Download 12.2 geometric proofs.notebook

12.2 geometric proofs.notebook
December 02, 2016
Friday 12/2/16
Construct a formal proof to prove the following:
1) Given: 1/3(3x ­12)­1 = 2x
Prove : x = ­5
Steps Reasons
1/3(3x ­12)­1 = 2x
Given
x ­ 4 ­ 1 = 2x
distributive
x ­ 5 = 2x
combine like terms
­ 5 = x
subtraction
x = ­5
symmetric
1
12.2 geometric proofs.notebook
December 02, 2016
Given
2
substitution
subtraction
12
12/4
x = 3
Given
subtraction
3x+8­8=2­8
3x = ­6
Dividing
x = ­2
2
12.2 geometric proofs.notebook
December 02, 2016
DEFINITIONS:
Congruence: if two object are congruent, then their measures EX: AB BC, then AB=BC. If AB = 20, BC = 20
Congruent Segments: If two segments are congruent, then their measures are equal
EX: AB CD, then AB = CD.
If AB = 15 feet, then CD = 15 feet
Congruent Angles: If two angles are congruent, then their measures are equal
EX: Angle 1 Angle 2, then Angle 1=Angle 2
If angle 1 = 30, then angle 2 = 30
Midpoint: any point that cuts a segment into two congruent halves
EX: A B
C, If AB BC, then AB=BC and B is the midpoint
Vertical Angles: Two pairs of congruent opposite angles are formed when two lines intersect
1=3
2
4=2
1
3
4
Complementary Angles: Angles whose sum equals 90 degrees
80
10
Supplementary angles: Angles whose sum equals 180 degrees
EX:
100
80
Perpendicular Lines: two lines that intersect at right angles
90
POSTULATES:
Ruler Postulate: The distance between any two point can be measured
Protractor Postulate: If point A is exterior to a given segment CB, then the angle between A and CB can be measured
EX: C B
A
Segment Addition Postulate: If B is between A and C, then AB+BC = AC
EX: A B C IF AB = 5 and BC = 10, AC = 15
Angle Addition Postulate: IF P lies interior to Angle RST, then RSP+PST=RST
EX:
IF Angle 1 = 20, and Angle 2 = 50
1
A
Angle A = 70
2
POINTS, LINES, AND PLANES POSTULATES:
1) Through any two points there is exactly one line
2) A line contains at least two points 3) Two lines intersect at EXACTLY one point
EX:
4) Linear Pair Postulate: If two angles form a linear pair, then they are supplementary 5) Two planes intersect at exactly ONE line
3
12.2 geometric proofs.notebook
December 02, 2016
4
12.2 geometric proofs.notebook
December 02, 2016
DEFINITIONS:
Congruence: If two objects are congruent, then their measures are equal
EX: AB BC, AB = BC (Definition of congruence)
Congruence of Segments: If two segments are congruent, then their measures are equal
EX: A B C D IF AB = 5, CD = 5
Congruent Angles: If two angles are congruent then their measures are equal
Angle A = Angle B, then Angle A B
Midpoint: The middle of a segment; divided into two congruent halves
EX: A B C AB BC, then B is the midpoint Angle Bisector: A segment that bifurcates (cuts in half) any angle creating two congruent interior angles
b
EX: segment ad bisects angle bac. bad, and dac are congruent
a
d
c
Vertical Angles: pair of congruent opposite angles formed by the intersection of two lines 3
1=2
1
3=4
2
4
Complementary Angles: angles whose sum is 90 degrees
EX: 10
80
Supplementary: angles whose sum is 180 degrees
EX: 100
80
Perpendicular Lines: two lines that intersect at right angles
EX: 90
POSTULATES:
Ruler Postulate: that you can measure the distance between any two points
Protractor Postulate: Give any point that is exterior to a given line segment, you can measure the angle between that point
EX: A B
C
Segment addition postulate: If B is between A and C, then AB+BC=AC.
EX: A 5 B 10 C AB + BC = 15
Angle addition postulate: If B is interior to angle ACD, then
ACB + BCD = ACD
EX:
3
1
2
Angle 1+2 = 3 5
12.2 geometric proofs.notebook
December 02, 2016
DEFINITIONS:
Congruence ­ if two objects are congruent, their measures are equal
EX: AB BC, therefore AB = BC (Definition of Congruence)
Congruent Segments: If two segments are congruent, then their measures are equal
A B If AB = CD, then AB CD. If AB = 10, then CD = 10
C D
Congruent Angles: If two angles are congruent, then their measures are equal
Angle 1 Angle 2. If Angle 1 = 60, then Angle 2 = 60
Midpoint: Point that bifurcates (cut in half) a segment into two equal halves
EX: A B C If AB = BC, then B is the midpoint.
