: are either both true or both false. The converse and inverse: are either both true or both false.

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Laws Laws of Logic Law of Detachment If the hypothesis of a true conditional statement is true, then the conclusion is also true. Law of Syllogism If these statements are true: If hypothesis (A), then conclusion (B). If hypothesis (B), then conclusion (C). Then this statement is true: If hypothesis (A), then conclusion (C).

Postulate 5 Point, Line, and Plane Postulates Through any two points there exists exactly one line.

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Postulate 6 Point, Line, and Plane Postulates A line contains at least two points.

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Postulate 7 Point, Line, and Plane Postulates If two lines intersect, then their intersection is exactly one point.

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Postulate 8 Point, Line, and Plane Postulates Through any three noncollinear points there exists exactly one plane.

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Postulate 9 Point, Line, and Plane Postulates A plane contains at least three noncollinear points.

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Postulate 10 Point, Line, and Plane Postulates If two points lie in a plane, then the line containing them lies in the plane.

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Postulate 11 Point, Line, and Plane Postulates If two planes intersect, then their intersection is a line.

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Classification Line Perpendicular to a Plane if and only if the line intersects the plane in a point and is perpendicular to every line in the plane that intersects it at that point. In a diagram: Line perpendicular to a plane must be marked with a right angle symbol.

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Properties of Equality (Let a, b, and c be real numbers) Addition Property If (a = b), then (a) + c = (b) + c. Subtraction Property If (a = b), then (a) - c = (b) – c. Multiplication Property If (a = b), then (ac) = (bc). Division Property If (a = b) and c ≠ 0, then (a) ÷ c = (b) ÷ c. Substitution Property If (a = b), then (a) can be substituted for (b) in any equation or expression. Distributive Property a(b + c) = (ab) + (ac).

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Properties of Equality Reflexive Property of Equality Real Numbers For any real number a, a = a Segment Length For any segment AB,

𝐴𝐵 = 𝐴𝐵 Angle Measure For any angle A, m ∠ A = m ∠ A

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Properties of Equality Symmetric Property of Equality Real Numbers For any real numbers a and b, if a = b then b = a. Segment Length For any segments AB and CD, if AB = CD then CD = AB. Angle Measure For any angles A and B, if m ∠ A = m ∠ B, then m ∠ B = m ∠ A.

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Properties of Equality Transitive Property of Equality Real Numbers For any real numbers a, b, and c, if a = b and b = c, then a = c. Segment Length For any segments AB, CD, and EF, if AB = CD and CD = EF then AB = EF. Angle Measure For any angles A, B, and C, if m ∠ A = m ∠ B and m ∠ B = m ∠ C then m ∠ A = m ∠ C.

Theorem 2.1 Congruence Of Segments Segment congruence is reflexive, symmetric, and transitive.

Reflexive: Given… 𝐴𝐵 ≅ 𝐴𝐵

Symmetric: If… 𝐴𝐵 ≅ 𝐶𝐷

Then… 𝐶𝐷 ≅ 𝐴𝐵

Transitive: If… 𝐴𝐵 ≅ 𝐶𝐷

And… 𝐶𝐷 ≅ 𝐸𝐹

Then… 𝐴𝐵 ≅ 𝐸𝐹

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Theorem 2.2 Congruence Of Angles Angle congruence is reflexive, symmetric, and transitive. Reflexive: Given… ∠ A ≅ ∠ A Symmetric: If… ∠ A ≅ ∠ B Then… ∠ B ≅ ∠ A Transitive: If… ∠ A ≅ ∠ B And… ∠ B ≅ ∠ C Then… ∠ A ≅ ∠ C

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Theorem 2.3 Right Angles Congruence Theorem All right angles are congruent.

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Theorem 2.4 Congruent Supplements Theorem If two angles are supplementary to the same angle (or to congruent angles), then they are congruent.

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Theorem 2.5 Congruent Complements Theorem If two angles are complementary to the same angle (or to congruent angles), then they are congruent.

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Postulate 12 Linear Pair Postulate If two angles form a linear pair, then they are supplementary. Angle (1) and angle (2) form a linear pair, so angle (1) and angle (2) are

supplementary and the measure of angle (1) and angle (2) equals 180 degrees.

