Download Honors Geometry 10-26-10 2.5 Postulates and Paragraph Proofs

Honors Geometry
2.5 Postulates
and Paragraph
Proofs
10-26-10
objective:
-identify and use basic postulates about points, lines and planes
Bell-ringer
Postulate (axiom) a statement that is accepted as true.







Postulate 2.1: Through any two points, there is exactly one line.
Postulate 2.2: Through any three points not on the same line, there is
exactly one plane.
Postulate 2.3: A line contains at least two points.
Postulate 2.4: A plane contains at least three points not on the same
line.
Postulate 2.5: If two points lie in a plane, then the line containing
those points lies in the plane.
Postulate 2.6: If two lines intersect, then their intersection is exactly
one point.
Postulate 2.7: If two planes intersect, then their intersection is a line.
Examples In the figure,
and
are in plane Q and
|| . State the postulate that can be used to
show each statement is true.
-
-
Exactly one plane contains points F, B and E.
lies in Q.
and
lie in plane J and H
In the figure,
lies on
. State the postulate that can be used
to show each statement is true.
-
G and P are collinear.
-
Points D, H and P are coplanar.
Honors Geometry
10-26-10
Theorem A statement that can be proved true
Midpoint Theorem If M is the midpoint of
Assignment 2.5
Pg 92, #16-27
Summary:
, then
≅
Honors Geometry
2.6 Algebraic
Proofs
10-26-10
objective:
-Use algebra to write 2 column proofs
Bell-ringer
Reflexive property
For every number a, __________.
Symmetric Property
For all numbers a and b, if _______then _______.
Transitive property
For all numbers a, b, and c, if _______ and _______ then ______.
Addition and
Subtraction Property
For all numbers a, b, and c, if _______ then ____________ and
____________.
Multiplication and
Division Property
For all numbers a, b, and c, if ______ then ____________, and if c ≠ 0
then _____________.
Substitution Property For all numbers a and b, if a = b then a may be replaced by b in any
equation or expression.
Distributive Property For all numbers a, b, and c, a(b + c) = ___________.
Segments
Reflexive property
AB = AB
Symmetric Property
If AB = CD, then CD = AB
If AB = CD and CD = EF,
then AB = EF
Example
Solve: 6x + 2(x-1) = 30.
Algebraic Steps
6
2
1
30
6
2
2 30
8
2 30
8
2 2 30 2
8
32
32
8
8
8
4
Angles
Transitive property
Properties
Given
Distributive Property
Substitution
Addition Property
Substitution
Division Property
Substitution
Two-Column Proof Contains statements and reasons organized in two columns. Each step is
called a statement and the properties, definitions, postulates, or theorems
that justify each step are called reasons
Honors Geometry
10-26-10
Examples State the property that justifies each statement
1. If 80 = mA, then mA = 80.
2. If RS = TU and TU = YP, the RS = YP.
3. If 7x = 28, then x=4.
4. If VR + TY = EN +TY, then VR = EN.
5. If m1 = 30 and m1 = m2, then m2 = 30.
Examples Complete each proof.
Given:
9
Prove: x = 3
Statements
Reasons
a. ____________________
9
a.
b. ___
b. Multiplication Property
2 9
c. 4
6
18
d. 4
6
6
e. 4
__________________
f.
c. ____________________
18
6
_______________
g. ___________________
d. ____________________
e. Substitution
f. Division Property
g. Substitution
Honors Geometry
Examples Given: 8
5 2
Prove: x = 1
Statements
Assignment 2.6
Pg 97: #14-27 all,
and 37, 38
a. 8
5
2
1
b. 8
5
2
2
10-26-10
1
Reasons
a. ____________________
1
2
b. ____________________
c. ______________________
c. Substitution Property
d. ______________________
d. Addition Property
e. 6
e. ____________________
6
f.
f. ____________________
g. ___________________
g. ____________________
Summary:
Honors Geometry
2.7 Proving
Segment
Relationships
10-27-10
objective:
-Write proofs involving segment addition and segment congruence
Bell-ringer
Segment addition B is between A and C if and only if AB+ BC = AC.
Postulate
Theorem 2.2 Congruence of segments is reflexive, symmetric, and transitive.
Reflexive Property:
Symmetric Property:
Transitive Property:
AB  AB
If AB  CD, then CD  AB.
If AB  CD and CD  EF, then AB  EF.
Examples Justify each statement with a property of equality, a property of
congruence, or a postulate.
1. QA = QA
2. If AB / BC and BC / CE, then AB / CE.
3. If Q is between P and R, then PR = PQ + QR.
4. If AB + BC = EF + FG and AB + BC = AC, then EF + FG = AC
.
Examples Complete the proof
Given: BC = DE
Prove: AB + DE = AC
Statements
Reasons
a. BC = DE
a. ______________________
b. __________________
b. Seg. Add. Post.
c. AB + DE = AC
c. _____________________
Honors Geometry
Examples Complete the proof.
Given:
≅
Prove:
≅
Statements
Assignment 2.7
Pg 104: #12-18, 21
10-27-10
Reasons
a.
≅
a. ____________________________
b.
≅
b. ____________________________
c.
c. ____________________________
d. __________________
d. Segment Addition Postulate
e.
e. ____________________________
f. __________________
f. Subtraction Property
g. __________________
g. Definition of congruence of segments
Summary:
Honors Geometry
2.8 Proving
Angle
Relationships
10-29-10
objective:
-Solve algebraic equations using theorem about angles
-Write proofs involving supplementary and complementary angles
-Write proofs involving congruent and right angles
Bell-ringer Write a 2-column proof: If 5
2
, then
6
Angle Addition R is in the interior of PQS if and only if
Postulate mPQR + mRQS = mPQS.
Supplement If two angles form a linear pair, then they are
Theorem supplementary angles.
(Theorem 2.3) If 1 and 2 form a linear pair, then
m1 + m2 = 180.
Complement If the noncommon sides of two adjacent angles
Theorem form a right angle, then the angles are
(Theorem 2.4) complementary angles.
If GF  GH, then m3 + m4 = 90.
Vertical Angle If two angles are vertical angles, then they are
Theorem congruent.
(Theorem 2.8) m6  m7
Examples Find the measure of each numbered angle
m7 = 5x + 5,
m8 = x - 5
m5=5x,
m6=4x+6,
m7 = 10x,
m8 = 12x - 12
m11 = 11x,
m12 = 10x + 10
Honors Geometry
10-29-10
Theorem 2.6 Angles supplementary to the same or congruent angles are congruent.
Theorem 2.7 Angles complementary to the same or congruent angles are congruent
Theorem 2.9 Perpendicular lines intersect to form four right angles.
Theorem 2.10 All right angles are congruent.
Theorem 2.11 Perpendicular lines form congruent adjacent angles.
Theorem 2.12 If two angles are congruent and supplementary, then each angle is a right
angle.
Theorem 2.13 If two congruent angles form a linear pair, then they are right angles.
Examples Complete the proof.
Given: 1 and 2 form a
linear pair.
m1 + m3 = 180
Prove: 2 ≅ 3
Statements
Assignment 2.8
Pg 112: #16-24,
27-32, 38, 39
Reasons
a. 1 and 2 form a
linear pair.
m1 + m3 = 180
a. Given
b. __________________
b. Suppl. Theorem
c. 1 is suppl. to 3
c. ________________________
d. __________________
d.  suppl. to the same  are ≅
Summary: