49 terms

Geometry Ch. 3

The real number attributed to every point on a line, which can form a ruler ___.2______.4___
The real number for the line connecting one point to another
___.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
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
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
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)
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
Two equal sections of something that is bisected
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) ./
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