Angle Bisector: segment that bifurcates any angle into two congruent halves
bd is an angle bisector:
a
abd = dbc
d
b
c
Vertical Angles: pairs of congruent angles formed by two intersecting lines
2
1 = 3
3
1
2 = 4
4
Complementary Angles: angles whose sum is 90 degrees
EX: 10
80
Supplementary Angles: angles whose sum is 180 degrees
EX:
80
100
Perpendicular Lines: two lines that intersect at a right angle
90
POSTULATES:
Ruler Postulate: the distance between any two points can be measured
EX: Given: segment AB = 10 inches
Prove: AB = 10 inches (The distance can be measured)
Protractor Postulate: given any point exterior to a segment can be measured between 0­180 degrees
EX: A B
C
Segment Addition Postulate: If B is between A and C, then AB+BC=AC
EX: A 5 B 10 C If AB=5 and BC = 10, AC=15
Given: AB=5 and BC = 10
Prove: AC = 15 Steps Reasons
AB=5 and BC = 10 Given
AB+BC=AC
Segment Addition
5+10=AC
Substitution
15=AC
Addition
AC=15
Symmetric
Angle Addition Postulate: If P is interior to RST, then RSP+PST=RST
R
Angle RSP+PST=RST
P
S
T
6
12.2 geometric proofs.notebook
December 02, 2016
Definitions:
Congruence ­ If two objects are congruent, then their measures are equal
EX: Given: AB = BC
Prove: AB BC (Definition of Congruence)
Congruent Segments ­ IF two segments are congruent, then their measures are equal
EX: A B B C
IF AB BC and AB = 10 feet, BC = 10 feet (Congruent segments)
Congruent Angles ­ IF two angles are congruent, then their measures are equal
EX: Angle A Angle B and Angle A = 20, then Angle B = 20
Midpoint ­ Cuts a segment into two equal segments
EX: A C B IF AC CB, then C is the MIDPOINT!!!
Angle Bisector ­ Segment that cuts any angle into two congruent halves IF Segment BD bisects Angle A
ABC, then Angles ABD and B
DBA are congruent
D
C
Vertical Angles ­ Any two lines intersecting form two pairs of congruent angles
c
a
a=b
d
c=d
b
Complementary Angles ­ Angles whose sum equals 90 degrees
Angle 1 = 50, Angle 2 = 40
50
1
(1 + 2 = 90)
40
2
Supplementary angles ­ Angles whose sum equals 180 degrees
100
80
Perpendicular Lines ­ two lines that intersect forming 4 right angles
90
POSTULATES:
Ruler Postulate: Any two points, the distance between them can be measured
Protractor Postulate: If a point is exterior to a line segment, then the angle between that point and the segment can be measured
EX: A B
C 7
12.2 geometric proofs.notebook
December 02, 2016
Definitions:
Congruence ­ If two objects are congruent, then their measures are equal
ex: Given: AB = BC
Prove: AB BC (Definition of congruence)
Congruent Segments ­ If two segments are congruent, then their measures are equal
Congruent Angles ­ If two angles are congruent, then their measures are equal
Midpoint ­ The middle of a line segment creating two congruent segments
Angle Bisector ­ a line segment that cuts an angle into two congruent angles (Measures are equal)
Vertical Angles ­ Opposite congruent angles formed by intersecting lines or segments
3
2
1 4
Complementary Angles ­ angle whose sum is 90 degrees
10
80
Supplementary angles ­ angle whose sum is 180 degrees
100
80
Perpendicular lines ­ Two lines that intersect forming 4 right angles
90
POSTULATES Segment addition postulate ­ If B is between A and C, then AB + BC = AC
A B C
AB + BC = AC
Angle Addition Postulate ­ If P is in the interior of RST, then RSP + PST = RST
40
RST = 60
20
Ruler Postulate ­ Any two points on a line can be measured.
Protractor Postulate ­ Any point exterior to a line segment can be measured between 0 ­ 180 degrees
C
A B
POINTS, LINES, AND PLANES POSTULATE:
1) Through any two points, there is EXACTLY one line
2) A line contains at least 2 points 3) Two line intersect at EXACTLY one point
8
12.2 geometric proofs.notebook
December 02, 2016
9
12.2 geometric proofs.notebook
December 02, 2016
IB4E: Write a 2 column proof for the following:
Given: 1/3(3x­12) ­ 1 = 2x
Prove: x = ­5
Steps Reasons
10
12.2 geometric proofs.notebook
December 02, 2016
11