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Coordinate

The real number attributed to every point on a line, which can form a ruler ___.2______.4___

Distance

The real number for the line connecting one point to another

___.2____.4___ 4-2=2

___.2____.4___ 4-2=2

The points on a line can be numbered so that positive number differences measure distances

The Ruler Postulate

A point is between two other points on the same line iff its coordinate is between theirs (A-B-C iff a>b>c or a<b<c)

Betweenness of Points Definition ___.A____.B____.C___

(a) 3 (b) 4 (c) 6

(a) 3 (b) 4 (c) 6

If A-B-C, then AB + BC = AC

Theorem 1; Betweenness of Points Theorem

Given; Betweenness of Points Definition; Ruler Postulate; Addition; Ruler Postulate; Substitution

Betweenness of Points Theorem Proof: A-B-C; a < b < c; AB= a- b, BC= b- c; AB + BC= (b- a) + (c- b)= c- a; AC= c- a; AB+ BC= AC

Reflexive Property

Any number is equal to itself; a= a

Substitution Property

If a= b, then 'a' can be substituted for 'b' in any expression

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 (a)(c)= (b)(c)

Division Property

If a= b, and c≠ 0, then a/ c= b/ c

Degree

The unit of measurement for any of the 360 equal parts of a protractor; __°

Rotation of rays

The amount of rays, one for every degree, of a full 360° protractor, all with an endpoint in the middle

Half- rotation rays

All rays of degrees that go within a semicircular protractor

Measure of the angle

The positive difference between coordinates of rays (sides) of an angle ___.25 35- 25= 10

\.35

\.35

The rays in a half- rotation can be numbered 0- 180 so positive differences

Protractor Postulate

Definition of acute angle

An angle that is less than 90° .______.

\.

\.

Definition of right angle

An angle that is equal to 90° |.

|.________.

|.________.

Definition of obtuse angle

An angle that is greater than 90° but less than 180° ._____.

./

./

Definition of straight angle

An angle that is equal to 180° ._____._____.

A ray is between two others in a half- rotation iff its coordinate is between their coordinates

Betweenness of Rays Definition; OA-OB-OC, a< b< c or a> b> c A.(a)_____.______C.(c)

A./ (b)

A./ (b)

If OA-OB-OC, then ∠AOB + ∠BOC= ∠AOC

Betweenness of Rays Theorem

Given; Definition of Betweenness of Rays ; Protractor Postulate; Addition; Protractor Postulate; Substitution

Betweenness of Rays Theorem Proof: OA-OB-OC; a> b> c; ∠AOB= a- b, ∠BOC= b- c; ∠AOB+ ∠BOC= (a- b)+ (b- c)= a- c; ∠AOC= a- c; ∠AOC= ∠AOB+ ∠BOC

Definition of Midpoint of a line segment

A point iff it divides the line segment into two equal segments

Definition of Bisect of an angle

A line iff it divides the angle into two equal angles

Congruent

Two equal sections of something that is bisected

Corollary

A theorem that, based on another theorem or postulate, is easy to prove

Corollary to Ruler Postulate

A line segment has exactly one midpoint

Corollary to Protractor Postulate

An angle has exactly one ray that bisects it

Definition of Complementary Angles

Two angles iff they are equal to 90°

Definition of Supplemenmtary Angles

Two angles if they are equal to 180°

Theorem (3)

Complements of the same angle are equal

Given; Definition of Complementary Angles; Substitution; Subtraction

Complements of same angle equal Theorem proof: ∠1 and ∠2 complements of ∠3; ∠1+ ∠3= 90°, ∠2+ ∠3= 90°; ∠1+ ∠3= ∠2+ ∠3; ∠1= ∠2

Theorem (4)

Supplements of the same angle are equal; proof similiar to complements

Opposite rays

Rays that point in opposite directions from one point

___.___

___.___

Linear pair

Two angles iff they have a common side and the other sides are opposite rays _____./_____

Vertical angles

Two angles iff the sides of one are opposite rays to the other's sides

Theorem (5)

The angles in a linear pair are supplementary

Given; Definition of Linear Pair; Protractor Postulate; Protractor Postulate; Addition; Definition of Supplementary Angles

Linear pair supplementary angles proof: ∠1 and ∠2 are a linear pair; rays OA and OB are opposite rays; OA= 0, OB= 'n', OC= 180; ∠1= n- 0= n°, ∠2= (180- n)°; ∠1+ ∠2= n°+ (180- n)°= 180°; ∠1 and ∠2 are supplementary

A. (0)---.O/----.C (180)

B (n) ./

A. (0)---.O/----.C (180)

B (n) ./

Theorem (6)

Vertical angles are equal

Given; Definition of Vertical Angles; Definition of Linear Pair; supplements of angles are equal Theorem

Vertical angles equal proof: ∠1 and∠3 are vertical angles; Sides of ∠1 are opposite to sides of ∠3; ∠1 and ∠2 are a linear pair, ∠2 and ∠3 are a linear pair; ∠1= ∠3

Definition of perpendicular (⊥)

Two lines iff they form a right angle

Theorem (7)

Perpendicular lines form four right angles

Given; Def. of Perprendicular; Def. of Right Angle; Def. of Linear Pair; Def. of Supplementary; Subtraction; Def. of Vertical Angles; Substitution

Perpendicular lines form four right angles proof: a⊥b; a⊥b forms a right angle (∠1); ∠1= 90°; ∠1 and ∠2 are a linear pair; ∠1+ ∠2= 180°; ∠2= 90°; ∠1 and ∠3, ∠2 and ∠4 are vertical angles, vertical angles are equal; ∠3= 90°, ∠4= 90°; all four angles are right angles

Corollary to Definition of Right Angles

All right angles are equal

Theorem (8)

If angles in a linear pair are equal, their sides are perpendicular

Proof; Def. of Linear Pair; Given; Substitution; Division; Def. of Right Angle; Def. of Perpendicular

Equal linear pair lines have perpendicular sides proof: ∠1 and ∠2 are a linear pair; ∠1 and ∠2 are supplementary/ = 180°; ∠1= ∠2; ∠1+ ∠1= 180°, 2∠1= 180°; ∠1= 90°; ∠1 is a right angle; OB⊥AC

Parallel (||)

Two lines iff they lie in the same plane and do not intersect

________________.B

________________.A

________________.B

________________.A