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Theorem 2.6 Vertical Angles Congruence Theorem Vertical angles are congruent. Angle (1) is congruent to angle (3), and angle (2) is congruent to (4).

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Postulate 13 Parallel Postulate If there is a line (l) and a point not on the line (P), then there is exactly one line through the point parallel to the given line. If: Then:

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Postulate 14 Perpendicular Postulate If there is a line (l) and a point not on the line (P), then there is exactly one line through the point perpendicular to the given line. If: Then:

Classification Angles Formed By Transversals corresponding alternate interior alternate exterior consecutive interior

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Postulate 15 Corresponding Angles Postulate If two parallel lines (p) and (q) are cut by a transversal (t), then the pairs of corresponding angles are congruent.

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Theorem 3.1 Alternate Interior Angles Theorem If two parallel lines (p) and (q) are cut by a transversal (t), then the pairs of alternate interior angles are congruent.

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Theorem 3.2 Alternate Exterior Angles Theorem If two parallel lines (p) and (q) are cut by a transversal (t), then the pairs of alternate exterior angles are congruent.

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Theorem 3.3 Consecutive Interior Angles Theorem If two parallel lines (p) and (q) are cut by a transversal (t), then the pairs of consecutive interior angles are supplementary.

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Postulate 16 Corresponding Angles Postulate Converse If two parallel lines (j) and (k) are cut by a transversal (t) so the corresponding angles are congruent, then the lines are parallel.

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Theorem 3.4 Alternate Interior Angles Theorem Converse If two parallel lines (j) and (k) are cut by a transversal (t) so the alternate interior angles are congruent, then the lines are parallel.

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Theorem 3.5 Alternate Exterior Angles Theorem Converse If two parallel lines (j) and (k) are cut by a transversal (t) so the alternate exterior angles are congruent, then the lines are parallel.

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Theorem 3.6 Consecutive Interior Angles Theorem Converse If two parallel lines (j) and (k) are cut by a transversal (t) so the consecutive interior angles are supplementary, then the lines are parallel.

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Theorem 3.7 Transitive Property of Parallel Lines Theorem If two lines (p) and (r) are parallel to the same line (q), then they are parallel to each other.

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Formula The Slope Formula If a line in the coordinate plane passes through points (x1, y1) and (x2, y2) then the slope (m) is:

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Postulate 17 Slopes of Parallel Lines Postulate In a coordinate plane, two non-vertical lines are parallel “if and only if” they have the same slope.

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Postulate 18 Slopes of Perpendicular Lines Postulate In a coordinate plane, two non-vertical lines are perpendicular “if and only if” the product of their slopes is (-1).

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Theorem 3.8 Theorem 3.8 If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. If ∠ 1 ≅ ∠ 2, then g ⊥ h.

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Theorem 3.9 Theorem 3.9 If two lines are perpendicular, then they intersect to form four right angles. If a ⊥ b, then ∠ 1, ∠ 2, ∠ 3, and ∠ 4 are right angles.

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Theorem 3.10 Theorem 3.10 If two sides of two adjacent acute angles are perpendicular, then the angles are complementary.

If 𝐵𝐴 ⊥ 𝐵𝐶 then ∠ 1 and ∠ 2 are complementary.

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Theorem 3.11 Perpendicular Transversal Theorem If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other. If h ǁ k and j ⊥ h, then j ⊥ k.

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Theorem 3.12 Lines Perpendicular to a Transversal Theorem In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. If m ⊥ p and n ⊥ p, then m i n.

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Classification Distance from a Point to a Line The distance from a point to a line is the length of the perpendicular segment (the shortest distance between the point and the line) from the point to the line. The distance between two parallel lines is the length of any perpendicular segment joining the two lines.

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Postulate 13 Parallel Postulate Iasdf

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Theorem 3.1 asdf Theorem Iasdf

Vocabulary Chapter 01

Point: has no dimension, is represented by a dot. Line: has one dimension, is represented by a line with two arrowheads that extends without end. Plane: has two dimensions, is represented by a parallelogram shape that extends without end. Collinear: on the same line. Coplanar: on the same plane. Segment: the segment of a line between two endpoints. Congruent Segment: segments that have the same length. Ray: starts at an endpoint and extends without end through another point. Opposite Rays: two rays on a line with the same endpoint that extend in opposite directions. Postulate (axiom): a rule that is accepted without proof. Theorem: a rule that is can be proved.

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Vocabulary Chapter 01 cont. Midpoint: the point that divides a segment into two congruent segments. Segment Bisector: intersects a segment at its midpoint. Angle: two different rays with the same endpoint.. Sides: the rays of an angle. Vertex: the endpoint of an angle. Congruent Angles: angles that have the same measure. Angle Bisector: a ray that divides an angle into two congruent angles. Construction: a geometric drawing usually made with a compass and straightedge. Complementary Angles: the sum of their measures is 90°. Supplementary Angles: the sum of their measures is 180°. Adjacent Angles: two angles that share a common side and vertex. Linear Pair: two adjacent angles with noncommon sides that are opposite rays.

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Vocabulary Chapter 01 cont. Vertical Angles: two angles with sides that form two pairs of opposite rays. Polygon: a closed plane figure that lies in a plane; it is formed by three or more line segments called sides; each side intersects exactly two sides, one at each endpoint, so that no two sides with a common endpoint are collinear. Convex: no line that contains a side of the polygon, contains a point in the interior of the polygon. Concave: any line that contains a side of the polygon, contains a point in the interior of the polygon. n-gon: (n) is the number of a polygon’s sides, and is used to name a polygon. Equilateral: all sides are congruent. Equiangular: all angles are congruent. Regular: a convex polygon that is both equilateral and equiangular. Polygons with (3,4,5,6,7,8,9,10,12,n) sides: triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, decagon, dodecagon, n-gon.

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Vocabulary Chapter 02 Conjecture: an unproven statement that is based on observations. Inductive Reasoning: when you find a pattern in specific cases and then write a conjecture for the general case. Counterexample: a specific case for which the conjecture is false. Conditional Statement: a logical statement that has two parts, a hypothesis and a conclusion. If-Then Form: uses “if” and “then” to write a conditional statement. Hypothesis: the “if” part of a conditional statement. Conclusion: the “then” part of a conditional statement. Negation: the opposite of the original statement. Converse: a conditional statement in which hypothesis and conclusion exchange. Inverse: a conditional statement that is negated. Contrapositive: a converse that is negated.

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Vocabulary Chapter 02 Equivalent Statements: when two statements are both true or both false. Perpendicular Lines: two lines that intersect to form a right angle. Biconditional Statement: a statement that contains the phrase “if and only if”. Deductive Reasoning: uses facts, definitions, accepted properties, and the laws of logic to form a logical argument. Truth Value: true or false. Truth Table: a table that shows the truth values for a hypothesis and conclusion. Proof: a logical argument that shows a statement is true. Two Column Proof: numbered statements and corresponding reasons that show an argument in a logical order.

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Vocabulary Chapter 03 Parallel Lines: two lines that do not intersect and are coplanar. Skew Lines: two lines that do not intersect and are not coplanar. Parallel Planes: two planes that do not intersect. Transversal: a line that intersects two or more coplanar lines at different points. Corresponding Angles: two angles that have corresponding (matching) positions. Alternate Interior Angles: two angles that lie between two lines and on opposite sides of the transversal. Alternate Exterior Angles: two angles that lie outside two lines and on opposite sides of the transversal. Consecutive Interior Angles: two angles that lie between the two lines and on the same side of the transversal. Same-Side Interior Angles: another name for consecutive interior angles. Paragraph Proof: The statements and reasons in a paragraph proof are written in sentences, using words to explain the logical flow of the argument.

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Vocabulary Chapter 03 Slope: the ratio of vertical change (rise) to horizontal change (run) between any two points on the line. Slope-Intercept Form: the general form of a linear equation is “y = mx + b”, where m is the slope and b is the y-intercept.. Standard Form: another form of a linear equation is “Ax + By = C”, where A and B are not both zero. Distance From a Point To a Line: the length of the perpendicular segment from the point to the line. This perpendicular segment is the shortest distance between the point and the line.

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